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Design of a wideband metamaterial absorber for optical wavelength applications based on the quantum-inspired Hadamard matrix

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Abstract

Attaining broadband absorption with a simple-designed photonic absorber is still considered a complex and daunting task. The Hadamard matrix has been a well-known concept for designing quantum gates in quantum mechanics and computing. Despite a series of noteworthy works being done with regard to the design of metamaterial absorbers, very few works exist in literature where a quantum-inspired design has been employed to design a metamaterial-based photonic absorber. In this work, we have demonstrated a polarization-insensitive ultrathin and wideband metamaterial (MM) absorber based on the pattern distribution of the Hadamard matrix. Within the optical domain, the demonstrated MM absorber showcases absorption of over 91% for both transverse electric (TE) mode and transverse magnetic (TM) mode. To explore the angular dependence on absorption features of our Hadamard matrix meta-absorber (HMMA), both TE and TM modes have been used at numerous incident-angles. Finite integration technique has been utilized to simulate the demonstrated MM absorber design and validated using the interference theory model to assure the simulated data. Moreover, electric and magnetic field characteristics, current distributions, and a plethora of parametric sweeps have also been investigated in order to better understand the suggested HMMA absorption mechanism. Because of its wideband absorption and polarization-insensitive characteristic, this MM absorber based on the Hadamard matrix arrangement permits a variety of applications such as light detectors, optical-sensors, magnetic resonance imaging, plasmonic-sensors, and thermal imaging applications.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Metamaterials are intentionally created materials composed of periodic sub-wavelength-sized unit-cell designs exhibiting exotic and unique features that are hard to acquire in natural materials, such as negative reflection or refraction, reversal of Doppler-effect, and Vavilov-Cerenkov radiation [18]. Because of these properties, many applications heretofore possible have been assembled, such as stealth cloaks [9,10]. Many other potential applications in perfect absorption, imaging, and so on [1118] have been explored. Landy and coworkers illustrated the first perfect MM absorber in 2008 [19], achieving absorption of 96% at 11.65 GHz for the very first time. Even more MM absorber research has been presented, with the operating band extending from microwave to invisible light because of the advantageous attributes of THz waves [20,21]. With FR4 and copper, Lee et al. [22] demonstrated the first broadband MM absorber in the GHz frequency range. Following that, in a series of research that has continued to this day, a broadband MM absorber in the optical domain was shown.

Hedayati et al. [23] demonstrated that a four-layer polarization-independent plasmonic MM absorber, made of SiO2 and Au nanocomposite, has an absorbance of over 85% between 400 and 750 nm. Luo et al. [24] used Si and Ni-based MM absorbers and found that they absorbed over 90% of the light in the optical spectrum and over 99 percent between 500 and 560 nm. Heidari and Sedighy [25] demonstrated an Au and SiO2 sandwich, and genetic algorithm (GA) based MM absorber with over 90% absorption and 40° polarization independence in both TE and TM modes. Zhou et al. [26] demonstrated a plasmonic broadband absorber with polarization independence up to 60° and 99.9% peak absorption at 498.25 nm. With a three-layered MM structure, Mahmud et al. [27] demonstrated 99.99% peak absorption and over 90% within the optical spectrum. Mahmud et al. reported another 99.99% peak absorption at 545.73 nm in [28], with over 90% absorption remaining within optical range. Zhang et al. [29] demonstrated a dual-band visible light absorber using an Au-based resonator, with an absorption level of more than 40%, a broad incidence angle of up to 80°, and a peak absorbance of 99.9%. In addition, Charola et al. presented in [30] a numerical investigation of an MM absorber based on tungsten and SiO2, which demonstrated an average absorption of over 90% between 404 and 631 nm with wide-angle stability up to 30°.

In this paper, we aim to investigate the Hadamard matrix as a geometry design basis and demonstrate this as a metasurface absorber for the optical regime. To generate wideband absorption, the Hadamard matrix of rank eight $({2^{n = 3}})$ has been selected after going through the matrix dimensions of two, four, and eight $({2^{n = 1,2,3}})$. The quantitative results show an optical absorption greater than 91% for both TE and TM modes for the entirety of the optical range, i.e., 380 to 700 nm, and a 99.99% peak absorbance at 495.16 nm, an average absorption of 97.15% for the TEM mode, and with polarization-insensitive to 90° as well. Such a MM absorber has been a viable candidate to be used as a solar cell, solar thermo-photovoltaics (STPV), and optical sensor, as HMMA shifts its resonance wavelength when the dielectric thickness has been changed. Changing the dielectric in the Pyrex will also allow the HMMA to be used as a light detector.

2. Designing and simulation

The architecture of the proposed MM absorber has been illustrated in Fig. 1(a) and 1(b), which is a three-layered MM with tungsten [W, (optical, Palik)] as the resonator and ground pad, and silicon dioxide [SiO2, (optical, Ghosh)] as the intermediate dielectric substrate [31,32]. These provide the birefringence of the optical materials, optical characteristics, and band structure. W has been selected as a metal and resonator layer despite its significant intrinsic losses owing to its excellent impedance match with free space in the optical domain. W is an excellent absorber in its own right, with a low ohmic loss. Because of its excellent temperature stability of 3422°C, W has been typically a good choice for MM absorbers in the optical domain. The rationale for using SiO2 as a dielectric spacer has been due to its lossless properties at the relevant wavelengths [33]. For its high melting point (1600°C), SiO2 has exceptional thermal stability, which helps the structure resist the immensely resilient electromagnetic waveforms. SiO2 also exhibits a comparatively negative real part of permittivity rather than a high non-real part of the dielectric constant at the visible spectrum [34]. Consequently, in a break-down state, the real component of permittivity diminishes, resulting in a more closed propagating wave for an evanescent wave characteristic. This anisotropic tendency also contributes to polarization and propagation regulation within the substrate since the birefringence characteristics of dispersion relation pairs with SiO2’s refractive index [35]. The dielectric layer’s low refractive index (Ri = 1.5) assists the structure in acquiring better absorbance with a wider bandwidth by aiding in the formation of the right coupling condition between the resonator and metal layer with respect to capacitance and inductance. Furthermore, because the melting points of both materials have been notably substantial, the suggested structure can tolerate high temperatures.

 figure: Fig. 1.

Fig. 1. Schematic of the (a) HMMA unit cell in a 6 × 6 array, (b) 3-D view of the unit cell and structural parameters, and (c) the top-front surface view of the unit cell and its surface design in relation to the Hadamard matrix to form the resonator, where “-1” represents W placements on the unit cell’s top layer and “+1” signifies placements where W was not placed on the top layer.

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Encapsulating excellent near-unity absorption has been strongly reliant on the physical dimensions of a structure and the resonator. To design the suggested HMMA, two square boxes of length a = 480 nm have been formed, with a thickness of tb = 150 nm and tt = 55 nm, respectively. In Fig. 1(c) the resonator geometry distribution has been showcased, and the formation of the structure has been via the Hadamard matrix [36] of rank eight $({2^{n = 3}})$, where ‘${\otimes} $’ is a recursive Kronecker product and has been noted in the following equations:

$${H_{{2^n}}} = {H_{_2}} \otimes {H_{{2^{n - 1}}}}$$
where,
$${H_2} = \left[ {\begin{array}{cc} { + 1}&{ + 1}\\ { + 1}&{ - 1} \end{array}} \right]$$

Thus,

$${H_4} = \left[ {\begin{array}{cccc} { + 1}&{ + 1}&{ + 1}&{ + 1}\\ { + 1}&{ - 1}&{ + 1}&{ - 1}\\ { + 1}&{ + 1}&{ - 1}&{ - 1}\\ { + 1}&{ - 1}&{ - 1}&{ + 1} \end{array}} \right]$$
$${H_8} = \left[ {\begin{array}{cccccccc} { + 1}&{ + 1}&{ + 1}&{ + 1}&{ + 1}&{ + 1}&{ + 1}&{ + 1}\\ { + 1}&{ - 1}&{ + 1}&{ - 1}&{ + 1}&{ - 1}&{ + 1}&{ - 1}\\ { + 1}&{ + 1}&{ - 1}&{ - 1}&{ + 1}&{ + 1}&{ - 1}&{ - 1}\\ { + 1}&{ - 1}&{ - 1}&{ + 1}&{ + 1}&{ - 1}&{ - 1}&{ + 1}\\ { + 1}&{ + 1}&{ + 1}&{ + 1}&{ - 1}&{ - 1}&{ - 1}&{ - 1}\\ { + 1}&{ - 1}&{ + 1}&{ - 1}&{ - 1}&{ + 1}&{ - 1}&{ + 1}\\ { + 1}&{ + 1}&{ - 1}&{ - 1}&{ - 1}&{ - 1}&{ + 1}&{ + 1}\\ { + 1}&{ - 1}&{ - 1}&{ + 1}&{ - 1}&{ + 1}&{ + 1}&{ - 1} \end{array}} \right]$$

Capturing the Hadamard matrix of rank eight $({2^{n = 3}})$ to construct one-fourth of the unit cell to inaugurate the four-fold symmetrical nature of the proposed unit cell.

When constructing the resonator, we assumed -1 to be the position where we would use W, leaving +1 without any metal. We also made sure to test for the opposite, where +1 has been in the position where W has been used and -1 was neglected. After seeing both results, we concluded that -1 with W produced better and more desirable results. Table 1 has a complete list of the optimized parameters used. While designing the MM absorber, firstly the back-layer metal thickness has been made greater than the MM absorber’s skin-depth in order to prevent EM wave propagation [37]. So as to guarantee near-zero transmission in the full optical domain. The rest of the parameters observed has been optimized through various parameter sweeps. As it will be shown below in Table 1, with adequate data, this design has a logical, symmetrical structure. The entire thickness of the unit cell is 216 nm, which is ultrathin. These ultrathin structures are readily adaptable to any STPV cell. Using the commercially available CST Microwave Studio (CST MWS) software, all simulations have been conducted for a single unit cell. In the x and y axis, unit cell boundary conditions have been used, but the z-axis was left with open add space. From the + z-axis, the TE and TM modes have been shined. A higher mesh order has been employed to render the simulation results more precise. During several passes, the simulation converged with a threshold of less than 0.01. All of the studies have been conducted using frequency-domain analysis based on the finite integration technique (FIT).

Tables Icon

Table 1. The Proposed Unit Cell Design’s Parameter List

3. Results analysis and discussion

3.1 Absorbance methodology

The Nicolson–Ross–Weir (NRW) technique has been used to calculate absorbance by extracting scattering parameters (S-parameters) [38]. Absorption is inversely proportional to the reflection coefficient $R(\omega )$ and the transmission coefficient $T(\omega )$, where $R(\omega )$ is the linear value of ${S_{11}}$ and $T(\omega )$ is the linear value of ${S_{21}}$ S-parameters respectively. Hence, the equation for $A(\omega )$ as shown in Eq. (5) can be written as [39]:

$$A(\omega )= 1 - R(\omega )- T(\omega )= 1 - |{S_{11}}(\omega ){|^2} - |{S_{21}}(\omega ){|^2}$$

Meanwhile, because the ground plate thickness has been greater than the skin-depth $(\delta )$ of W expressed in Eq. (6), where $\rho $ represents the density of W, $T(\omega )$ may be represented as close to zero.

$$\delta = \sqrt {2\rho /2\pi f\mu {}_r{\mu _o}} $$

Finally, the resulting absorption formula will be as follows:

$$A(\omega ) = 1 - R(\omega ) = 1 - |{S_{11}}(\omega ){|^2}$$

Because absorbance relies primarily on impedance matching with respect to the unit cell's impedance.

$$Z(\omega )= \sqrt {\left[ {\frac{{{\mu_r}(\omega ){\mu_o}}}{{{\varepsilon_r}(\omega ){\varepsilon_o}}}} \right]} $$

The impedance characteristics, ${Z_o} = \sqrt {{{{\mu _o}} / {{\varepsilon _o}}}} = 376.73 \approx 377\Omega $

Where, ${\mu _o} = $ permeability of free space, ${\mu _r} = $ relative permeability, ${\varepsilon _o} = $ permittivity of free space, and ${\varepsilon _r} = $ relative permittivity.

By adjusting the design's physical dimensions, we can achieve the condition $[{Z(\omega )= {Z_o}} ]$. And because the absorption has been largely dependent on wavelength, the design at that wavelength will offer virtually near unity-absorption. Here, $Z(\omega )\approx {Z_o}$ which has been quite near to our desired value; if the value can be equaled, perfect absorption can be achieved with the design. The impedance $Z(\omega )$ is dependent on the ${\mu _r}$ and ${\varepsilon _r}$.

The complex permeability and permittivity of all materials may be electrically represented in the frequency domain. The constitutive parameters of the materials determine the feedback to the EM radiation. The following Eqs. (9) and (10) can be used to show ${\mu _r}$ and ${\varepsilon _r}$ in terms of S-parameters. The NRW technique has been utilized to derive effective permeability and permittivity from the S-parameters.

$${\mu _r} = 2[{1 - ({S_{21}} - {S_{11}})} ]/j{k_0}d[{1 + ({{S_{21}} - {S_{11}}} )} ]$$
$${\varepsilon _r} = 2[{1 - ({S_{21}} + {S_{11}})} ]/j{k_0}d[{1 + ({{S_{21}} + {S_{11}}} )} ]$$

The relative refractive index, ${n_r}$ can be seen in Eq. (11) in terms of S-parameters as:

$${n_r} = c[{{{({{S_{21}} - 1} )}^2} - S_{11}^2} ]/j\pi fd[{{{({{S_{21}} - 1} )}^2} - S_{11}^2} ]$$

Here, ${k_o}$ is the wave number, which can be expressed by ${k_o} = \omega /c = 2\pi f/c$, $d$ the substrate thickness, and $c$ being the speed of light. To obtain a near-unity absorption, it is strongly reliant on the physical dimensions of a structure and the resonator. The ground pad and the W-based resonator array can have a strong near-field interaction. In combination with zero transmission owing to the ground pad, its operational principle has been an impedance matching to free space due to destructive interference in reflection. The proposed HMMA is comparable to that of a lossy high impedance surface that has been proven capable of high absorption. Further details on the equations used are available in Supplement 1.

Figure 2(a) demonstrates the absorbance, and reflectance of transverse electromagnetic (TEM), TE, and TM modes for the optical domain of 380 to 700 nm using our proposed MM absorber with optimized parameters, as shown before in Table 1. As stated earlier, the absorption curve has been obtained from (5) and (7). The proposed HMMA structure has 99.99% absorption at 495.16 nm and above 99% absorption from 448.52 to 550.94 nm, with a wide range of 102.42 nm. Its absorption starts at 92.29% at 380 nm and 91.40% at 700 nm and has an average absorption of 97.15% for the entirety of the optical spectrum. Figure 2(b) illustrates real and imaginary parts of the normalized impedance for TEM mode. The real values of normalized impedance should ideally be near to one; meanwhile, imaginary portions should be close to zero in order to achieve a high impedance match. These two events have been facilitated by the architecture of the unit cell, which has been evident in Fig. 2(b). The decent coupling capacitive and inductance values of the metal and the resonator surface with the assistance of the dielectric has also been a key factor for high absorption over the whole domain with a near-unity apex. Consequently, unless the resonator is a symmetrical resonator with a commendable structure, it is possible that these occurrences would not be able to catch the falling waves. The illustrations in Fig. 2(c) and 2(d) show both TE and TM modes, co-polarization and cross-polarization, and polarization conversion ratio in both magnitude (dB) and the linear scales. These co-polarization and cross-polarization phenomena are needed to answer whether or not the proposed HMMA is the MM absorber converting the polarization wave rather than of good absorptivity. Co-polarization and cross-polarization can be calculated from Eq. (12) and Eq. (13) as:

$$|{\textrm S_{11}}(\omega ){|^2} = |{\textrm R_{yy}}{|^2} + |{\textrm R_{yx}}{|^2}$$
$$|{\textrm S_{11}}(\omega ){|^2} = |{\textrm R_{xx}}{|^2} + |{\textrm R_{xy}}{|^2}$$

 figure: Fig. 2.

Fig. 2. (a) Absorbance and reflectance of the design in the range 380 to 700 nm in TEM, TE, and TM modes of the proposed unit cell structure, (b) real and imaginary parts of the normalized impedance, (c) magnitude of co-polarization and cross-polarization in dB of the reflection coefficient for both TE and TM-polarization modes, (d) PCR of the proposed unit cell for both TE and TM modes, EM wave properties in which (e) relative permeability, (f) relative permittivity, (g) refractive index vs. wavelength of the proposed HMMA unit cell, (h) absorbance comparison of a full, half, quarter size HMMA unit cell, (i), representing of a parametric sweep for the parameter “td” in TEM mode, and (j) the effect of different dielectric materials on absorbance.

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Here, ${R_{yx}}$ and ${R_{xy}}$ are the cross-polarization component, ${R_{xx}}$ and ${R_{yy}}$ are the co-polarization component. Polarization conversion ratio (PCR) can be can be calculated using the linear value of ${R_{xx}}$, ${R_{yy}}$, ${R_{xy}}$ and ${R_{yx}}$ with the following Eq. (14) and Eq. (15).

$$PC{R_{TE}} = \frac{{R_{yx}^2}}{{(R_{yy}^2 + R_{yx}^2)}}$$
$$PC{R_{TM}} = \frac{{R_{xy}^2}}{{(R_{xx}^2 + R_{xy}^2)}}$$

From Fig. 2(d) it has been observed that the components of the cross-polarization component in the linear magnitude scale are near zero; this verifies that the design does not transform the waves in the region of consideration. The illustrative visualization discussed in equations from (9)-(11) is available in Fig. 2(e)–2(g). It can be evident from the figures that the HMMA possesses a negative value near the middle of the visible wavelength realm. Therefore, the proposed HMMA design holds the MM properties with negative permeability, permittivity, and refractive index [40,41].

While comparing different geometry and various dielectric substrate on the unit cell, we applied a scaled configuration to the HMMA to half its scale and then further scaled it to a quarter of the original design. The HMMA showed an outstanding average absorption of 97.04% and a peak of 99.87% for the half-scaled HMMA structure, while the quarter-scaled resulted in an average absorption of 96.37% and a peak of 99.39%, as shown in Fig. 2(h). A MM unit cell may be made considerably smaller than the wavelength's free space, resulting in considerable phase variation at practical operating frequencies [40]. This has been a key cause of the structure's absorption. In this case, the resonator, dielectric, and metal layers are all the same thickness. However, at half (a = 240 nm) and quarter (a = 120 nm) sizes, the parameter “L” has been modified from 30 to 15 and 7.5 nm, respectively. Given the small size of the structure, the design might be scaled down to half or a quarter of its original size. It has been illustrated in Fig. 2(i) a major and important sweep of the parameter “td”, which refers to the thickness of the dielectric material. A substantial change in absorbance has been detected after changing the thickness from 45 to 75 nm, and the resonance wavelength has also shifted significantly. The resonance wavelength changed linearly at 426.17, 495.16, 567.21, and 638.69 nm while still maintaining 99.89, 99.99, 99.89, and 99.70%, respectively. When the structure has been exposed to an EM wave, it generates capacitance with the metal layer and a resonator with the help of the dielectric layer. The capacitance of a structure, on the other hand, has been dependent significantly on the dielectric depth and has been inversely related to it. As the depth has been increased, the capacitance dropped, and the responsive wavelength shifts right linearly. For this significant phenomenon, this structure could be employed as an optical sensor. We also simulated our proposed HMMA structure with different dielectric materials, which have been shown in Fig. 2(j) along with SiO2; the dielectric component has been replaced with two other different compounds, as in silicon nitride (Si3N4), and amorphous silicon (A-Si), which were the new materials. In this case, we could see that SiO2 has a significantly higher average absorption than that of the other materials. The various Ri of such materials accounts for this type of fluctuation. Since we already know, the smaller the Ri, the greater the absorption and the wider the band. The Ri of SiO2, Si3N4, and A-Si are 1.5, 2.0, and 4.4, respectively. That being said, because the absorption amount has been linearly raised by Si3N4, it is possible to employ it as a light detector within the optical domain. Furthermore, A-Si can be employed as half-power absorbers since they display half-absorption for nearly the entire optical domain.

3.2 Theoretical verification

Interference theory has been employed to comprehend the absorption process of HMMA and to validate the results of our simulation. For verification, decoupled simulated data has been compared to calculated data produced from the interference theory model. Figure 3 depicts the graph highlighting the difference in absorbance between the interference theory and the decoupling model. According to interference theory [4244], with the ground plate, the total reflection ${S_{11(Total)}}$ for layer one can be calculated from the equation as:

$${S_{11(Total)}} = |{\textrm S_{11}}|\exp (j{\theta _{11}}) + \frac{{|{S_{12}}||{S_{21}}|\exp[j({\theta _{12}} + {\theta _{21}} - 2\beta - \pi )]}}{{1 - |{S_{22}}|\exp[j({\theta _{22}} - 2\beta - \pi )]}}$$

Here,

  • ${\textrm S_{11}} = |{\textrm S_{11}}|\exp (j{\theta _{11}})$, $R(\omega )$ for layer one from area one to two.
  • ${\textrm S_{21}} = |{\textrm S_{21}}|\exp (j{\theta _{21}})$, $T(\omega )$ for layer one from area one to two.
  • ${\textrm S_{12}} = |{\textrm S_{12}}|\exp (j{\theta _{12}})$, $T(\omega )$ for layer one from area two to one.
  • ${\textrm S_{22}} = |{\textrm S_{22}}|\exp (j{\theta _{22}})$, $R(\omega )$ for layer one from area two to one.

Moreover, $\beta = {k_0}d$ where, $\beta $= complex propagation phase, ${k_o}$= wave-number in area 2, and $d$= propagation distance. The symmetrical nature of the design has been demonstrated in the preceding section. Hence, $[{{S_{12}} = {S_{21}}} ]$ so we can re-write Eq. (17) as:

$${S_{11(Total)}} = |{\textrm S_{11}}|\exp (j{\theta _{11}}) + \frac{{|{\textrm S_{12}}{|^2}\exp[j(2{\theta _{12}} - 2\beta - \pi )]}}{{1 - |{\textrm S_{22}}|\exp[j({\theta _{22}} - 2\beta - \pi )]}}$$

 figure: Fig. 3.

Fig. 3. (a) A side-by-side view of the simulated absorbance with the calculated values, and (b) a representational illustration of the interference theory structure.

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Ultimately, absorption may be calculated theoretically using the simple formula $A(\omega ) = 1 - |{S_{11(Total)}}{|^2}$. The simulated absorbance values have been quite comparable to the estimated absorbance we got from the interference theory, as shown in Fig. 3(a). Both findings demonstrated remarkable absorption, with peak values above 99.9% for the simulated and calculated results.

3.3 Incident and polarization angle stability

As described in earlier sections, the HMMA seems to have been a wide-incidence and polarization angle stable structure in both TE and TM modes. Because of its symmetrical construction, the HMMA design has been completely polarization angle insensitive from 0° to 90°, as shown in Fig. 4(a) and 4(b). However, an ideal MM absorber must maintain a constant level of absorbance for changing incident angles in order to be used as a solar cell, for harvesting energy, as a solar detector or sensor, or for a number of other critical use cases. Figure 4(c) and 4(d) highlights changing incident angles from 0° to 70° with increments of 10° for various incidence angles in both TE and TM modes. The average absorbance in TE mode has been more than 71.58% for all incidence angles. Because the wave's path length has been exactly proportional to the angle of incidence, the greater the angle, the greater the path length. For the weaker magnetic dipolar resonance, the coupling effect of the resonator and metal reduces due to the long route. The effect affects wave confinement in the dielectric layer, resulting in reduced absorptivity.

 figure: Fig. 4.

Fig. 4. Effect of changing polarization angles (φ) from 0-90° on absorbance in (a) TE mode, (b) TM mode, and for incident angles (θ) in the range 0-70° in (c) TE mode, (d) TM mode.

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A distinct situation arises for the TM mode shown in Fig. 4(d) as it demonstrated superior stability with an average absorptivity that varied by more than 84.00% for an incidence angle of 70°. In TM mode, the level of absorptivity of the MM absorber does not vary as much in comparison to the TE mode. Better stability in the TM mode has been due to the higher electric field involving the perpendicular incidence angle component instead of the parallel component. Excellent coupling combined with excellent wave confinement makes the HMMA model a polarization and incident angle stable MM absorber, broadening its application requirements in a variety of domains.

3.4 Mechanical stress of the metamaterial absorber

Several mechanical deformities can be induced during the fabrication and imposition of the underlying MM absorber [45,46]. Therefore, as a result, it has been vital to examine the HMMA's behavior in the presence of aberrations. The aim has been to create a near-unity MM absorber in which the electro-optical properties of the proposed structure have been evaluated in relation to mechanical deformation effects. The objective of testing absorption qualities has been to achieve a desirable level, particularly when two types of mechanically warped bending — concave (positive) and convex (negative) — have been utilized. Tensile tension causes convex deformation, whereas compression causes concave deformation. The underlying MM absorber has been bent from -30° to 30° in 5° increments with non-uniform mechanical stress. In the case of 0°, no mechanical tension has been exerted. The maximum $vM$ stress has also been computed using [4749] for numerical simulation.

Figure 5(a) illustrates the numerical findings for convex bending. Redshift occurs when the bending moment has been altered in 5° increments from -30° to 0°. Peak and average absorption both drop significantly as convex bending increases. This has been due to variations in coupling capacitance (C) and inductance (L) because when C and L vary, the resonant frequency alternates as well. Likewise, the average absorption at -30° to -5° is 96.71-97.17%. As a result, the suggested HMMA exhibits a minimum of 96.71% absorption under convex deformation. The corresponding $vM$ stress for the specified bending angles has been shown in Table 2.

 figure: Fig. 5.

Fig. 5. Absorption phenomenon of the HMMA unit cell under (a) negative/convex bending from 0° to 30°, and (b) positive/concave bending from 0° to 30°. Insets: (a) representation of negative bending with θ degree, and (b) positive bending with θ degree.

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Tables Icon

Table 2. Bending angle in degrees associated with maximum vM stress in GPa for proposed HMMA

The concave distortion in the MM absorber can be caused by unanticipated compression during the fabrication or application process. Figure 5(b) depicts the numerical results in the unit cell under non-uniform mechanical load. During concave bending, the absorption rates rise as the bending moment rises. The bending moment has been altered in 5° increments from 5° to 30°. This is related to the C and L resonance coupling. The mean absorption at 5° to 30° is 97.41-98.12%. The HMMA shows good mean absorption of 98.12% at a 30° severe bending force.

3.5 Electric field, magnetic field, and surface current distribution

The EM field and surface charge distribution influence absorption properties in both TE and TM modes. In this part, E-field, H-field, and surface current will be simulated at three distinct wavelengths $\lambda $= 420, 495.16, and 690 nm, which correspond to three different absorption levels of $A(\omega )$= 94.88, 99.99, and 91.88%, respectively, as illustrated in Fig. 6 and 7.

 figure: Fig. 6.

Fig. 6. Distribution of the E-field (a)-(c) for TE polarization at 420, 495.16, and 690 nm in the y-x axis, (d) the cross-sectional view of the middle of the proposed structure at 495.16 nm (peak) resonance wavelength in TE mode in the z-x axis, (e)-(g) E-field distribution in TM polarization at 420, 495.16, 690 nm in the y-x axis, (h) the cross-sectional view of the middle of the structure at 495.16 nm (peak) resonance wavelength in TM mode in the z-x axis. The H-field distribution (i)-(k) for TE polarization at 420, 495.16, and 690 nm for the unit cell in the y-x axis, (l) middle cross-sectional view of the proposed structure at 495.16 nm (peak) resonance wavelength in TE mode in the z-x axis, (m)-(o) H-field distribution in TM polarization at 420, 495.16 and 690 nm in the y-x axis, and (p) the cross-sectional view of the middle of the structure at 495.16 nm (peak) resonance wavelength in TM mode in the z-x axis.

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 figure: Fig. 7.

Fig. 7. Distribution of the surface charge of the unit cell (a)-(c) for TE polarization at 420, 495.16, and 690 nm in the y-x axis, (d)-(f) TM polarization at 420, 495.16, and 690 nm in the y-x axis for normal incident angle. The cross-section has been done with an x-y axis.

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At a particular wavelength, the EM field has been resonantly fixated and strengthened at particular portions of the absorber, and the field distribution for the E-field, H-field, and surface current density changes with the polarization mode. As previously stated, the bottom ground pad inhibits incident EM wave transmission, which elevates dielectric resonance - which would be a key factor of broadband absorption - and induces a further surface plasmon that enhances E-field growth. It should be noted that while the polarization modes are shifted, the EM propagation changes. The dispersed E-field has been located predominantly at the cell center. Figure 6(a) to 6(h) illustrates a powerful E-field for both TE and TM modes at the three previously mentioned wavelengths. The dielectric SiO2, which energizes the electric resonant dipolar moment, has been shown to considerably restrict the E-field in Fig. 6(d) and Fig. 6(h) for both TE and TM. Such high E-field regions have been mostly seen at the interfaces of the metal resonator or dielectric material that characterized the developing surface plasmon effect shown in Fig. 6(a) to 6(c), and Fig. 6(e) to 6(g). These surface plasmons combined to produce a perfectly resonant dipole, magnifying the concentrated E-field [50,51]. As a consequence of the greater surface plasmons at the interfaces, paired with the strong optical properties of W, the MM absorber exhibited high absorption at such wavelengths. In addition, since our proposed design has perfect symmetry geometrically, the H-field follows the same distribution as E-field for both TE and TM mode with stipulated wavelength. The anti-parallel circulating current induced at the interface of the resonator layer and base metal layer generates a magnetic moment that has been immersed on the dielectric layer, thus reducing the reflections [52]. The confined surface plasmons resonance across the entire operational spectrum causes a greater H-field. Furthermore, a strong H-field occurrence signifies that the intended meta-surface traps the EM incident wave, assisting wideband absorption. In Fig. 6(i) to 6(p), H-field can be seen firmly centered on the metal resonator and significantly constrained by the dielectric material. As demonstrated in Fig. 6(j) and 6(n), the H-field has been exceptionally strong in both TE and TM modes at the maximal absorption stage of $\lambda $= 495.16 nm. A cross-sectional view at $\lambda $= 495.16 nm demonstrates that the H-field level has been high near the core of the HMMA. The Hadamard matrix-based surface, which is perfectly distributed with E-field as well as H-field charges, is also another factor of perfect absorption. Further details are available in Supplement 1.

In Fig. 7, three distinct wavelengths have been used to represent the surface charge distribution for both the TE and TM modes to better link the fields, much as in the E-field or H-field. For both TE and TM polarization, the utmost highly scattered surface has been localized at resonance wavelength 495.16 nm. Due to the consistent anti-parallel circulation of the surface charge, the structure creates a large EM field in the dielectric spacer [53].

3.6 Comparative study

Table 3 illustrates that our suggested HMMA might be extremely desired. After comparing our HMMA design to various previous studies with comparable characteristics and domains, it has been clear that it outperforms other designs reported to date. In the end, our study demonstrated an ultra-thin MM absorber with very good temperature stability. It has also been worth noting that the materials utilized in the production of our HMMA are less expensive than other materials like gold, silver, silicon, and so on. Our HMMA may also be scaled down to not just half the unit cell size but a quarter unit cell size and still be effective as an optical absorber. The design has also been highly functional at incident angles of up to 70°, which has been optimal for harvesting energy and for applications such as sensors. Furthermore, since W has been used as the base metal, it does not require the use of quartz or a glass layer, as it has been the case with other designs. As a consequence of this feature, the cost of fabricating this structure will be reduced. Because of its large bandwidth, the high absorption peak of 99.99%, and the aforementioned features, the suggested MM absorber is a powerful solution in the field of optical applications.

Tables Icon

Table 3. Comparison of the proposed unit cell's characteristics and bandwidth with that of prior studies

4. Conclusion

We constructed and theoretically validated a novel three-layer broadband, polarization-independent, ultra-thin MM absorber consisting of W and SiO2, with a metasurface structure based on the pattern distribution of the Hadamard matrix. As a result of the simulations, we demonstrated that our HMMA displays good absorption across a wide spectrum spanning the optical domain in both the TE and TM modes, with a peak value of 99.99% for both modes. Furthermore, with an average absorption of 97.15%, which can be very effective. Because of the strong demand in the optical sector and the simplicity of the geometry, the proposed design might be used for light detectors, a true invisibility cloak, optical sensors, magnetic resonance imaging, plasmonic sensors, and other high thermal imaging applications.

Funding

CTRGC grant 2021-22 of North South University (approved by the members of BOT).

Disclosures

Authors of this article declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.

Supplemental document

See Supplement 1 for supporting content.

References

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References

  • View by:

  1. V. G. Veselago, “The electrodynamic of substances with simultaneous negative values of e and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
    [Crossref]
  2. N. I. Zheludev, “The road ahead for metamaterials,” Science 328(5978), 582–583 (2010).
    [Crossref]
  3. Y. Liu and X. Zhang, “Metamaterials: A new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
    [Crossref]
  4. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using µ-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [Crossref]
  5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref]
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184 (2000).
  7. S. H. Lee, C. M. Park, Y. M. Seo, and C. K. Kim, “Reversed Doppler effect in double negative metamaterials,” Phys. Rev. B 81(24), 241102 (2010).
    [Crossref]
  8. Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8(1), 14901–7 (2017).
    [Crossref]
  9. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer New York, 2010).
  10. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15(18), 11133 (2007).
    [Crossref]
  11. A. Sarsen and C. Valagiannopoulos, “Robust polarization twist by pairs of multilayers with tilted optical axes,” Phys. Rev. B 99(11), 115304 (2019).
    [Crossref]
  12. D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. De Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
    [Crossref]
  13. S. H. Raad and Z. Atlasbaf, “Tunable optical meta-surface using graphene-coated spherical nanoparticles,” AIP Adv. 9(7), 075224 (2019).
    [Crossref]
  14. A. Hoque, M. T. Islam, A. F. Almutairi, M. E. H. Chowdhury, and M. Samsuzzaman, “SNG and DNG meta-absorber with fractional absorption band for sensing application,” Sci. Rep. 10(1), 13086–17 (2020).
    [Crossref]
  15. M. A. Baqir and P. K. Choudhury, “On the VO2 metasurface-based temperature sensor,” J. Opt. Soc. Am. B 36(8), F123–F130 (2019).
    [Crossref]
  16. R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
    [Crossref]
  17. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4×2.4µm 2 footprint,” Nat. Photonics 9(6), 378–382 (2015).
    [Crossref]
  18. L. Luo, I. Chatzakis, J. Wang, F. B. P. Niesler, M. Wegener, T. Koschny, and C. M. Soukoulis, “Broadband terahertz generation from metamaterials,” Nat. Commun. 5(1), 3055 (2014).
    [Crossref]
  19. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [Crossref]
  20. S. Ju, S. H. A. Shah, M. A. Javed, J. Li, G. Palteru, J. Robin, Y. Xing, O. Kanhere, and T. S. Rappaport, “Scattering mechanisms and modeling for terahertz wireless communications,” ICC 2019-2019 IEEE International Conference on Communications (ICC) (IEEE, 2019), pp. 1–7.
  21. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
    [Crossref]
  22. Y. P. Lee, P. V. Tuong, H. Y. Zheng, J. Y. Rhee, and W. H. Jang, “An application of metamaterials: Perfect absorbers,” J. Korean Phys. Soc. 60(8), 1203–1206 (2012).
    [Crossref]
  23. M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater. 23(45), 5410–5414 (2011).
    [Crossref]
  24. M. Luo, S. Shen, L. Zhou, S. Wu, Y. Zhou, and L. Chen, “Broadband, wide-angle, and polarization- independent metamaterial absorber for the visible regime,” Opt. Express 25(14), 16715–16724 (2017).
    [Crossref]
  25. M. H. Heidari and S. H. Sedighy, “Broadband wide-angle polarization-insensitive metasurface solar absorber,” J. Opt. Soc. Am. A 35(4), 522 (2018).
    [Crossref]
  26. Y. Zhou, M. Luo, S. Shen, H. Zhang, D. Pu, and L. Chen, “Cost-effective near-perfect absorber at visible frequency based on homogenous meta-surface nickel with two-dimension cylinder array,” Opt. Express 26(21), 27482 (2018).
    [Crossref]
  27. S. Mahmud, S. S. Islam, A. F. Almutairi, and M. T. Islam, “A wide incident angle, ultrathin, polarization-insensitive metamaterial absorber for optical wavelength applications,” IEEE Access 8, 129525–129541 (2020).
    [Crossref]
  28. S. Mahmud, S. S. Islam, K. Mat, M. E. H. Chowdhury, H. Rmili, and M. T. Islam, “Design and parametric analysis of a wide-angle polarization-insensitive metamaterial absorber with a star shape resonator for optical wavelength applications,” Results Phys. 18, 103259 (2020).
    [Crossref]
  29. Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, M. Pan, and Y. Yi, “Dual band visible metamaterial absorbers based on four identical ring patches,” Physica E: Low-dimensional Systems and Nanostructures 127, 114526 (2021).
    [Crossref]
  30. S. Charola, S. K. Patel, K. Dalsaniya, R. Jadeja, T. K. Nguyen, and V. Dhasarathan, “Numerical investigation of wideband L-shaped metasurface based solar absorber for visible and ultraviolet region,” Physica B: Condensed Matter 601, 412503 (2021).
    [Crossref]
  31. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163(1-3), 95–102 (1999).
    [Crossref]
  32. P. M. Schneidert and W. B. Fowler, “Band structure and optical properties of silicon dioxide,” Phys. Rev. Lett. 36(8), 425 (1976).
  33. A. K. Azad, W. J. M. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Metasurface broadband solar absorber,” Sci. Rep. 6(1), 6–11 (2016).
    [Crossref]
  34. E. G. Gamaly and A. V. Rode, “Transient optical properties of dielectrics and semiconductors excited by an ultrashort laser pulse,” J. Opt. Soc. Am. B 31(11), C36 (2014).
    [Crossref]
  35. M. M. K. Shuvo, M. I. Hossain, S. Mahmud, S. Rahman, M. T. H. Topu, A. Hoque, S. S. Islam, M. S. Soliman, S. H. A. Almalki, M. S. Islam, and M. T. Islam, “Polarization and angular insensitive bendable metamaterial absorber for UV to NIR range,” Sci. Rep. 12(1), 1–16 (2022).
    [Crossref]
  36. R. K. Yarlagadda and J. E. Hershey, Hadamard Matrix Analysis and Synthesis: With Applications to Communications and Signal/Image Processing, Vol. 383 (Springer Science and Business Media, 2012).
  37. M. Rauf, A. M. Khan, A. Ansari, M. T. Jilani, and T. Shahzeb, “Skin depth verification of the electromagnetic waves for hydrocarbon detection,” JAE 60(3), 313–326 (2019).
    [Crossref]
  38. E. J. Rothwell, J. L. Frasch, S. M. Ellison, P. Chahal, and R. O. Ouedraogo, “Analysis of the Nicolson-Ross-Weir method for characterizing the electromagnetic properties of engineered materials,” Prog. Electromagn. Res. 157(July), 31–47 (2016).
    [Crossref]
  39. D. Lee, J. G. Hwang, D. Lim, T. Hara, and S. Lim, “Incident angle- and polarization-insensitive metamaterial absorber using circular sectors,” Sci. Rep. 6(1), 1–8 (2016).
    [Crossref]
  40. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
  41. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
    [Crossref]
  42. T. Wanghuang, W. Chen, Y. Huang, and G. Wen, “Analysis of metamaterial absorber in normal and oblique incidence by using interference theory,” AIP Adv. 3(10), 102118 (2013).
    [Crossref]
  43. H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express 20(7), 7165 (2012).
    [Crossref]
  44. H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010).
    [Crossref]
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2022 (1)

M. M. K. Shuvo, M. I. Hossain, S. Mahmud, S. Rahman, M. T. H. Topu, A. Hoque, S. S. Islam, M. S. Soliman, S. H. A. Almalki, M. S. Islam, and M. T. Islam, “Polarization and angular insensitive bendable metamaterial absorber for UV to NIR range,” Sci. Rep. 12(1), 1–16 (2022).
[Crossref]

2021 (3)

Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, M. Pan, and Y. Yi, “Dual band visible metamaterial absorbers based on four identical ring patches,” Physica E: Low-dimensional Systems and Nanostructures 127, 114526 (2021).
[Crossref]

S. Charola, S. K. Patel, K. Dalsaniya, R. Jadeja, T. K. Nguyen, and V. Dhasarathan, “Numerical investigation of wideband L-shaped metasurface based solar absorber for visible and ultraviolet region,” Physica B: Condensed Matter 601, 412503 (2021).
[Crossref]

I. Hossain, M. Samsuzzaman, M. Moniruzzaman, B. B. Bais, M. S. J. Singh, and M. T. Islam, “Polarization-independent broadband optical regime metamaterial absorber for solar harvesting: a numerical approach,” Chinese Journal of Physics 71(April), 699–715 (2021).
[Crossref]

2020 (4)

A. Hoque, M. T. Islam, A. F. Almutairi, M. E. H. Chowdhury, and M. Samsuzzaman, “SNG and DNG meta-absorber with fractional absorption band for sensing application,” Sci. Rep. 10(1), 13086–17 (2020).
[Crossref]

S. Mahmud, S. S. Islam, A. F. Almutairi, and M. T. Islam, “A wide incident angle, ultrathin, polarization-insensitive metamaterial absorber for optical wavelength applications,” IEEE Access 8, 129525–129541 (2020).
[Crossref]

S. Mahmud, S. S. Islam, K. Mat, M. E. H. Chowdhury, H. Rmili, and M. T. Islam, “Design and parametric analysis of a wide-angle polarization-insensitive metamaterial absorber with a star shape resonator for optical wavelength applications,” Results Phys. 18, 103259 (2020).
[Crossref]

R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
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2019 (6)

A. Sarsen and C. Valagiannopoulos, “Robust polarization twist by pairs of multilayers with tilted optical axes,” Phys. Rev. B 99(11), 115304 (2019).
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S. H. Raad and Z. Atlasbaf, “Tunable optical meta-surface using graphene-coated spherical nanoparticles,” AIP Adv. 9(7), 075224 (2019).
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M. A. Baqir and P. K. Choudhury, “On the VO2 metasurface-based temperature sensor,” J. Opt. Soc. Am. B 36(8), F123–F130 (2019).
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M. Rauf, A. M. Khan, A. Ansari, M. T. Jilani, and T. Shahzeb, “Skin depth verification of the electromagnetic waves for hydrocarbon detection,” JAE 60(3), 313–326 (2019).
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N. Gasimov, M. Karaaslan, C. Sabah, and F. Karadag, “Some aspects of mass-energy equivalence which appears in left-handed metamaterials,” EPJ Appl. Metamat. 6, 16 (2019).
[Crossref]

H. Xu, L. Hu, Y. Lu, J. Xu, and Y. Chen, “Dual-band metamaterial absorbers in the visible and near-infrared regions,” J. Phys. Chem. C 123(15), 10028–10033 (2019).
[Crossref]

2018 (4)

S. Suzuki, T. Doi, A. Masuda, and T. Tanahashi, “Bending cyclic load test for crystalline silicon photovoltaic modules,” Jap. J. Appl. 57, 02CE05 (2018).

T. Zhang, L. Xie, Y. Li, T. K. Mallick, Q. Wei, X. Hao, and B. He, “Experimental and theoretical research on bending behavior of photovoltaic panels with a special boundary condition,” Energies 11(12), 3435 (2018).

M. H. Heidari and S. H. Sedighy, “Broadband wide-angle polarization-insensitive metasurface solar absorber,” J. Opt. Soc. Am. A 35(4), 522 (2018).
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Y. Zhou, M. Luo, S. Shen, H. Zhang, D. Pu, and L. Chen, “Cost-effective near-perfect absorber at visible frequency based on homogenous meta-surface nickel with two-dimension cylinder array,” Opt. Express 26(21), 27482 (2018).
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2017 (3)

2016 (3)

A. K. Azad, W. J. M. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Metasurface broadband solar absorber,” Sci. Rep. 6(1), 6–11 (2016).
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E. J. Rothwell, J. L. Frasch, S. M. Ellison, P. Chahal, and R. O. Ouedraogo, “Analysis of the Nicolson-Ross-Weir method for characterizing the electromagnetic properties of engineered materials,” Prog. Electromagn. Res. 157(July), 31–47 (2016).
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D. Lee, J. G. Hwang, D. Lim, T. Hara, and S. Lim, “Incident angle- and polarization-insensitive metamaterial absorber using circular sectors,” Sci. Rep. 6(1), 1–8 (2016).
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2015 (2)

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. De Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
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B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4×2.4µm 2 footprint,” Nat. Photonics 9(6), 378–382 (2015).
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2014 (3)

L. Luo, I. Chatzakis, J. Wang, F. B. P. Niesler, M. Wegener, T. Koschny, and C. M. Soukoulis, “Broadband terahertz generation from metamaterials,” Nat. Commun. 5(1), 3055 (2014).
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E. G. Gamaly and A. V. Rode, “Transient optical properties of dielectrics and semiconductors excited by an ultrashort laser pulse,” J. Opt. Soc. Am. B 31(11), C36 (2014).
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X. Duan, S. Chen, W. Liu, H. Cheng, Z. Li, and J. Tian, “Polarization-insensitive and wide-angle broadband nearly perfect absorber by tunable planar metamaterials in the visible regime,” J. Opt. 16(12), 125107 (2014).
[Crossref]

2013 (1)

T. Wanghuang, W. Chen, Y. Huang, and G. Wen, “Analysis of metamaterial absorber in normal and oblique incidence by using interference theory,” AIP Adv. 3(10), 102118 (2013).
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2012 (3)

H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express 20(7), 7165 (2012).
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H. M. Lee and J. C. Wu, “A wide-angle dual-band infrared perfect absorber based on metal dielectric metal split square-ring and square array,” J. Appl. Phys. D 45(20), 205101 (2012).
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Y. P. Lee, P. V. Tuong, H. Y. Zheng, J. Y. Rhee, and W. H. Jang, “An application of metamaterials: Perfect absorbers,” J. Korean Phys. Soc. 60(8), 1203–1206 (2012).
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2011 (2)

M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater. 23(45), 5410–5414 (2011).
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Y. Liu and X. Zhang, “Metamaterials: A new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
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2010 (3)

N. I. Zheludev, “The road ahead for metamaterials,” Science 328(5978), 582–583 (2010).
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S. H. Lee, C. M. Park, Y. M. Seo, and C. K. Kim, “Reversed Doppler effect in double negative metamaterials,” Phys. Rev. B 81(24), 241102 (2010).
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H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010).
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2008 (1)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
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2007 (2)

2006 (2)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using µ-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
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Q. Wei, H. T. Zhang, B. E. Schuster, K. T. Ramesh, R. Z. Valiev, L. J. Kecskes, R. J. Dowding, L. Magness, and K. Cho, “Microstructure and mechanical properties of super-strong nanocrystalline tungsten processed by high-pressure torsion,” Acta Mater. 54(15), 4079–4089 (2006).
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2005 (2)

Q. Wei, T. Jiao, K. T. Ramesh, E. Ma, L. J. Kecskes, L. Magness, R. Dowding, V. U. Kazykhanov, and R. Z. Valiev, “Mechanical behavior and dynamic failure of high-strength ultrafine grained tungsten under uniaxial compression,” Acta Mater. 54(1), 77–87 (2005).
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D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).

2001 (2)

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D. Dragoi, E. Üstündag, B. Clausen, and M. A. M. Bourke, “Investigation of thermal residual stresses in tungsten-fiber/bulk metallic glass matrix composites,” Scr. Mater. 45(2), 245–252 (2001).
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2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
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1999 (1)

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1968 (1)

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M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater. 23(45), 5410–5414 (2011).
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Ali, M. M.

R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
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Almalki, S. H. A.

M. M. K. Shuvo, M. I. Hossain, S. Mahmud, S. Rahman, M. T. H. Topu, A. Hoque, S. S. Islam, M. S. Soliman, S. H. A. Almalki, M. S. Islam, and M. T. Islam, “Polarization and angular insensitive bendable metamaterial absorber for UV to NIR range,” Sci. Rep. 12(1), 1–16 (2022).
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Almutairi, A. F.

A. Hoque, M. T. Islam, A. F. Almutairi, M. E. H. Chowdhury, and M. Samsuzzaman, “SNG and DNG meta-absorber with fractional absorption band for sensing application,” Sci. Rep. 10(1), 13086–17 (2020).
[Crossref]

S. Mahmud, S. S. Islam, A. F. Almutairi, and M. T. Islam, “A wide incident angle, ultrathin, polarization-insensitive metamaterial absorber for optical wavelength applications,” IEEE Access 8, 129525–129541 (2020).
[Crossref]

Altug, H.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. De Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref]

Ansari, A.

M. Rauf, A. M. Khan, A. Ansari, M. T. Jilani, and T. Shahzeb, “Skin depth verification of the electromagnetic waves for hydrocarbon detection,” JAE 60(3), 313–326 (2019).
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Atlasbaf, Z.

S. H. Raad and Z. Atlasbaf, “Tunable optical meta-surface using graphene-coated spherical nanoparticles,” AIP Adv. 9(7), 075224 (2019).
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A. K. Azad, W. J. M. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Metasurface broadband solar absorber,” Sci. Rep. 6(1), 6–11 (2016).
[Crossref]

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010).
[Crossref]

Bais, B. B.

I. Hossain, M. Samsuzzaman, M. Moniruzzaman, B. B. Bais, M. S. J. Singh, and M. T. Islam, “Polarization-independent broadband optical regime metamaterial absorber for solar harvesting: a numerical approach,” Chinese Journal of Physics 71(April), 699–715 (2021).
[Crossref]

Baqir, M. A.

R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
[Crossref]

M. A. Baqir and P. K. Choudhury, “On the VO2 metasurface-based temperature sensor,” J. Opt. Soc. Am. B 36(8), F123–F130 (2019).
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R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
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Bourke, M. A. M.

D. Dragoi, E. Üstündag, B. Clausen, and M. A. M. Bourke, “Investigation of thermal residual stresses in tungsten-fiber/bulk metallic glass matrix composites,” Scr. Mater. 45(2), 245–252 (2001).
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E. J. Rothwell, J. L. Frasch, S. M. Ellison, P. Chahal, and R. O. Ouedraogo, “Analysis of the Nicolson-Ross-Weir method for characterizing the electromagnetic properties of engineered materials,” Prog. Electromagn. Res. 157(July), 31–47 (2016).
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Chakravadhanula, V. S. K.

M. K. Hedayati, M. Javaherirahim, B. Mozooni, R. Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula, V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri, “Design of a perfect black absorber at visible frequencies using plasmonic metamaterials,” Adv. Mater. 23(45), 5410–5414 (2011).
[Crossref]

Charola, S.

S. Charola, S. K. Patel, K. Dalsaniya, R. Jadeja, T. K. Nguyen, and V. Dhasarathan, “Numerical investigation of wideband L-shaped metasurface based solar absorber for visible and ultraviolet region,” Physica B: Condensed Matter 601, 412503 (2021).
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Chatzakis, I.

L. Luo, I. Chatzakis, J. Wang, F. B. P. Niesler, M. Wegener, T. Koschny, and C. M. Soukoulis, “Broadband terahertz generation from metamaterials,” Nat. Commun. 5(1), 3055 (2014).
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Chen, F.

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010).
[Crossref]

Chen, H. T.

A. K. Azad, W. J. M. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Metasurface broadband solar absorber,” Sci. Rep. 6(1), 6–11 (2016).
[Crossref]

H. T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad, and A. J. Taylor, “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010).
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Chen, H.-T.

Chen, L.

Chen, M.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8(1), 14901–7 (2017).
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Chen, S.

X. Duan, S. Chen, W. Liu, H. Cheng, Z. Li, and J. Tian, “Polarization-insensitive and wide-angle broadband nearly perfect absorber by tunable planar metamaterials in the visible regime,” J. Opt. 16(12), 125107 (2014).
[Crossref]

Chen, W.

T. Wanghuang, W. Chen, Y. Huang, and G. Wen, “Analysis of metamaterial absorber in normal and oblique incidence by using interference theory,” AIP Adv. 3(10), 102118 (2013).
[Crossref]

Chen, X.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8(1), 14901–7 (2017).
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Chen, Y.

H. Xu, L. Hu, Y. Lu, J. Xu, and Y. Chen, “Dual-band metamaterial absorbers in the visible and near-infrared regions,” J. Phys. Chem. C 123(15), 10028–10033 (2019).
[Crossref]

Cheng, H.

X. Duan, S. Chen, W. Liu, H. Cheng, Z. Li, and J. Tian, “Polarization-insensitive and wide-angle broadband nearly perfect absorber by tunable planar metamaterials in the visible regime,” J. Opt. 16(12), 125107 (2014).
[Crossref]

Cheng, S.

Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, M. Pan, and Y. Yi, “Dual band visible metamaterial absorbers based on four identical ring patches,” Physica E: Low-dimensional Systems and Nanostructures 127, 114526 (2021).
[Crossref]

Cho, K.

Q. Wei, H. T. Zhang, B. E. Schuster, K. T. Ramesh, R. Z. Valiev, L. J. Kecskes, R. J. Dowding, L. Magness, and K. Cho, “Microstructure and mechanical properties of super-strong nanocrystalline tungsten processed by high-pressure torsion,” Acta Mater. 54(15), 4079–4089 (2006).
[Crossref]

Choudhury, P. K.

R. M. H. Bilal, M. A. Saeed, P. K. Choudhury, M. A. Baqir, W. Kamal, M. M. Ali, and A. A. Rahim, “Elliptical metallic rings-shaped fractal metamaterial absorber in the visible regime,” Sci. Rep. 10(1), 1–12 (2020).
[Crossref]

M. A. Baqir and P. K. Choudhury, “On the VO2 metasurface-based temperature sensor,” J. Opt. Soc. Am. B 36(8), F123–F130 (2019).
[Crossref]

Chowdhury, M. E. H.

A. Hoque, M. T. Islam, A. F. Almutairi, M. E. H. Chowdhury, and M. Samsuzzaman, “SNG and DNG meta-absorber with fractional absorption band for sensing application,” Sci. Rep. 10(1), 13086–17 (2020).
[Crossref]

S. Mahmud, S. S. Islam, K. Mat, M. E. H. Chowdhury, H. Rmili, and M. T. Islam, “Design and parametric analysis of a wide-angle polarization-insensitive metamaterial absorber with a star shape resonator for optical wavelength applications,” Results Phys. 18, 103259 (2020).
[Crossref]

Chu, P.

Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, M. Pan, and Y. Yi, “Dual band visible metamaterial absorbers based on four identical ring patches,” Physica E: Low-dimensional Systems and Nanostructures 127, 114526 (2021).
[Crossref]

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D. Dragoi, E. Üstündag, B. Clausen, and M. A. M. Bourke, “Investigation of thermal residual stresses in tungsten-fiber/bulk metallic glass matrix composites,” Scr. Mater. 45(2), 245–252 (2001).
[Crossref]

Dalsaniya, K.

S. Charola, S. K. Patel, K. Dalsaniya, R. Jadeja, T. K. Nguyen, and V. Dhasarathan, “Numerical investigation of wideband L-shaped metasurface based solar absorber for visible and ultraviolet region,” Physica B: Condensed Matter 601, 412503 (2021).
[Crossref]

Dalvit, D. A. R.

A. K. Azad, W. J. M. Kort-Kamp, M. Sykora, N. R. Weisse-Bernstein, T. S. Luk, A. J. Taylor, D. A. R. Dalvit, and H. T. Chen, “Metasurface broadband solar absorber,” Sci. Rep. 6(1), 6–11 (2016).
[Crossref]

De Abajo, F. J. G.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. De Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref]

Dhasarathan, V.

S. Charola, S. K. Patel, K. Dalsaniya, R. Jadeja, T. K. Nguyen, and V. Dhasarathan, “Numerical investigation of wideband L-shaped metasurface based solar absorber for visible and ultraviolet region,” Physica B: Condensed Matter 601, 412503 (2021).
[Crossref]

Doi, T.

S. Suzuki, T. Doi, A. Masuda, and T. Tanahashi, “Bending cyclic load test for crystalline silicon photovoltaic modules,” Jap. J. Appl. 57, 02CE05 (2018).

Dowding, R.

Q. Wei, T. Jiao, K. T. Ramesh, E. Ma, L. J. Kecskes, L. Magness, R. Dowding, V. U. Kazykhanov, and R. Z. Valiev, “Mechanical behavior and dynamic failure of high-strength ultrafine grained tungsten under uniaxial compression,” Acta Mater. 54(1), 77–87 (2005).
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Supplementary Material (1)

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Supplement 1       Supplemental Material 1

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the (a) HMMA unit cell in a 6 × 6 array, (b) 3-D view of the unit cell and structural parameters, and (c) the top-front surface view of the unit cell and its surface design in relation to the Hadamard matrix to form the resonator, where “-1” represents W placements on the unit cell’s top layer and “+1” signifies placements where W was not placed on the top layer.
Fig. 2.
Fig. 2. (a) Absorbance and reflectance of the design in the range 380 to 700 nm in TEM, TE, and TM modes of the proposed unit cell structure, (b) real and imaginary parts of the normalized impedance, (c) magnitude of co-polarization and cross-polarization in dB of the reflection coefficient for both TE and TM-polarization modes, (d) PCR of the proposed unit cell for both TE and TM modes, EM wave properties in which (e) relative permeability, (f) relative permittivity, (g) refractive index vs. wavelength of the proposed HMMA unit cell, (h) absorbance comparison of a full, half, quarter size HMMA unit cell, (i), representing of a parametric sweep for the parameter “td” in TEM mode, and (j) the effect of different dielectric materials on absorbance.
Fig. 3.
Fig. 3. (a) A side-by-side view of the simulated absorbance with the calculated values, and (b) a representational illustration of the interference theory structure.
Fig. 4.
Fig. 4. Effect of changing polarization angles (φ) from 0-90° on absorbance in (a) TE mode, (b) TM mode, and for incident angles (θ) in the range 0-70° in (c) TE mode, (d) TM mode.
Fig. 5.
Fig. 5. Absorption phenomenon of the HMMA unit cell under (a) negative/convex bending from 0° to 30°, and (b) positive/concave bending from 0° to 30°. Insets: (a) representation of negative bending with θ degree, and (b) positive bending with θ degree.
Fig. 6.
Fig. 6. Distribution of the E-field (a)-(c) for TE polarization at 420, 495.16, and 690 nm in the y-x axis, (d) the cross-sectional view of the middle of the proposed structure at 495.16 nm (peak) resonance wavelength in TE mode in the z-x axis, (e)-(g) E-field distribution in TM polarization at 420, 495.16, 690 nm in the y-x axis, (h) the cross-sectional view of the middle of the structure at 495.16 nm (peak) resonance wavelength in TM mode in the z-x axis. The H-field distribution (i)-(k) for TE polarization at 420, 495.16, and 690 nm for the unit cell in the y-x axis, (l) middle cross-sectional view of the proposed structure at 495.16 nm (peak) resonance wavelength in TE mode in the z-x axis, (m)-(o) H-field distribution in TM polarization at 420, 495.16 and 690 nm in the y-x axis, and (p) the cross-sectional view of the middle of the structure at 495.16 nm (peak) resonance wavelength in TM mode in the z-x axis.
Fig. 7.
Fig. 7. Distribution of the surface charge of the unit cell (a)-(c) for TE polarization at 420, 495.16, and 690 nm in the y-x axis, (d)-(f) TM polarization at 420, 495.16, and 690 nm in the y-x axis for normal incident angle. The cross-section has been done with an x-y axis.

Tables (3)

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Table 1. The Proposed Unit Cell Design’s Parameter List

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Table 2. Bending angle in degrees associated with maximum vM stress in GPa for proposed HMMA

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Table 3. Comparison of the proposed unit cell's characteristics and bandwidth with that of prior studies

Equations (17)

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H 2 n = H 2 H 2 n 1
H 2 = [ + 1 + 1 + 1 1 ]
H 4 = [ + 1 + 1 + 1 + 1 + 1 1 + 1 1 + 1 + 1 1 1 + 1 1 1 + 1 ]
H 8 = [ + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 + 1 1 1 + 1 + 1 1 1 + 1 1 1 + 1 + 1 1 1 + 1 + 1 + 1 + 1 + 1 1 1 1 1 + 1 1 + 1 1 1 + 1 1 + 1 + 1 + 1 1 1 1 1 + 1 + 1 + 1 1 1 + 1 1 + 1 + 1 1 ]
A ( ω ) = 1 R ( ω ) T ( ω ) = 1 | S 11 ( ω ) | 2 | S 21 ( ω ) | 2
δ = 2 ρ / 2 π f μ r μ o
A ( ω ) = 1 R ( ω ) = 1 | S 11 ( ω ) | 2
Z ( ω ) = [ μ r ( ω ) μ o ε r ( ω ) ε o ]
μ r = 2 [ 1 ( S 21 S 11 ) ] / j k 0 d [ 1 + ( S 21 S 11 ) ]
ε r = 2 [ 1 ( S 21 + S 11 ) ] / j k 0 d [ 1 + ( S 21 + S 11 ) ]
n r = c [ ( S 21 1 ) 2 S 11 2 ] / j π f d [ ( S 21 1 ) 2 S 11 2 ]
| S 11 ( ω ) | 2 = | R y y | 2 + | R y x | 2
| S 11 ( ω ) | 2 = | R x x | 2 + | R x y | 2
P C R T E = R y x 2 ( R y y 2 + R y x 2 )
P C R T M = R x y 2 ( R x x 2 + R x y 2 )
S 11 ( T o t a l ) = | S 11 | exp ( j θ 11 ) + | S 12 | | S 21 | exp [ j ( θ 12 + θ 21 2 β π ) ] 1 | S 22 | exp [ j ( θ 22 2 β π ) ]
S 11 ( T o t a l ) = | S 11 | exp ( j θ 11 ) + | S 12 | 2 exp [ j ( 2 θ 12 2 β π ) ] 1 | S 22 | exp [ j ( θ 22 2 β π ) ]
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