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Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma

Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma applied sciences Article Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma 1 , 2 , 3 , 2 , 4 Tatjana Gric * and Edik Rafailov Department of Electronic Systems, VILNIUS TECH, LT-10223 Vilnius, Lithuania Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK; e.rafailov@aston.ac.uk Center for Physical Sciences and Technology, Semiconductor Physics Institute, LT-02300 Vilnius, Lithuania Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia * Correspondence: tatjana.gric@vilniustech.lt Featured Application: Antenna design. Abstract: The main features of surface plasmon polaritons (SPPs) that can propagate in a metamaterial– magnetic plasma structure are studied from theoretical perspectives. Both the conventional and imaginary parts of the dispersion relation of SPPs are demonstrated considering transverse magnetic (TM) polarization. We examine and discuss the influence of the external magnetic field. The results demonstrate that this factor dramatically alters the nature of SPPs. It is concluded that the positions and propagation lengths of SPPs can be engineered. Moreover, we present an approach allowing for an absorption enhancement that is a pivotal factor in antenna design. A unified insight into the practical methods aiming to attain hyperbolic dispersion by means of nanostructured and nanowire metamaterials is demonstrated. Keywords: metamaterial; hyperbolic; absorption Citation: Gric, T.; Rafailov, E. Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma. Appl. Sci. 2021, 11, 4720. https://doi.org/10.3390/ 1. Introduction app11114720 Surface plasmon polaritons (SPPs) are treated as the electromagnetic excitations occur- ring at the boundary separating two different substances. It is worthwhile mentioning that Academic Editor: Theodore the real part of the dielectric function changes the signs on the interface. A conventional E. Matikas boundary separating a conducting and a dielectric medium was considered in [1]. Recently, astonishing advancement has been obtained in the field of SPPs. In particular, many in- Received: 27 April 2021 vestigations implying SPPs in metamaterials and photonic crystals have been conducted Accepted: 19 May 2021 because of the latter ’s ability to dramatically tune SPPs’ features [2]. Published: 21 May 2021 Metamaterial research has attracted the attention of optical engineers and material scientists due to the variety of possible applications such as imaging [3], cloaking [4], Publisher’s Note: MDPI stays neutral sensing [5], waveguiding [6], and simulating space–time phenomena [7]. Hyperbolic meta- with regard to jurisdictional claims in materials (HMMs) are named because of the topology of the isofrequency surface. The published maps and institutional affil- linear dispersion and isotropic performance of propagating plasmons cause a spherical iations. isofrequency surface. The exotic behavior of modes possessing large-magnitude wavevec- tors is considered as the most fascinating feature of such substrate. In vacuum, such large wavevector waves are evanescent with exponential decay. The tunable properties of plasmas offering some peculiar advantages paved the way Copyright: © 2021 by the authors. for the increasing interest in plasma photonic crystals and plasma composites. Plasmas Licensee MDPI, Basel, Switzerland. possess major advantages in comparison with other conventional materials. Doing so, one This article is an open access article may use the applied power supply for producing plasmas and adjusting the gas pressure or distributed under the terms and temperature of plasmas, aiming to engineer the dynamic shift in permittivity and calibrate conditions of the Creative Commons the amplitude on the complex plane. Attribution (CC BY) license (https:// Herein, we consider an interface of a hyperbolic metamaterial and magnetic plasma, creativecommons.org/licenses/by/ aiming to give rise to tunable properties including absorption enhancement. To attain 4.0/). Appl. Sci. 2021, 11, 4720. https://doi.org/10.3390/app11114720 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 7 Appl. Sci. 2021, 11, 4720 2 of 7 Herein, we consider an interface of a hyperbolic metamaterial and magnetic plasma, aiming to give rise to tunable properties including absorption enhancement. To attain the mentioned goal, we consider two different types of hyperbolic metamaterials, i.e., the mentioned goal, we consider two different types of hyperbolic metamaterials, i.e., nanolayered metamaterials and nanowire metamaterials. The inclusion of a plasma layer nanolayered metamaterials and nanowire metamaterials. The inclusion of a plasma layer opens wide avenues for investigations. The influences of the applied magnetic field, col- opens wide avenues for investigations. The influences of the applied magnetic field, lision frequency of plasma, background material dielectric constant, and thickness on the collision fr dispersion equency relations of of plasma, SPPs backgr are ound explo material red and discussed. Th dielectric constant, is paper is structured and thickness as fol- on lows. Firstly, we present the theoretical background, followed by the Results section. the dispersion relations of SPPs are explored and discussed. This paper is structured as follows. Firstly, we present the theoretical background, followed by the Results section. 2. Materials and Methods 2. Materials and Methods The schematic illustrations of the metamaterial samples chosen for the considerations are demonstrated in Figure 1. An obliquely incident EM wave of TM polarization was The schematic illustrations of the metamaterial samples chosen for the considerations assumed. It is worthwhile mentioning that we are dealing with the propagation of waves are demonstrated in Figure 1. An obliquely incident EM wave of TM polarization was at the interface separating the magnetized plasma layer and the metamaterial layer. A assumed. It is worthwhile mentioning that we are dealing with the propagation of waves nanostructured metamaterial composed of exchanging sheets of metal and dielectric re- at the interface separating the magnetized plasma layer and the metamaterial layer. A sults in the required extreme anisotropy [8] (Figure 1a). Seeking for the valid homogeni- nanostructured metamaterial composed of exchanging sheets of metal and dielectric results zation approach, the thicknesses of the sheet should be smaller than the size of the oper- in the required extreme anisotropy [8] (Figure 1a). Seeking for the valid homogenization ating wavelength. Metallic nanowires embedded in the dielectric material might be em- approach, the thicknesses of the sheet should be smaller than the size of the operating ployed as an alternative approach seeking a hyperbolic behavior [9] (Figure 1b). Silver wavelength. Metallic nanowires embedded in the dielectric material might be employed and gold are usually chosen as the possible metals that can be embedded in a nanoporous as an alternative alumina t appr empoach late. A seeking iming toa h hyp ave erbolic a deep ins behavior ight into[ t 9h ] e m (Figur ain ch e 1ar b). act Silver eristicsand of sur gold face waves, we employed a Drude model for the metal (i.e., silver) characterization. In this are usually chosen as the possible metals that can be embedded in a nanoporous alumina template. Aiming to have a deep insight into the main characteristics of surface waves, εω =− ε () m ∞ we employed a Drude model for the metal (i.e., silver) characterization. In this relation, ωδ + iω relation, permittivity is calculated as . The properties are found by permittivity fitting this permi is calculated ttias vity functi # (w) on to a = # particular . fre The quepr ncy ran operties ge of ar th ee bulk mate found by rial fitting [10]. m ¥ w +idw ω =9.5eV this permittivity function to a particular frequency range ε =5 of the p bulk material [10]. It is ∞ δ = 0.0987 eV It is concluded [11] that for silver, the values of , , and concluded [11] that for silver, the values of # = 5, w = 9.5 eV, and d = 0.0987 eV provide ¥ p provide a reasonable fit. a reasonable fit. (a) (b) (c) (d) Figure 1. Engineering hyperbolic metamaterials. (a) Nanostructured metamaterial comprising os- cillating metallic and dielectric layers resulting in a metal–dielectric nanostructure. (b) Nanowire metamaterial composed of metallic nanorods implanted in a dielectric material. (c) Enlarged view of the nanowire metamaterial unit cell. In both (a,b), the constituent components are subwave- length, permitting application of effective medium theory. Herein, d is the diameter of nanowires. (d) Structure under consideration with the SPPs propagating along the z axis. Appl. Sci. 2021, 11, 4720 3 of 7 The dielectric function of magnetic plasma is a permittivity tensor and is presented as follows: 2 3 # 0 ia 4 5 # = 0 g 0 , (1) ia 0 # where # , g, and a are the dielectric components perpendicular and parallel to the mag- netization, and the magneto-optical component, respectively. Moreover, the magnetized 2 2 # a plasma is expressed by the effective dielectric function as follows: # = , where 2 2 w (w+iv) w w p p # = 1 , a = , where v is the collision frequency in plasma, w 1 p 2 2 2 2 w (w+iv) w w (w+iv) w [ ] [ ] c c is the bulk plasma frequency, and w = eB /m is the cyclotron frequency [12]. Herein, B c 0 is the amplitude of the external magnetic field, e is the absolute charge of the electron, and m is the mass of the electron. One may obtain a dispersion expression for the surface plasmons propagating at the boundary between two anisotropic media. It is worthwhile obtaining a single surface mode with the propagation constant [13] by calculating the tangential components of the electric and magnetic fields at the interface: 0 1 1/2 # # # # v v jj ? @ A b = k , (2) # # # jj where b is the propagation constant, k is the wave number in vacuum, and # , # are jj the frequency-dependent permittivities of the metamaterial, being a highly anisotropic medium, in the parallel and perpendicular directions to the wave propagation. It is important that the result (2) is valid only under the condition of surface confinement, which can be presented in the following form: 2 2 2 k = k b /# # < 0 v v x,I (3) 2 2 2 II II k = k b /# # < 0 x,II jj ? Following the effective medium approximation approach, one may calculate the effective permittivities of the nanowire metamaterial as follows: # (1 + r) + # (1 r) # = # (4) ? d # (1 r) + # (1 + r) # = # r + # (1 r) (5) m d jj Here, r is the metal filling fraction ratio which is calculated as nanowire area r = (6) unit cell area 3. Results and Discussion Herein, we analytically investigate SPPs excited by a slit waveguide structure. The waveguide is formed by two semi-infinite plasma layers and a thin background dielectric layer, in which one plasma layer is in the presence of the external magnetic field and the other plasma layer is in the absence of a magnetic field. Two magnetic-optical effects are presented, the Faraday effect, and the Voigt effect if the plasma is involved with the external magnetic field. However, in this analysis, only the Voigt effect is taken into consideration. The plasma cannot be magnetized or influenced by the applied magnetic field under the transverse magnetic polarization case. Herein, we present a theoretical study by employing Equation (2) and consider the main characteristics of the SPPs. Equation (2) was solved with respect to b, aiming to obtain the results. The impact of the applied magnetic field, collision Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 7 3. Results and Discussion Herein, we analytically investigate SPPs excited by a slit waveguide structure. The waveguide is formed by two semi-infinite plasma layers and a thin background dielectric layer, in which one plasma layer is in the presence of the external magnetic field and the other plasma layer is in the absence of a magnetic field. Two magnetic-optical effects are presented, the Faraday effect, and the Voigt effect if the plasma is involved with the ex- ternal magnetic field. However, in this analysis, only the Voigt effect is taken into consid- eration. The plasma cannot be magnetized or influenced by the applied magnetic field under the transverse magnetic polarization case. Herein, we present a theoretical study Appl. Sci. 2021, 11, 4720 4 of 7 by employing Equation (2) and consider the main characteristics of the SPPs. Equation (2) was solved with respect to β, aiming to obtain the results. The impact of the applied mag- netic field, collision frequency, the dielectric constant, and the thickness of the dielectric layer embedded into the nanostructured metamaterial on the main characteristics of SPPs frequency, the dielectric constant, and the thickness of the dielectric layer embedded into is studied. A magnetic field B0 is applied parallel to the interface separating two regions. the nanostructured metamaterial on the main characteristics of SPPs is studied. A magnetic Figure 2 demonstrates the impact of the applied magnetic field on the dispersion di- field B is applied parallel to the interface separating two regions. agrams of SPPs. The displayed dispersion curves tend to a stable frequency. The men- Figure 2 demonstrates the impact of the applied magnetic field on the dispersion dia- tioned pheno grams of SPPs. menon takes The displayed place dispersion as the SPP mo curves de tend s prop to agat a stable e at t fr hequency e interface . The unmentioned der study. The dra phenomenon matic shi takes ft in place the SPPs to the higher as the SPP modes fre propagate quency rang at the e is o interface bserved, enh under study ancin.g the The dramatic shift in the SPPs to the higher frequency range is observed, enhancing the external external magnetic field. As it is seen in Figure 2b, employment of the nanowire metamate- rial med magnetic ia causes the exotic be field. As it is seen in Figur havior e of 2b, the disper employment sion of curves. Bet the nanowir we een the reg metamaterial ime of the media causes the exotic behavior of the dispersion curves. Between the regime of the bound and bound and radiative modes, a frequency gap region with purely imaginary β prohibiting radiative modes, a frequency gap region with purely imaginary b prohibiting propagation propagation exists. As we can clearly observe in Figure 2b, the case of ωc/ωp = 2 possesses exists. As we can clearly observe in Figure 2b, the case of w /w = 2 possesses some some discrepancies in this specific region between bound and racdiatpive modes with β be- discrepancies in this specific region between bound and radiative modes with b being not ing not purely imaginary. Moreover, it can clearly be observed from Figure 2b that β is the purely imaginary. Moreover, it can clearly be observed from Figure 2b that b is the complex complex number with Re(β) ≠ 0 around ω = 6.2 × 10 Hz. The presence of the real part of number with Re(b) 6= 0 around w = 6.2  10 Hz. The presence of the real part of the the propagation constant between regimes of the bound and radiative modes is treated as propagation constant between regimes of the bound and radiative modes is treated as the the exotic behavior of the proposed structure. Aiming to have a closer look at the nature exotic behavior of the proposed structure. Aiming to have a closer look at the nature of of the propagating waves, we introduce electric field distributions. Doing so, the case of the propagating waves, we introduce electric field distributions. Doing so, the case of the the nanowires is presented in Figure 3. nanowires is presented in Figure 3. (a) (b) Figure 2. Dispersion diagrams of SPPs for the case of different external magnetic fields. Other parameters are chosen as Figure 2. Dispersion diagrams of SPPs for the case of different external magnetic fields. Other parameters are chosen as follows, i.e., nanolayers (a): d1 = 10 nm; d2 = 20 nm; nanowires (b): d = 30 nm, S = 70 nm. follows, i.e., nanolayers (a): d = 10 nm; d = 20 nm; nanowires (b): d = 30 nm, S = 70 nm. 1 2 We will further examine this case by engineering dielectric properties of the host media of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plotted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligible effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot have an impact on the absorption. As it is seen in Figure 4b, the presence of the magnetic plasma significantly enhances the absorption. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 7 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 7 Appl. Sci. 2021, 11, 4720 5 of 7 (a) (b) Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) ωc/ωp = 1, (b) ωc/ωp = 2. It is assumed that the SPPs propagate at the interface of the plasma and nanowire metamaterial case, ω = 200 THz. We will further examine this case by engineering dielectric properties of the host me- dia of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plot- ted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligi- (a) (b) ble effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) ωc/ωp = 1, (b) ωc/ωp = Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) w /w = 1, c p 2. It is assumed that the SPPs hapropagate at the inte ve an impact on thrface e ab of sortp ht e plasma and ion. As it is s nanowire metamaterial ca een in Figure 4b, the pre se, sence o ω = 200 THz f the . magnetic (b) w /w = 2. It is assumed that the SPPs propagate at the interface of the plasma and nanowire metamaterial case, c p plasma significantly enhances the absorption. w = 200 THz. We will further examine this case by engineering dielectric properties of the host me- dia of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plot- ted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligi- ble effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot have an impact on the absorption. As it is seen in Figure 4b, the presence of the magnetic plasma significantly enhances the absorption. (a) (b) Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance of the nanowire composite interface, if ωc/ωp = 1, d = 30 nm, S = 70 nm. of the nanowire composite interface, if w /w = 1, d = 30 nm, S = 70 nm. c p Lastly, we introduce the numerical results of the effect of the geometry of the nanowires on the properties of SPPs, as shown in Figure 5. It can be observed that geometrical changes do not have a dramatic impact on the dispersion curves. (a) (b) Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance of the nanowire composite interface, if ωc/ωp = 1, d = 30 nm, S = 70 nm. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 7 Lastly, we introduce the numerical results of the effect of the geometry of the nan- owires on the properties of SPPs, as shown in Figure 5. It can be observed that geometrical Appl. Sci. 2021, 11, 4720 6 of 7 changes do not have a dramatic impact on the dispersion curves. (a) (b) Figure 5. Dispersion relations of SPPs for changeable nanowire metamaterial geometry: (a) S = 70 nm, (b) d = 30 nm. Figure 5. Dispersion relations of SPPs for changeable nanowire metamaterial geometry: (a) S = 70 nm, (b) d = 30 nm. 4. Conclusions 4. Conclusions To conclude, we studied the main characteristics of SPPs that can be excited in a met- To conclude, we studied the main characteristics of SPPs that can be excited in a amaterial–magnetic plasma structure. Taking on board two different types of hyperbolic metamaterial–magnetic plasma structure. Taking on board two different types of hyper- metamaterials, i.e., nanolayered and nanowire structures, we depicted both normal and bolic metamaterials, i.e., nanolayered and nanowire structures, we depicted both normal absorbing dispersion relations of SPPs for TM polarization. We conclude that employment and absorbing dispersion relations of SPPs for TM polarization. We conclude that employ- of magnetic plasmas into the structure under investigation gives rise to the tunable intri- ment of magnetic plasmas into the structure under investigation gives rise to the tunable guing features of SPPs. It can be concluded that changes in the external magnetic field intriguing features of SPPs. It can be concluded that changes in the external magnetic allow for the shift in the dispersion curves to the higher-frequency range. The former takes field allow for the shift in the dispersion curves to the higher-frequency range. The former place by increasing the value of the ωc/ωp ratio. Moreover, SPPs at the boundary of nan- takes place by increasing the value of the w /w ratio. Moreover, SPPs at the boundary of c p owire composites possess an intriguing behavior. Between the regime of the bound and nanowire composites possess an intriguing behavior. Between the regime of the bound and radiative modes, a frequency gap region with purely imaginary β prohibiting propagation radiative modes, a frequency gap region with purely imaginary b prohibiting propagation exists. The case of ωc/ωp = 2 possesses some discrepancies in this specific region with β exists. The case of w /w = 2 possesses some discrepancies in this specific region with b c p being not being not pur pure ely ly im imaginary aginary.. Mo Mor reover, eover, it it can can c clearly learly b be e observe observed d fr from Figur om Figure e 2b 2b that thatβb is is the complex the complex number w number with ith Re( Re(βb ) )≠ 0 a 6= 0round aroundω = w6.2 × = 6.2 10 10 Hz. The Hz. pr The esence presence of the ofreal p the ra eal rt of the propag part of the pr ation const opagationa constant nt between reg between imes o regimes f the bo ofund the bound and rad and iative mode radiative s is treate modes is d as the exotic treated as the behavior of the proposed exotic behavior of the proposed structstr ure. It c ucture. an be concluded that It can be concluded that inclusion inclusion of of magnetic plasma gives rise to the absorption enhancement that is pivotal and desirable magnetic plasma gives rise to the absorption enhancement that is pivotal and desirable for antenna design applications. Absorption enhancement can be controlled by varying the for antenna design applications. Absorption enhancement can be controlled by varying permittivity of the dielectric medium. the permittivity of the dielectric medium. Author Contributions: Conceptualization, T.G. and E.R.; methodology, T.G.; software, T.G.; vali- Author Contributions: Conceptualization, T.G. and E.R.; methodology, T.G.; software, T.G.; vali- dation, T.G. and E.R.; formal analysis, T.G.; investigation, T.G.; resources, E.R.; data curation, E.R.; dation, T.G. and E.R.; formal analysis, T.G.; investigation, T.G.; resources, E.R.; data curation, E.R.; writing—original draft preparation, T.G. and E.R.; writing—review and editing, T.G. and E.R.; visual- writing—original draft preparation, T.G. and E.R.; writing—review and editing, T.G. and E.R.; vis- ization, T.G.; supervision, E.R.; project administration, T.G. and E.R.; funding acquisition, T.G. and ualization, T.G.; supervision, E.R.; project administration, T.G. and E.R.; funding acquisition, T.G. E.R. Both authors have read and agreed to the published version of the manuscript. and E.R. Both authors have read and agreed to the published version of the manuscript. Fund Funding: ing: Th This is project has received funding from th project has received funding from the e European Union’s European Union’s H Horizon orizon 2020 research and 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement No 713694 and from innovation programme under the Marie Sklodowska Curie grant agreement No 713694 and from Engineering and Physical Scie Engineering and Physical Sciences nces Resear Research ch Council Council (EPSR (EPSRC) C) (Grant No. (Grant No. EP/R024898/1). EP/R024898/1). The work The work of E.U. Rafailov was partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-934 dated 17 November 2020). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appl. 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Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma

Gric, Tatjana; Rafailov, Edik
Applied Sciences , Volume 11 (11) – May 21, 2021
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applied sciences Article Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma 1 , 2 , 3 , 2 , 4 Tatjana Gric * and Edik Rafailov Department of Electronic Systems, VILNIUS TECH, LT-10223 Vilnius, Lithuania Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK; e.rafailov@aston.ac.uk Center for Physical Sciences and Technology, Semiconductor Physics Institute, LT-02300 Vilnius, Lithuania Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia * Correspondence: tatjana.gric@vilniustech.lt Featured Application: Antenna design. Abstract: The main features of surface plasmon polaritons (SPPs) that can propagate in a metamaterial– magnetic plasma structure are studied from theoretical perspectives. Both the conventional and imaginary parts of the dispersion relation of SPPs are demonstrated considering transverse magnetic (TM) polarization. We examine and discuss the influence of the external magnetic field. The results demonstrate that this factor dramatically alters the nature of SPPs. It is concluded that the positions and propagation lengths of SPPs can be engineered. Moreover, we present an approach allowing for an absorption enhancement that is a pivotal factor in antenna design. A unified insight into the practical methods aiming to attain hyperbolic dispersion by means of nanostructured and nanowire metamaterials is demonstrated. Keywords: metamaterial; hyperbolic; absorption Citation: Gric, T.; Rafailov, E. Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma. Appl. Sci. 2021, 11, 4720. https://doi.org/10.3390/ 1. Introduction app11114720 Surface plasmon polaritons (SPPs) are treated as the electromagnetic excitations occur- ring at the boundary separating two different substances. It is worthwhile mentioning that Academic Editor: Theodore the real part of the dielectric function changes the signs on the interface. A conventional E. Matikas boundary separating a conducting and a dielectric medium was considered in [1]. Recently, astonishing advancement has been obtained in the field of SPPs. In particular, many in- Received: 27 April 2021 vestigations implying SPPs in metamaterials and photonic crystals have been conducted Accepted: 19 May 2021 because of the latter ’s ability to dramatically tune SPPs’ features [2]. Published: 21 May 2021 Metamaterial research has attracted the attention of optical engineers and material scientists due to the variety of possible applications such as imaging [3], cloaking [4], Publisher’s Note: MDPI stays neutral sensing [5], waveguiding [6], and simulating space–time phenomena [7]. Hyperbolic meta- with regard to jurisdictional claims in materials (HMMs) are named because of the topology of the isofrequency surface. The published maps and institutional affil- linear dispersion and isotropic performance of propagating plasmons cause a spherical iations. isofrequency surface. The exotic behavior of modes possessing large-magnitude wavevec- tors is considered as the most fascinating feature of such substrate. In vacuum, such large wavevector waves are evanescent with exponential decay. The tunable properties of plasmas offering some peculiar advantages paved the way Copyright: © 2021 by the authors. for the increasing interest in plasma photonic crystals and plasma composites. Plasmas Licensee MDPI, Basel, Switzerland. possess major advantages in comparison with other conventional materials. Doing so, one This article is an open access article may use the applied power supply for producing plasmas and adjusting the gas pressure or distributed under the terms and temperature of plasmas, aiming to engineer the dynamic shift in permittivity and calibrate conditions of the Creative Commons the amplitude on the complex plane. Attribution (CC BY) license (https:// Herein, we consider an interface of a hyperbolic metamaterial and magnetic plasma, creativecommons.org/licenses/by/ aiming to give rise to tunable properties including absorption enhancement. To attain 4.0/). Appl. Sci. 2021, 11, 4720. https://doi.org/10.3390/app11114720 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 7 Appl. Sci. 2021, 11, 4720 2 of 7 Herein, we consider an interface of a hyperbolic metamaterial and magnetic plasma, aiming to give rise to tunable properties including absorption enhancement. To attain the mentioned goal, we consider two different types of hyperbolic metamaterials, i.e., the mentioned goal, we consider two different types of hyperbolic metamaterials, i.e., nanolayered metamaterials and nanowire metamaterials. The inclusion of a plasma layer nanolayered metamaterials and nanowire metamaterials. The inclusion of a plasma layer opens wide avenues for investigations. The influences of the applied magnetic field, col- opens wide avenues for investigations. The influences of the applied magnetic field, lision frequency of plasma, background material dielectric constant, and thickness on the collision fr dispersion equency relations of of plasma, SPPs backgr are ound explo material red and discussed. Th dielectric constant, is paper is structured and thickness as fol- on lows. Firstly, we present the theoretical background, followed by the Results section. the dispersion relations of SPPs are explored and discussed. This paper is structured as follows. Firstly, we present the theoretical background, followed by the Results section. 2. Materials and Methods 2. Materials and Methods The schematic illustrations of the metamaterial samples chosen for the considerations are demonstrated in Figure 1. An obliquely incident EM wave of TM polarization was The schematic illustrations of the metamaterial samples chosen for the considerations assumed. It is worthwhile mentioning that we are dealing with the propagation of waves are demonstrated in Figure 1. An obliquely incident EM wave of TM polarization was at the interface separating the magnetized plasma layer and the metamaterial layer. A assumed. It is worthwhile mentioning that we are dealing with the propagation of waves nanostructured metamaterial composed of exchanging sheets of metal and dielectric re- at the interface separating the magnetized plasma layer and the metamaterial layer. A sults in the required extreme anisotropy [8] (Figure 1a). Seeking for the valid homogeni- nanostructured metamaterial composed of exchanging sheets of metal and dielectric results zation approach, the thicknesses of the sheet should be smaller than the size of the oper- in the required extreme anisotropy [8] (Figure 1a). Seeking for the valid homogenization ating wavelength. Metallic nanowires embedded in the dielectric material might be em- approach, the thicknesses of the sheet should be smaller than the size of the operating ployed as an alternative approach seeking a hyperbolic behavior [9] (Figure 1b). Silver wavelength. Metallic nanowires embedded in the dielectric material might be employed and gold are usually chosen as the possible metals that can be embedded in a nanoporous as an alternative alumina t appr empoach late. A seeking iming toa h hyp ave erbolic a deep ins behavior ight into[ t 9h ] e m (Figur ain ch e 1ar b). act Silver eristicsand of sur gold face waves, we employed a Drude model for the metal (i.e., silver) characterization. In this are usually chosen as the possible metals that can be embedded in a nanoporous alumina template. Aiming to have a deep insight into the main characteristics of surface waves, εω =− ε () m ∞ we employed a Drude model for the metal (i.e., silver) characterization. In this relation, ωδ + iω relation, permittivity is calculated as . The properties are found by permittivity fitting this permi is calculated ttias vity functi # (w) on to a = # particular . fre The quepr ncy ran operties ge of ar th ee bulk mate found by rial fitting [10]. m ¥ w +idw ω =9.5eV this permittivity function to a particular frequency range ε =5 of the p bulk material [10]. It is ∞ δ = 0.0987 eV It is concluded [11] that for silver, the values of , , and concluded [11] that for silver, the values of # = 5, w = 9.5 eV, and d = 0.0987 eV provide ¥ p provide a reasonable fit. a reasonable fit. (a) (b) (c) (d) Figure 1. Engineering hyperbolic metamaterials. (a) Nanostructured metamaterial comprising os- cillating metallic and dielectric layers resulting in a metal–dielectric nanostructure. (b) Nanowire metamaterial composed of metallic nanorods implanted in a dielectric material. (c) Enlarged view of the nanowire metamaterial unit cell. In both (a,b), the constituent components are subwave- length, permitting application of effective medium theory. Herein, d is the diameter of nanowires. (d) Structure under consideration with the SPPs propagating along the z axis. Appl. Sci. 2021, 11, 4720 3 of 7 The dielectric function of magnetic plasma is a permittivity tensor and is presented as follows: 2 3 # 0 ia 4 5 # = 0 g 0 , (1) ia 0 # where # , g, and a are the dielectric components perpendicular and parallel to the mag- netization, and the magneto-optical component, respectively. Moreover, the magnetized 2 2 # a plasma is expressed by the effective dielectric function as follows: # = , where 2 2 w (w+iv) w w p p # = 1 , a = , where v is the collision frequency in plasma, w 1 p 2 2 2 2 w (w+iv) w w (w+iv) w [ ] [ ] c c is the bulk plasma frequency, and w = eB /m is the cyclotron frequency [12]. Herein, B c 0 is the amplitude of the external magnetic field, e is the absolute charge of the electron, and m is the mass of the electron. One may obtain a dispersion expression for the surface plasmons propagating at the boundary between two anisotropic media. It is worthwhile obtaining a single surface mode with the propagation constant [13] by calculating the tangential components of the electric and magnetic fields at the interface: 0 1 1/2 # # # # v v jj ? @ A b = k , (2) # # # jj where b is the propagation constant, k is the wave number in vacuum, and # , # are jj the frequency-dependent permittivities of the metamaterial, being a highly anisotropic medium, in the parallel and perpendicular directions to the wave propagation. It is important that the result (2) is valid only under the condition of surface confinement, which can be presented in the following form: 2 2 2 k = k b /# # < 0 v v x,I (3) 2 2 2 II II k = k b /# # < 0 x,II jj ? Following the effective medium approximation approach, one may calculate the effective permittivities of the nanowire metamaterial as follows: # (1 + r) + # (1 r) # = # (4) ? d # (1 r) + # (1 + r) # = # r + # (1 r) (5) m d jj Here, r is the metal filling fraction ratio which is calculated as nanowire area r = (6) unit cell area 3. Results and Discussion Herein, we analytically investigate SPPs excited by a slit waveguide structure. The waveguide is formed by two semi-infinite plasma layers and a thin background dielectric layer, in which one plasma layer is in the presence of the external magnetic field and the other plasma layer is in the absence of a magnetic field. Two magnetic-optical effects are presented, the Faraday effect, and the Voigt effect if the plasma is involved with the external magnetic field. However, in this analysis, only the Voigt effect is taken into consideration. The plasma cannot be magnetized or influenced by the applied magnetic field under the transverse magnetic polarization case. Herein, we present a theoretical study by employing Equation (2) and consider the main characteristics of the SPPs. Equation (2) was solved with respect to b, aiming to obtain the results. The impact of the applied magnetic field, collision Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 7 3. Results and Discussion Herein, we analytically investigate SPPs excited by a slit waveguide structure. The waveguide is formed by two semi-infinite plasma layers and a thin background dielectric layer, in which one plasma layer is in the presence of the external magnetic field and the other plasma layer is in the absence of a magnetic field. Two magnetic-optical effects are presented, the Faraday effect, and the Voigt effect if the plasma is involved with the ex- ternal magnetic field. However, in this analysis, only the Voigt effect is taken into consid- eration. The plasma cannot be magnetized or influenced by the applied magnetic field under the transverse magnetic polarization case. Herein, we present a theoretical study Appl. Sci. 2021, 11, 4720 4 of 7 by employing Equation (2) and consider the main characteristics of the SPPs. Equation (2) was solved with respect to β, aiming to obtain the results. The impact of the applied mag- netic field, collision frequency, the dielectric constant, and the thickness of the dielectric layer embedded into the nanostructured metamaterial on the main characteristics of SPPs frequency, the dielectric constant, and the thickness of the dielectric layer embedded into is studied. A magnetic field B0 is applied parallel to the interface separating two regions. the nanostructured metamaterial on the main characteristics of SPPs is studied. A magnetic Figure 2 demonstrates the impact of the applied magnetic field on the dispersion di- field B is applied parallel to the interface separating two regions. agrams of SPPs. The displayed dispersion curves tend to a stable frequency. The men- Figure 2 demonstrates the impact of the applied magnetic field on the dispersion dia- tioned pheno grams of SPPs. menon takes The displayed place dispersion as the SPP mo curves de tend s prop to agat a stable e at t fr hequency e interface . The unmentioned der study. The dra phenomenon matic shi takes ft in place the SPPs to the higher as the SPP modes fre propagate quency rang at the e is o interface bserved, enh under study ancin.g the The dramatic shift in the SPPs to the higher frequency range is observed, enhancing the external external magnetic field. As it is seen in Figure 2b, employment of the nanowire metamate- rial med magnetic ia causes the exotic be field. As it is seen in Figur havior e of 2b, the disper employment sion of curves. Bet the nanowir we een the reg metamaterial ime of the media causes the exotic behavior of the dispersion curves. Between the regime of the bound and bound and radiative modes, a frequency gap region with purely imaginary β prohibiting radiative modes, a frequency gap region with purely imaginary b prohibiting propagation propagation exists. As we can clearly observe in Figure 2b, the case of ωc/ωp = 2 possesses exists. As we can clearly observe in Figure 2b, the case of w /w = 2 possesses some some discrepancies in this specific region between bound and racdiatpive modes with β be- discrepancies in this specific region between bound and radiative modes with b being not ing not purely imaginary. Moreover, it can clearly be observed from Figure 2b that β is the purely imaginary. Moreover, it can clearly be observed from Figure 2b that b is the complex complex number with Re(β) ≠ 0 around ω = 6.2 × 10 Hz. The presence of the real part of number with Re(b) 6= 0 around w = 6.2  10 Hz. The presence of the real part of the the propagation constant between regimes of the bound and radiative modes is treated as propagation constant between regimes of the bound and radiative modes is treated as the the exotic behavior of the proposed structure. Aiming to have a closer look at the nature exotic behavior of the proposed structure. Aiming to have a closer look at the nature of of the propagating waves, we introduce electric field distributions. Doing so, the case of the propagating waves, we introduce electric field distributions. Doing so, the case of the the nanowires is presented in Figure 3. nanowires is presented in Figure 3. (a) (b) Figure 2. Dispersion diagrams of SPPs for the case of different external magnetic fields. Other parameters are chosen as Figure 2. Dispersion diagrams of SPPs for the case of different external magnetic fields. Other parameters are chosen as follows, i.e., nanolayers (a): d1 = 10 nm; d2 = 20 nm; nanowires (b): d = 30 nm, S = 70 nm. follows, i.e., nanolayers (a): d = 10 nm; d = 20 nm; nanowires (b): d = 30 nm, S = 70 nm. 1 2 We will further examine this case by engineering dielectric properties of the host media of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plotted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligible effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot have an impact on the absorption. As it is seen in Figure 4b, the presence of the magnetic plasma significantly enhances the absorption. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 7 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 7 Appl. Sci. 2021, 11, 4720 5 of 7 (a) (b) Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) ωc/ωp = 1, (b) ωc/ωp = 2. It is assumed that the SPPs propagate at the interface of the plasma and nanowire metamaterial case, ω = 200 THz. We will further examine this case by engineering dielectric properties of the host me- dia of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plot- ted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligi- (a) (b) ble effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) ωc/ωp = 1, (b) ωc/ωp = Figure 3. Electric field distribution in y direction for the case of different external magnetic fields: (a) w /w = 1, c p 2. It is assumed that the SPPs hapropagate at the inte ve an impact on thrface e ab of sortp ht e plasma and ion. As it is s nanowire metamaterial ca een in Figure 4b, the pre se, sence o ω = 200 THz f the . magnetic (b) w /w = 2. It is assumed that the SPPs propagate at the interface of the plasma and nanowire metamaterial case, c p plasma significantly enhances the absorption. w = 200 THz. We will further examine this case by engineering dielectric properties of the host me- dia of the nanowire metamaterial along with the metamaterial geometry. Figure 4 is plot- ted aiming to take into account the impact of the permittivity of the host material when the collision is considered for the plasma. One can conclude that the rise in the dielectric constant will result in the shift in the propagating plasmons to the lower-frequency range. The former phenomenon can be described by the variational principle [14]. In other words, an increase in the dielectric constant ends with the shift in the modes to the lower- frequency range. It should be mentioned that for the instance of planar nanostructured hyperbolic metamaterials, one should expect the absorption enhancement to be a negligi- ble effect because of the lack of any localized plasmons resulting in the field hotspots [15]. Additionally, the high k-modes cannot be excited by free space illumination and cannot have an impact on the absorption. As it is seen in Figure 4b, the presence of the magnetic plasma significantly enhances the absorption. (a) (b) Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance of the nanowire composite interface, if ωc/ωp = 1, d = 30 nm, S = 70 nm. of the nanowire composite interface, if w /w = 1, d = 30 nm, S = 70 nm. c p Lastly, we introduce the numerical results of the effect of the geometry of the nanowires on the properties of SPPs, as shown in Figure 5. It can be observed that geometrical changes do not have a dramatic impact on the dispersion curves. (a) (b) Figure 4. Dependences of the real (a) and imaginary (b) parts of the propagation constant versus frequency for the instance of the nanowire composite interface, if ωc/ωp = 1, d = 30 nm, S = 70 nm. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 7 Lastly, we introduce the numerical results of the effect of the geometry of the nan- owires on the properties of SPPs, as shown in Figure 5. It can be observed that geometrical Appl. Sci. 2021, 11, 4720 6 of 7 changes do not have a dramatic impact on the dispersion curves. (a) (b) Figure 5. Dispersion relations of SPPs for changeable nanowire metamaterial geometry: (a) S = 70 nm, (b) d = 30 nm. Figure 5. Dispersion relations of SPPs for changeable nanowire metamaterial geometry: (a) S = 70 nm, (b) d = 30 nm. 4. Conclusions 4. Conclusions To conclude, we studied the main characteristics of SPPs that can be excited in a met- To conclude, we studied the main characteristics of SPPs that can be excited in a amaterial–magnetic plasma structure. Taking on board two different types of hyperbolic metamaterial–magnetic plasma structure. Taking on board two different types of hyper- metamaterials, i.e., nanolayered and nanowire structures, we depicted both normal and bolic metamaterials, i.e., nanolayered and nanowire structures, we depicted both normal absorbing dispersion relations of SPPs for TM polarization. We conclude that employment and absorbing dispersion relations of SPPs for TM polarization. We conclude that employ- of magnetic plasmas into the structure under investigation gives rise to the tunable intri- ment of magnetic plasmas into the structure under investigation gives rise to the tunable guing features of SPPs. It can be concluded that changes in the external magnetic field intriguing features of SPPs. It can be concluded that changes in the external magnetic allow for the shift in the dispersion curves to the higher-frequency range. The former takes field allow for the shift in the dispersion curves to the higher-frequency range. The former place by increasing the value of the ωc/ωp ratio. Moreover, SPPs at the boundary of nan- takes place by increasing the value of the w /w ratio. Moreover, SPPs at the boundary of c p owire composites possess an intriguing behavior. Between the regime of the bound and nanowire composites possess an intriguing behavior. Between the regime of the bound and radiative modes, a frequency gap region with purely imaginary β prohibiting propagation radiative modes, a frequency gap region with purely imaginary b prohibiting propagation exists. The case of ωc/ωp = 2 possesses some discrepancies in this specific region with β exists. The case of w /w = 2 possesses some discrepancies in this specific region with b c p being not being not pur pure ely ly im imaginary aginary.. Mo Mor reover, eover, it it can can c clearly learly b be e observe observed d fr from Figur om Figure e 2b 2b that thatβb is is the complex the complex number w number with ith Re( Re(βb ) )≠ 0 a 6= 0round aroundω = w6.2 × = 6.2 10 10 Hz. The Hz. pr The esence presence of the ofreal p the ra eal rt of the propag part of the pr ation const opagationa constant nt between reg between imes o regimes f the bo ofund the bound and rad and iative mode radiative s is treate modes is d as the exotic treated as the behavior of the proposed exotic behavior of the proposed structstr ure. It c ucture. an be concluded that It can be concluded that inclusion inclusion of of magnetic plasma gives rise to the absorption enhancement that is pivotal and desirable magnetic plasma gives rise to the absorption enhancement that is pivotal and desirable for antenna design applications. Absorption enhancement can be controlled by varying the for antenna design applications. Absorption enhancement can be controlled by varying permittivity of the dielectric medium. the permittivity of the dielectric medium. Author Contributions: Conceptualization, T.G. and E.R.; methodology, T.G.; software, T.G.; vali- Author Contributions: Conceptualization, T.G. and E.R.; methodology, T.G.; software, T.G.; vali- dation, T.G. and E.R.; formal analysis, T.G.; investigation, T.G.; resources, E.R.; data curation, E.R.; dation, T.G. and E.R.; formal analysis, T.G.; investigation, T.G.; resources, E.R.; data curation, E.R.; writing—original draft preparation, T.G. and E.R.; writing—review and editing, T.G. and E.R.; visual- writing—original draft preparation, T.G. and E.R.; writing—review and editing, T.G. and E.R.; vis- ization, T.G.; supervision, E.R.; project administration, T.G. and E.R.; funding acquisition, T.G. and ualization, T.G.; supervision, E.R.; project administration, T.G. and E.R.; funding acquisition, T.G. E.R. Both authors have read and agreed to the published version of the manuscript. and E.R. Both authors have read and agreed to the published version of the manuscript. Fund Funding: ing: Th This is project has received funding from th project has received funding from the e European Union’s European Union’s H Horizon orizon 2020 research and 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement No 713694 and from innovation programme under the Marie Sklodowska Curie grant agreement No 713694 and from Engineering and Physical Scie Engineering and Physical Sciences nces Resear Research ch Council Council (EPSR (EPSRC) C) (Grant No. (Grant No. EP/R024898/1). EP/R024898/1). The work The work of E.U. Rafailov was partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-934 dated 17 November 2020). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appl. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: May 21, 2021

Keywords: metamaterial; hyperbolic; absorption

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