Dodecanacci superconductor-metamaterial photonic quasicrystal
Chittaranjan Nayak, Alireza Aghajamali, Mehdi Solaimani, Jayanta K. Rakshit, Damodar Panigrahy, Kanaparthi V. P. Kumar, Bandaru Ramakrishna
DDodecanacci superconductor-metamaterial photonic quasicrystal
Chittaranjan Nayak, ∗ Alireza Aghajamali, Mehdi Solaimani, Jayanta K. Rakshit, Damodar Panigrahy, Kanaparthi V. P. Kumar, and Bandaru Ramakrishna Department of Electronics and Communication Engineering,SRM Institute of Science and Technology, Chennai, 603202, Tamilnadu, India Department of Physics and Astronomy, Curtin University, Perth, Western Australia 6102, Australia Department of Physics, Faculty of science, Qom University of Technology, Qom, Iran Department of Electronics and Instrumentation Engineering National Institute of technology Agartala, Agartala, India (Dated: March 3, 2020)Using the transfer matrix method, the present paper attempt to determine the properties of thephotonic spectra of the Dodecanacci superconductor-metamaterial one-dimensional quasiperiodicmultilayer. The numerical calculation is supported by using the transfer matrix method. At first,we analyze the transmission for Dodecanacci quasicrystal for different generations. After that, weanalyze the effect of the thickness of the building blocks and the operating temperature. We observedthat a vast number of forbidden bandgaps and transmission pecks are developed in its transmissionspectra up to a certain generation number of Dodecanacci quasiperiodic sequence. If the generationnumber increases further, then the bandgaps become wider. According to the obtained results,depending on its generation, this structure can be used as an optical reflector or narrowband filter.
I. INTRODUCTION
Photonic crystal (PC) is a type of artificial compositematerial with photonic bandgap (PBG), the frequencyranges over which electromagnetic modes are forbidden,created by periodical modulated dielectric functions inthe spatial domain. Since first proposed by John [1] andYablonovitch [2], their potential scientific and technolog-ical applications have inspired great interest among re-searchers. In particular, tunable PCs [3], which havethe capability of tuning the bandgap by external stim-uli such as temperature, pressure, electric field, magneticfield, and many more, have attracted increasing interestbecause of their potential applications in optical devices.Therefore, the optical properties of the PCs containingvarious kinds of materials including dielectrics, metals,metamaterials, semiconductors, plasma, and supercon-ductors (SCs) have been investigated [4–17]. When theconstituent materials of PCs are superconductor, or evenonly a defect layer is a superconductor, superconductingphotonic crystals (SC-PCs) are formed [17–23]. The useof the superconducting photonic crystals has an advan-tage over conventional metal-dielectric PCs; for example,the loss issue of the PCs can be remedied by the uti-lization of superconductors, dielectric function of the su-perconductor depends on the external temperature, etc.Therefore, SC-PCs have an excellent opportunity for ap-plication in the tunable photonic crystal.Over 30 years ago, Veselago [24] theoretically stud-ied the propagation of electromagnetic waves in ma-terials characterized by simultaneously having a nega-tive permittivity ( (cid:15) ) and permeability ( µ ) and referredto such materials as left-handed materials to emphasizethe fact that the electric field ( E ), the magnetic field ∗ ( H ) and the propagation wave vector ( k ) are related bythe left-handed rule. These materials, which are nowcalled double-negative (DNG) materials (or metamate-rial), have received extensive attention due to their ex-perimental validation and unusual electromagnetic prop-erties. Recently, by the possibility of manufacturingPCs with metamaterials, called metamaterial photoniccrystals (MPCs), a new research area has emerged, andnumerous interesting results have been reported by re-searchers so far. The inclusion of metamaterial in PCled to the emergence of new mechanisms to produce pho-tonic gaps, which helps to design dielectric mirrors, effec-tive waveguides, filters, perfect lenses, etc. Based on theproperties of metamaterials, superconductor, and PCs,researchers now intend to investigate the transmittanceof one-dimensional PCs in the visible and microwave re-gions. It was suggested that the combination of super-conductor NbN with DNG metamaterials is possible todesign the metamaterial superconductor PC because ofthe feasibility of coating [25].Moreover, artificially fabricated deterministic or so-called quasiperiodic structures constitute a separate fieldof research. These quasiperiodic photonic structures areformed by the superposition of two (or more) incom-mensurate periods so that they can be defined as in-termediate systems between a photonic crystal and therandom photonic multilayer. Compared to the periodicand random photonic multilayers, photonic quasicrystals(PQCs) share distinctive physical properties with bothperiodic media, i.e., the formation of well-defined PBGs,and disordered random media, i.e., the presence of lo-calized states, thus offering an almost unexplored poten-tial for the control and manipulation of localized fieldstates. This conception of quasicrystal was first intro-duced by Kohmoto et al. [26] in the field photonicswith a Fibonacci arrangement of two dielectric materials.Then after various quasicrystals including, Thue-Morse[14, 27, 28], double-periodic [14, 28], Rudin-Shapiro [28], a r X i v : . [ phy s i c s . c l a ss - ph ] F e b Octonocci [14, 29, 30], and many more are presented withinteresting and useful results. Brief reports on this topicwere presented by Bellingeri [31], Vardeny et al. [32],and Edagawa [33] in which various PQCs structures werediscussed. Recently it was presented that the cutoff fre-quency and the bandwidth of superconducting photonicquasicrystals (SC-PQCs) are remarkably sensitive to thetemperature of superconducting material, periodic shortorder of the quasiperiodic system, incident angles, andtype of polarization [28, 34, 35]. The addition of metama-terial as a constituent material in the SC-PQCs is quitefeasible and be an interesting topic in the field of photonicmultilayer design.Recently, few studies on Dodecannaci photonic qua-sicrystals [36] have been carried out using dielectric [37]and magnetized plasma [38]. Silva et al. presented thedielectric Dodecannaci [37] with the SiO /TiO multi-layers, where a graphene monolayer at the interfacesbetween distinct layers. The result indicates that thewhole optical spectrum becomes affected by the pres-ence of graphene in the interfaces including the shift ofbandgaps to high-frequency regions, the emergence ofa graphene induced bandgap at low-frequency regions,and the decreasing of transmittance in the whole fre-quency range. In another investigation, Nayak [38] pre-sented Dodecannaci extrinsic magnetized plasma qua-sicrystal, which evident the robust against layer positionphotonic bandgap aroused above the plasma frequency.Motivated by these distinctive results offered by the Do-decannaci quasicrystal and the ability to manage theelectromagnetic waves by superconductor as well as themetamaterial, we now present the study of the photonicbandgap characteristics of one-dimensional Dodecannacisuperconductor-metamaterial quasicrystal.This manuscript is organized as follows. In Section II,we describe the basic Dodecanacci quasi-sequence, thepermittivity of the constituent materials, and summa-rize the numerical methods. In Section III, we presentthe numerical results and discuss many physical param-eters that could tune the transmittance of our proposedsuperconductor metamaterial quasicrystal. Finally, theconclusions are summarized in Section IV. II. METHODOLOGY
Before proceeding to the results and simulated data, inthis section, we are briefly describing the fundamentalsof the proposed study. This section is organized in thefollowing three sub-sections: • Dodecanacci quasi-sequence • Constituent materials • Transfer-matrix method
A. Dodecanacci quasi-sequence
A generalized Dodecanacci quasi-sequence [36–38] canbe stated by the deterministic rule, which is definedas D x = { AD x − D x − } D x − , for x ≥
3, with ini-tial condition: D = AAB , D = { AD } D = AAABAAABAAB . Here x is the generation number.The total number of blocks A and B in D x can be cal-culated by the recurrence relation, P x = 4 P x − − P x − for x ≥
3, where P = 3 and P = 11. Here, A is thebuilding block modelling of one of the constituent ma-terial having thickness, d A with permittivity ε A whereas B is the building block modelling of other constituentmaterial having thickness, d B with permittivity ε B . TheDodecanacci superlattice can also be generated by theinflation rule A → AAAB , and B → AAB . In Table I,we illustrate the three distributed chain D , D , and D of two building blocks A and B organized according toDodecanacci quasiperiodic sequence having x ≥
3. Theother two Dodecanacci quasiperiodic sequence used inthe work are D and D having 2131 and 7953 numberof building layers, respectively. B. Constituent materials
The proposed quasicrystal composed by double-negative metamaterial (DNG) and lossless superconduc-tor is arranged as per the Dodecanacci quasiperiodic se-quence. The description of frequency dependent dielec-tric properties of the DNG, represents layer A and thelossless superconductor represents layer B in the Dode-canacci quasi sequence are appended below. The fre-quency dependent numerical formula for calculating thepermittivity, ε A ( f ) and the permeability, µ A ( f ) of theDNG material are given by [5, 16, 25]. ε A ( f ) = 1 + 5 . − f − if γ e + 10 . − f − if γ e (1) µ A ( f ) = 1 + 3 . − f − i πωγ m (2)where, γ e and γ m represent electric damping frequencyand magnetic damping frequency, respectively.The frequency dependent dielectric constant of a loss-less superconductor is given by [17, 25] ε B ( f ) = 1 − (cid:32) π f µ ε λ l (cid:33) (3)The permittivity and permeability of air are noted in theabove equations with ε and µ respectively. Whereas λ l stands for the temperature-dependent London penetra-tion depth, which is given as λ l = λ / (cid:113) (1 − ( T /T c ) ) (4) TABLE I. Organized blocks (
A, B ) follow Dodecanacci quasiperiodic sequence.
Sequence: Distributed
A, B chain of Dodecanacci Total number D x = { AD x − D x − } D x − , for x ≥ D AAABAAABAAABAABAAABAAABAAABAABAAABAAABAAB D AAAABAAABAABAAABAAABAAABAABAAABAAABAAABAA
BAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAABAAABAAABAABAAAABAAABAAABAABAAABAAABAABD AAAABAAABAAABAABAAABAAABAAABAABAAABAAABAA
ABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAAABAAABAAABAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAAABAAABAABAAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAABAAABAABAAABAAABAAABAABAAABAAABAAABAABAAAB here, London penetration depth at absolute temperature,operating temperature (K) and critical temperature (K)is mentioned as λ , T , and T c , respectively. C. Transfer-matrix method
To investigate the deferent aspects of the one-dimensional Dodecannaci quasicrystal here we take thehelp of transfer matrix method [39]. The characteristicmatrices for layer A and layer B of the Dodecannaci qua-sicrystal is written as M A = (cid:20) cos ϕ A − ip A sin ϕ A − ip A sin ϕ A cos ϕ A (cid:21) (5)and M B = (cid:20) cos ϕ B − ip A sin ϕ B − ip B sin ϕ B cos ϕ B (cid:21) (6)In Eq. 5, ϕ A = ( ω/c ) n A d A cos θ A , c is the speed of lightin vacuum, n A is the refractive index of the A layer, θ A is the ray angle inside the layer A , p A = (cid:112) ε A /µ A cos θ A and cos θ A = (cid:113) − ( n sin θ /n A ), in which n is therefractive index of the air. In Eq. 6 the notations aresame Eq. 5 but for layer B .The transfer matrix of the Dodecannaci quasicrystal, M x for x ≥ M x = { M A M x − M x − } M x − = (cid:20) m m m m (cid:21) , (7)here, m , m , m and m are the matrix elements ofthe multilayer system M x with the initial conditions of M = M A M A M B and M = { M A M } M . The trans-mission coefficient ( t D ) of the Dodecannaci quasicrystalis given by t D = E O E i = 2 P in ( m + m P out ) P in + ( m + m P out )(8)Where, E O , and E i are the input and output electricfield intensity, respectively. P in = n cos θ , and P out = n out cos θ out in which n out is the refractive index of theenvironment having ray angle of θ out . The transmittance( T D ) of the Dodecannaci quasicrystal is represented by T D = t D • t D ∗ (9) III. RESULT AND DISCUSSIONS
We have studied in detail different realizations ofthe Dodecannaci superconductor-metamaterial photonicquasicrystals, composed of DNG metamaterial and low-temperature superconductor materials, NbN ( T C = 16 Kand λ L (0)=200 nm) [25, 40], by varying the generation FIG. 1. Transmittance spectrum from 1D multilayered stackarranged according Dodecannaci sequence at generation num-ber, x = 1 to 7, while d A and d B are fixed to be 10 mm and4 nm. number, x , thickness of DNG metamaterial, d A , thick-ness of superconductor material, d B , and the operat-ing temperature, T . The considered generation number, x of Dodecannaci quasicrystal, are from 1 to 7. Thefrequency range is fixed 1.5 to 3 GHz. The specificfrequency range was chosen to fit our objective to de-sign Dodecannaci superconductor-metamaterial photonicquasicrystals [41]. The medium surrounding the sup-posed structures is a vacuum. The parameters of DNGmetamaterial, electric damping frequency, γ e and mag-netic damping frequency, γ m are assumed to be the sameand are equal to 2 × − GHz [5, 17, 25, 41–43]. Atthis starting point, the thicknesses d A and d B , and op-erating temperatures are set to 10 mm, 4 nm and 4.2 Kas in previous reports, otherwise, it is mentioned in therespective caption of figures.Using expressions (1) to (9) transmission spectra of Do-decannaci superconductor-metamaterial photonic qua-sicrystal at normal incidence is plotted in Fig. 1. Here,seven sub-plots presented the transmission spectra of de-sired quasicrystals having for generation numbers from x = 1 to 7 and are clearly represented with an appropri-ate legend. On increasing the generation number, x , it isunderstood that the average transmission for the consid-ered range of frequency is decreased. However, the rateof decrease in the average transmission is region-specific.More clearly, the average transmission of the first half ofthe considered frequency (1.5 to 2.25 GHz) is very firstlydecaying as compared to the second half (2.25 to 3 GHz).This is mainly possible because of the magnitude of the FIG. 2. Thickness of the DNG metamaterial dependent trans-mittance spectrum from 1D multilayered stack arranged ac-cording Dodecannaci sequence at generation number, x = 3(a), x = 4 (b), x = 5 (c), and x = 7 (d) while d B is fixed tobe 4 nm. negative refractive index and the number of DNG meta-material layers in the considered stratified structures.From the transmission Dodecannaci superconductor-metamaterial photonic quasicrystal having x = 2, itis clearly understood that four-measure bandgaps areappeared in between 1.5 to 2.4 GHz. These arousedbandgaps are marginally affected with an increase in x ,whereas the transmission regions are splits and form nar-row transmission regions. This trend of splitting of trans-mission is appeared up to x = 4 after that the count ofthe transmission peaks decrease with increase in x . At, x = 7, we observe a very sharp transmission peak with al-most unity in magnitude around 2.1 GHz, whereas a peakhaving nearly 5% of transmission is also noted. Thesetransmission peaks are because of the multiple trans-mission and reflection from the individual elements ofthe stratified structure, there spatial arrangement, andmaterial properties. These results may be suitable fordesigning of narrow band filter for different microwavephotonics applications.As we know the transmission spectra is a function ofthe thickness of building blocks, here the investigation isperformed over a range of thicknesses of DNG metamate-rial, d A as well as thicknesses of superconductor material, d B . Fig. 2 represents the thickness of the DNG metama-terial dependent transmittance spectrum from 1D mul-tilayered stack arranged according to Dodecannaci se-quence at generation number, x = 3 (a), x = 4 (b), x = 5(c), and x = 7 (d). Here, excluding d A all other param-eters that are used to compute the transmission spectrain Fig. 1 are not changed. By observing the results, it is FIG. 3. Thickness of the superconductor dependent transmit-tance spectrum from 1D multilayered stack arranged accord-ing Dodecannaci sequence at generation number, x = 3 (a), x = 4 (b), x = 5 (c), and x = 7 (d) while d A is fixed to be10 mm. noted that the transmission spectra are the function ofthe thicknesses of the DNG metamaterial, d A and blueshifted. This observation is obvious and resemblance tothe previous findings [25]. However, it is also interestingto note that the transmission peaks have become moreextensive as the thicknesses of the DNG metamaterial, d A increases.In Fig. 3 we present the thickness of the superconduc-tor layers dependent transmittance spectrum from multi-layered stack arranged according Dodecannaci sequenceat generation number, x = 3 (a), x = 4 (b), x = 5 (c),and x = 7(d). As like our first case of thickness variationfor DNG metamaterial, here, excluding d B all other pa-rameters that are used to compute the transmission spec-tra in Fig. 1 are not changed. By observing the results, itis noted that the response of the shifting of transmittancespectra is similar to the change in layer thickness DNGmetamaterial, d A but with a reduced rate of blue shift.While observing the width of the transmission peak, herewe got an inverse effect i.e. , the width of the transmis-sion peak is decrease with an increase in the thickness ofthe superconductor layers. The physics behind these ob-served results is the effective optical path length causedby the constituent materials of the proposed supper lat-tice.As discussed in the preceding section, there is a cleardependence of dielectric constant of the superconduct-ing material (Eq. 3) on the operating temperature. Inthis regard, in this section, we address the influenceof the operating temperature on the transmission spec-tra Dodecannaci superconductor-metamaterial photonic FIG. 4. Temperature dependent transmittance spectrumfrom 1D multilayered stack arranged according Dodecannacisequence at generation number, x = 3 (a), x = 4 (b), x = 5(c), and x = 7 (d) while d A and d B are fixed to be 10 mmand 4 nm. quasicrystal. To evaluate the thickness of the supercon-ductor dependent transmittance spectrum from multilay-ered stack arranged according to Dodecannaci sequenceat generation number, x = 3 (a), x = 4 (b), x = 5 (c),and x = 7 (d) and plotted in Fig. 4. Here, excluding tem-perature, all other parameters that are used to computethe transmission spectra in Fig. 1 are not changed. Weobserved a significant change in the transmission char-acteristics due to the variation of T . The results haveclearly shown that the increase in the operating tem-perature leads to a redshift the transmission response.Moreover, the degree of redshift also increases with anincrease in temperature. The rate of red shift may bebecause of the decrease in the number of the super elec-trons about the number of the normal electrons. IV. CONCLUSIONS
In this work, we presented a general theory for thepropagation of electromagnetic waves in one-dimensionalsuperconductor-metamaterial superlattice, which is fab-ricated in a quasiperiodic fashion in accordance with thegeneralized Dodecannaci recurrence relation. It is shownthat the transmission response can be tuned efficientlyby the operating temperature as well as by the thick-nesses of the constituent materials. When the temper-ature or the thickness of the DNG metamaterial layeris increased, the transmission response is blue shifted,whereas the transmission peaks become wider. A lowrate of blue shift is observed in case of change in thicknessof the superconducting layers with narrowing the trans-mission peaks. The influence of temperature on Dode-cannaci superconductor-metamaterial quasicrystal is alsodiscussed and found that the transmission response isred shifted and response is not linear. Our simulationresults show that the proposed one-dimensional Dode-cannaci superconductor-metamaterial superlattice wouldbe a promising device with satisfying performance towork as tunable perfect narrowband filters and may havemany other potential applications.
ACKNOWLEDGEMENTS
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