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Dodecanacci extrinsic magnetized plasma multilayer
Optical Materials 100 (2020) 109653
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Dodecanacci extrinsic magnetized plasma multilayer Chittaranjan Nayak Department of Electronics and Communication Engineering, SRM Institute of Science and Technology, Chennai, 603203, Tamilnadu, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Extrinsic magnetized plasma Quasiperiodic structures Dodecanacci sequence Structure-property relationship
An extrinsic magnetized plasma quasi-multilayer is demonstrated using the transfer-matrix method with an extrinsic bulk plasma inspired structure. The transmittance of the extrinsic bulk plasma quasi-system has been shown using the Dodecanacci quasi-sequence. For the periodic structure, two significant gaps emerge and only one of them, the bandgap above the plasma frequency, appears in any of the proposed Dodecanacci systems and it is magnetically tunable and robust against the position of the layers. The other bandgap, observed below the plasma frequency is also magnetically tunable but it vanishes when the layers are quasiperiodically disposed because the emergence of several resonant peaks. The count and width of the transmission window in such vanished bandgaps decrease as the Dodecanacci generation number increases. The results also show that the electron density can control the gap location and gap width for all Dodecanacci structures. These listed results ensure that the approach of Dodecanacci extrinsic magnetized plasma multilayers can be applied to the design of future architectures for the control and the manipulation of the electromagnetic wave-matter interaction in microwave frequencies.
1. Introduction Photonic crystals (PhCs) are artificial periodic structures with two kinds of materials [1]. Indeed, periodic modulation of the dielectric constant leads to the appearance of a frequency range in which the electromagnetic wave is forbidden, called the photonic bandgap (PBG) [2]. A new trend in PhC research is to explore the active interaction of photonic structure with light emission and is very much familiar as the photonic quasicrystal [3]. The crystallographic and physical properties of photonic quasicrystal following some deterministic rules, for example, Fibonacci [4,5], Thue-Morse [4,6], double period [6], Octo nacci [5,7–9], Rudin-Shapiro [6], Cantor [10] have been widely studied and they are well known by the scientific community. The quasiperiodic structures are of special interest because their complex symmetries make them suitable for the fabrication of omnidirectional reflectors and fil ters, for instance Refs. [4–9]. The structures discussed above can be categorized as intrinsic photonic multilayer because of its design strategy. On the other hand, different efforts have been made to design the extrinsic PhCs composed of a bulk material whose dielectric function was altered by applying some external entities [5,7]. Besides being a unique platform in exploring the properties of the emerging physics of photonic bandgap, extrinsic PhCs, thanks to its ability to give an extra degree of freedom to control the electromagnetic wave. Historically, the
concept of such an extrinsic magnetized photonic crystal was first introduced by Xu et al. [11] in 2007, where a piece of n-GaAs semi conductor is influenced by an external and periodically applied mag netic field. Since then, several groups have proposed extrinsic PhC using other available doped semiconductors [12–15], and bulk plasma system [16–20]. In previous works, plasma multilayer structures have been estab lished from GHz to THz frequencies. For example, Yao et al. design a 1D photonic crystal filled with low-temperature plasma for controlling broadband microwave transmission [21]. Sakaguchi et al. given the proposal of bandgaps observed in millimeter and sub-THz ranges in two-dimensional micro-plasma arrays [22]. In all theoretical models discuss here, the plasma has been introduced in terms of the simplest approach available, namely the cold-plasma model. Recently, Qi et al. [15] and King et al. [16] theoretically investigated the propagation of the electromagnetic wave through an extrinsic magnetized plasma multilayer using the same model and suggested that this kind of system have photonic bandgaps in the microwave region. Although extrinsic magnetized plasma multilayer properties have been extensively studied in the literature, the interest has been shifted to how it can work with the quasiperiodic external magnetic field: one of the goals is to enhance the performance of photonic multilayers. In this context, Nayak et al. showed that extrinsic Octonacci magnetized plasma multilayer form four forbidden bands, instead of a single forbidden band while
E-mail address: [email protected]. https://doi.org/10.1016/j.optmat.2020.109653 Received 28 November 2019; Received in revised form 25 December 2019; Accepted 1 January 2020 Available online 16 January 2020 0925-3467/© 2020 Elsevier B.V. All rights reserved.
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Optical Materials 100 (2020) 109653
shown that two measure bandgaps, one appeared above the plasma frequency and the second appeared above the plasma frequency, are found in the Dodecanacci extrinsic magnetized plasma multilayer structure. Both the aroused bandgaps are dependent on external mag netic field and energy density. It is also found that, distinct from the bandgap appeared below the plasma frequency, the bandgap above the plasma frequency is also insensitive to Dodecanacci quasiperiodic sequence. These interesting results will offer many prospects for mi crowave photonic switches, filters and other photonic devices. The plan of this paper is as follows: in Sec. 2, we present the theory of Dodecanacci extrinsic magnetized plasma quasicrystal, in Subsecs. 2.1 and 2.2, and the method of calculation employed here, which is based on the transfer-matrix approach, in Subsec. 2.3; Sec. 3 is devoted to the discussion of this transmission spectra and dispersion relation for the Dodecanacci multilayer structures; and the conclusions of this work are presented in Sec. 4.
Fig. 1. (Color online) Schematic representation of a one-dimensional extrinsic magnetized plasma photonic crystal. Here we consider N ¼ 4. (For interpre tation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2. Theory of Dodecanacci extrinsic magnetized plasma quasicrystal
considering the periodic structure [8]. In another recent report, Nayak et al. demonstrated that the deterministic rule of the considered quasi periodic sequences specifying the layer count, type, and location could not able to change the bandgap, appeared above the plasma frequency, offered by such extrinsic magnetized plasma multilayer [5]. Since such interesting bandgap properties of the presented photonic extrinsic quasicrystals depend on the substitution rules, obtaining different characteristics is contingent on implementing different possible deter ministic extrinsic photonic multilayer. In the context of experimental investigations of the plasma photonic crystal, recently, the light propagation on plasma photonic systems has been investigated experimentally by Fan and Dong [23], by using the dielectric barrier discharge from two liquid electrodes, and by Zhang and Ouyang [24] that used a series of gas discharge tubes. However, extrinsic plasma photonic crystal is still undiscovered. Today, the gen eration of magnetized plasma is not a huge obstacle but the difficulties may arise to design the spatially varying external magnetic field. However, if the reference photonics structure is implemented then they can add an extra degree of freedom to the device which is expected quite effective in the control of the electromagnetic wave. In view of the impotence of the subject, based on previous works [5, 8] and in a recent report presented by Silva et al. [25], in this study, we consider an extrinsic magnetized plasma multilayer composed of an external magnetic field following Dodecanacci quasicrystal. We have
To study the propagation of light waves through extrinsic Dodeca nacci quasicrystal structure, we have assumed that material of the pro posed multilayered structure to be magnetized plasma and select a particular axis as the x-axis which is along the direction normal to the layers. The complex permittivity profile of the magnetized plasma is given by Ref. [26]. # "
ε�Plasma ðω; Be Þ ¼ 1
� ω2 1
ω2p
i ωγ � ωωl
� ;
(1)
where ωp ¼ ðne e2 =me ε0 Þ1=2 is the plasma frequency, ωl ¼ eBðxÞ=me is the gyromagnetic frequency, and γ is the effective collision frequency, ne being the electron concentration or called as energy density. e, me and Be denote elementary electron charge, electron mass, and external mag netic field intensity, respectively. The - (þ) sign in ωl refer to the positive (negative) magnetic field and is called the right (left)-hand polarization or RHP (LHP). 2.1. Extrinsic magnetized plasma multilayer The extrinsic magnetized plasma multilayer structure is a full plasma photonic crystal, which is created by the bulk plasma embedded into the external spatially alternating static magnetic field. As a first approxi mation, it is assumed that the magnitude of the magnetic field is
Fig. 2. (Color online) The transmittance spectra of the (from bottom to top) periodic and the 3rd, 6th, 9th and 12th generations of the (a) case-I and (b) case-II Dodecanacci extrinsic magnetized plasma multilayer as a function of the frequency for normal incidence with an external magnetic field jBe j ¼ 1 T and energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
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Optical Materials 100 (2020) 109653
Fig. 3. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from LHP plasma (layer A)- RHP plasma (layer B) as a function of wavelength and an external magnetic field, jBe j for normal incident at energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
homogeneous everywhere in the applied region, but its polarity changes periodically. In Fig. 1 we have a schematic representation of the system where layer A (thickness dA ) is the region under the negative magnetic field of Be , i.e., the region of bulk plasma system which is left-hand polarized. Similarly, layer B (thickness dB ) denotes the region under the positive magnetic field þBe and this region of the bulk plasma system is right-hand polarized. Therefore, according to Eq. (1), the permittiv ities of layers A and B are ε Plasma ðω; Be Þ and εþPlasma ðω;Be Þ, respectively. The spatial period of the extrinsic magnetized plasma photonic crystal is defined as d ¼ dA þ dB , the number of periods is N, and thus the length of the extrinsic magnetized plasma multilayer structure, L ¼ Nd.
2.3. Mathematical modeling of proposed structure The transmittance T of an N layered one-dimensional system is given � � by T ¼ �tj2 , where t ¼ 1=M11 is the transmission coefficient of the sys tem. To calculate the parameter M11 of the random system, we should use the transfer-matrix M of the system, and it is given by Ref. [28]: � � �YN � M11 M12 M¼ Dr Pr Dr 1 D0 ; (2) ¼ D0 1 r¼1 M21 M22 where M11 , M12 , M21 and M22 are the transfer-matrix elements. Pr and Dr stand for propagation matrix and dynamic matrix of the r-th slab having thickness dr and permittivity εr . D0 is the dynamic matrix for vacuum. At normal incidence, Pr and Dr are expressed as (c is the light speed in vacuum) [13]. � pffiffiffiffi . � " # 0 exp iω εr dr c � . � Pr ¼ ; (3) pffiffiffiffi 0 exp iω εr dr c
2.2. Dodecanacci sequence A generalized Dodecanacci sequence [25,27] can be obtained by the deterministic mathematical principle, which is defined as Sg ¼
fASg 2 Sg 1 g2 Sg 1 , where S1 ¼ fA; A; Bg, S2 ¼ fAS1 g2 S1 ¼ fA; A; A; B; A; A;A;B;A;A;Bg, and g � 3 is the generation number. The number of layers N for a particular value of generation number can be calculated by Pg ¼ 4Pg 1 Pg 2 , where P1 ¼ 3 and P2 ¼ 11. Here, A and B are the building blocks representing the materials having permittivities εA and εB , respectively. An alternative way to obtain this sequence is by using the substitution rule A→AAAB and B→AAB. One may obtain the generations Sg of the Dodecanacci sequence, for example: S3 ¼ fA; A; A; B; A; A; A; B; A; A; A; B; A; A; B; A; A; A; B; A; A; A; ⋯;B;A;A;A;B;A;A;B;A;A;A;B;A;A;A; B; A; A; A; B; A; A; Bg. The number of the building blocks for the higher generations of the Dodecanacci sequence are P4 ¼ 153, P5 ¼ 571, P6 ¼ 2131, and so on.
� 1 Dr ¼ pffiffiffiffi
εr
� 1 pffiffiffiffi :
εr
(4)
3. Result analysis In order to investigate the transmission characteristics of the pro posed structure in the microwave frequency region throughout the manuscript, we choose one of the magnetized plasma parameters, i.e., effective collision frequency, as being γ ¼ 4π � 102 GHz. This exclusion is because of the frequency range of the bandgap can not be changed by changing plasma collision frequency [29]. Consequently, the plasma collision frequency does not affect the bandgap. Therefore, in this 3
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Optical Materials 100 (2020) 109653
Fig. 4. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from RHP plasma (layer A)- LHP plasma (layer B) as a function of wavelength and an external magnetic field, jBe j for normal incident at energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
investigation, we are not introducing the effect of the plasma parameter collision frequency on the transmittance spectrum. The incident field is always assumed to be injected normally. The thickness of each segment of the plasma system is constant, i.e., dA ¼ dB ¼ 15 mm. The considered periodic structure has N ¼ 32, whereas the considered Dodecanacci sequence generations are 3rd, 6th, 9th, and 12th. Here, we focus only on the bandgaps that appear in the frequency range from 0.1 to 40 GHz. It should be noted that the reason for selecting such a wide range of fre quencies is to analyze the behavior of the forbidden regions and the localization of electromagnetic waves, which are expected to be below as well as above the plasma frequency. To have a good analysis of the transmission properties of Dodeca nacci extrinsic magnetized plasma quasicrystal, in the following, two cases have been considered: the case-I exhibits the LHP as layer A, and RHP as layer B; and the case-II is taken in the opposite way, i.e., the layer A represents RHP and layer B represents LHP. Moreover, we know that the constituent material of the proposed system is sensitive to the magnitude external magnetic field jBe j and energy density ne , indicating other possibilities of manipulating the transmission spectra. Therefore, after analyzing our main goal, i.e., the impact of generation number, in Subsec. 3.1, we investigate the effect of transmission spectra of Dodec anacci extrinsic magnetized plasma multilayer as a function of jBe j as well as ne in the Subsecs. 3.2 and 3.3, respectively.
spectra patterns, it is implicit that the bandgap observed above the plasma frequency (labeled as bandgap #2) is not at all affected by the considered quasi-sequence. Recently, the same kind of observation was presented by C. Nayak et al. [5] by considering Fibonacci, Thue-Morse, double-period and Octonacci extrinsic magnetized plasma multilayers. These observations inform the deterministic rule of the quasi-sequences specifying the layer count, type, and location regardless they do not affect this bandgap. Therefore, it may state that such kind of structure has a robust bandgap above the plasma frequency. On the other hand, the major bandgap below the plasma frequency (labeled bandgap #1), which appears for the periodic extrinsic magne tized plasma multilayer, vanishes for almost all the generations of the Dodecanacci sequence investigated. This modification of the forbidden region is because of the sequence under consideration instead of period one. Another notable impact is that with an increase in the generation number from 3rd to 12th, the average transmittance between the spe cific bandgap is decreasing. Since resonant peaks are aroused at a moderate value of generation number, therefore, these structures can be effectively employed as multi-channel filters. Only for the 12th gener ation, the extrinsic magnetized plasma again behave as a perfect reflector. The appearance of such modification in the transmission spectra is due to the long term periodic effect of the considered sequence. Moreover, for the case-I, the results also show that there are two more bandgaps are appeared below to 4.3 GHz [5]. It happens due to the presence of a comparably very higher layer count of the RHP plasma layers than the LHP plasma layers. The only significant change that we observed in these two cases is, as like case-I, case-II Dodecanacci does not have extremely low GHz bandgap. It is clear from Fig. 2 of Ref. [5]. In a general way, the considered cases show the qualitative difference because of the geometry of the structure and the polarization of the bulk plasma region.
3.1. Effect of generation number At first, we study the effect of generation numbers on both the cases discussed in this current section. Fig. 2 illustrates the spectral properties of the periodic and the 3rd, 6th, 9th, and 12th generations of the Dodecanacci quasiperiodic photonic crystals as a function of the fre quency for: (a) case-I and (b) case-II. The physical parameters used here are jBe j ¼ 1 T and ne ¼ 8 � 1017 m 3. By observing both transmittance 4
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Optical Materials 100 (2020) 109653
Fig. 5. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from LHP plasma (layer A)- RHP plasma (layer B) as a function of wavelength and energy density ne for normal incident at external magnetic field jBe j ¼ 1 T. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
3.2. Effect of external magnetic field
correspond to the transmissions through the case-II Dodecanacci extrinsic magnetized plasma multilayers with the generation number of 3rd, 6th, 9th, and 12th, respectively. From Figs. 3 and 4, we can see a very similar kind of response for the bandgap above the plasma fre quency, like the discussed bandgap is blue-shifted, and the width of the bandgap becomes narrower when the magnitude external field jBe j in creases. Moreover, the bandgap is also not affected by the generation number of the Dodecanacci sequence. On the other hand, the bandgap, which appears below to the plasma frequency, as we expected, the transmission coefficients for the case-II systems are not similar to the case-I systems. The bandgap appeared for the case-I structure is splits by the moderate value of the transmission region. From Fig. 4(b)–(d), we can observe some narrow transmission peaks as the generation number gown up. The number of narrow transmission peaks also decreases with an increase in generation num ber, but the count of narrow transmission peaks is always higher for case-II structures. As a consequence, the average transmission of this frequency region decreases with an increase in generation number. The reason for the same is the consequence of the effective permittivity of magnetized plasma in the considered frequency range.
Now we investigate the dependence of the transmission spectra on the variation of jBe j for the Dodecanacci extrinsic magnetized plasma multilayer. Fig. 3 shows the effect of jBe j on electromagnetic wave transmissivities for the case-I structures with energy density ne ¼ 8� 1017 m 3. Here, the dark and bright areas denote the bandgap and the allowed band, respectively. Fig. 3(a)–(d) correspond to the trans missions through the Dodecanacci extrinsic magnetized plasma multi layers with the generation number of 3rd, 6th, 9th, and 12th, respectively. From them, it is clear that the significant bandgap above the plasma frequency is blue-shifted, and the width of the bandgap be comes narrower. In the intraband transition, when jBe j increases, only electromagnetic waves with higher frequencies can be absorbed by electrons efficiently; therefore, the bandgap moves toward higher fre quencies, proving that the bandgap can be easily tuned by jBe j. It is also clearly seen that the bandgap is insensitive to the generation number, for the entire range of external magnetic fieldsjBe j, i.e., from 0 to 1.25 T. On the other hand, in Fig. 3, as the value of generation number in creases, the bandgap below the plasma frequency becomes broader with some intermediate localized states. The intermediate localized state comes from the fact that the long-range order of the arrangement of the layers in the Dodecanacci quasi-sequence. Again the count of the same localized states decreases with an increase in generation number. Therefore, as stated before, the localization states for a moderate value of Dodecanacci generations may be accepted as an effective narrow band filter design. For a comparison of case-II with case-I type structures, we have calculated the transmission at the same set of quasiperiodic structures. Fig. 4 displays the same as Fig. 3 but for the case-II, i.e., Fig. 4(a)–(d)
3.3. Effect of electron density Let us now examine the change of transmission spectra, for an external magnetic field jBe j ¼ 1 T, due to one of the plasma parameters, i.e., energy density ne . The obtained results for case-I type structures are plotted in Fig ?? Fig. 5(a)-(d) correspond to the transmissions through the case-I Dodecanacci extrinsic magnetized plasma multilayers with the, and 12th, respectively. From these figures, it can observe that with an increase in electron density, the behavior of the transmittance 5
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Optical Materials 100 (2020) 109653
Fig. 6. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from RHP plasma (layer A)- LHP plasma (layer B) as a function of wavelength and energy density ne for normal incident at external magnetic field jBe j ¼ 1 T. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
spectrum indicates an opposite trend to that of Fig. 4. In brief, the aroused bandgaps become wider with an increase in electron density ne . It is also revealed that, as compared to Fig. 3, the shift of bandgap to ward higher frequency is very nominal. This observation is significantly different from our recent studies [8], in terms of the shifting in the frequency axis, where we are practicing an Octonacci magnetized cold plasma quasicrystal. On the other hand, by observing Fig. 5(a)–(d), we can note the same kind of response from 3(a)-(d). In exact, the effective transmission of the observed bandgap range decreases with an increase in the generation number of Dodecanacci extrinsic magnetized plasma multilayers. Be sides, a marginal increase in the width of the bandgap is also noticed. These observations are obvious and happen due to the relevance dimension of the structure under the trail. In Fig. 6, we plotted the same as Fig. 5, but for the case-II structures. It is easy to observe that a similar result occurs when we analyze the behavior of case-II structures. In particular, as the generation index increases, the number of transmission peaks decreases, and they become narrower, which is a clear indication of the reduction in the count of localized states. Iit is significant to note that the effective average transmission of the bandgap regions for case-II drops as compared to case-I.
multilayers simultaneously. For this Dodecanacci structure, it was demonstrated that increasing the magnetic field shrinks the gap widths and change the gap location to higher frequencies. A dissimilar result is observed in increasing the electron density, whereas the gap location is still changed to the higher frequencies. As the generation number in creases, unlike the Bragg photonic band gaps get narrower and narrower [30] here, the bandgap above the plasma frequency remains unchanged. In contrast, the bandgap below the plasma frequency is a border with some intermediate resonant peaks. A very interesting fact is that the count of these spiky transmission spectra is decreased with an increase in generation number. We believe that similar to conventional photonic quasicrystals, this type of extrinsic quasicrystal could have potential applications in designing tunable photonic crystal devices.
4. Conclusions
Chittaranjan Nayak: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Writing - review & editing.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement
In this paper, we have theoretically investigated the tunable reflec tion properties of Dodecanacci extrinsic magnetized plasma multilayer in the presence of a spatially varying external magnetic field. Without spatially varying external magnetic fields, they are bulk plasma systems; when spatially varying external magnetic fields arranged in dodeca naccically, they are Dodecanacci extrinsic magnetized plasma
Acknowledgements The author acknowledges the HOD, Department of Electronics and Communication Engineering, the Director of Engineering and Technol ogy, and the Vice-Chancellor, SRM Institute of Science and Technology, 6
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Optical Materials 100 (2020) 109653
Chennai, for their continuous encouragement. The author also ac �, for knowledges Dr. C. H. Costa, from Federal University of Ceara managing the language of the manuscript.
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Contents lists available at ScienceDirect
Optical Materials journal homepage: http://www.elsevier.com/locate/optmat
Dodecanacci extrinsic magnetized plasma multilayer Chittaranjan Nayak Department of Electronics and Communication Engineering, SRM Institute of Science and Technology, Chennai, 603203, Tamilnadu, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Extrinsic magnetized plasma Quasiperiodic structures Dodecanacci sequence Structure-property relationship
An extrinsic magnetized plasma quasi-multilayer is demonstrated using the transfer-matrix method with an extrinsic bulk plasma inspired structure. The transmittance of the extrinsic bulk plasma quasi-system has been shown using the Dodecanacci quasi-sequence. For the periodic structure, two significant gaps emerge and only one of them, the bandgap above the plasma frequency, appears in any of the proposed Dodecanacci systems and it is magnetically tunable and robust against the position of the layers. The other bandgap, observed below the plasma frequency is also magnetically tunable but it vanishes when the layers are quasiperiodically disposed because the emergence of several resonant peaks. The count and width of the transmission window in such vanished bandgaps decrease as the Dodecanacci generation number increases. The results also show that the electron density can control the gap location and gap width for all Dodecanacci structures. These listed results ensure that the approach of Dodecanacci extrinsic magnetized plasma multilayers can be applied to the design of future architectures for the control and the manipulation of the electromagnetic wave-matter interaction in microwave frequencies.
1. Introduction Photonic crystals (PhCs) are artificial periodic structures with two kinds of materials [1]. Indeed, periodic modulation of the dielectric constant leads to the appearance of a frequency range in which the electromagnetic wave is forbidden, called the photonic bandgap (PBG) [2]. A new trend in PhC research is to explore the active interaction of photonic structure with light emission and is very much familiar as the photonic quasicrystal [3]. The crystallographic and physical properties of photonic quasicrystal following some deterministic rules, for example, Fibonacci [4,5], Thue-Morse [4,6], double period [6], Octo nacci [5,7–9], Rudin-Shapiro [6], Cantor [10] have been widely studied and they are well known by the scientific community. The quasiperiodic structures are of special interest because their complex symmetries make them suitable for the fabrication of omnidirectional reflectors and fil ters, for instance Refs. [4–9]. The structures discussed above can be categorized as intrinsic photonic multilayer because of its design strategy. On the other hand, different efforts have been made to design the extrinsic PhCs composed of a bulk material whose dielectric function was altered by applying some external entities [5,7]. Besides being a unique platform in exploring the properties of the emerging physics of photonic bandgap, extrinsic PhCs, thanks to its ability to give an extra degree of freedom to control the electromagnetic wave. Historically, the
concept of such an extrinsic magnetized photonic crystal was first introduced by Xu et al. [11] in 2007, where a piece of n-GaAs semi conductor is influenced by an external and periodically applied mag netic field. Since then, several groups have proposed extrinsic PhC using other available doped semiconductors [12–15], and bulk plasma system [16–20]. In previous works, plasma multilayer structures have been estab lished from GHz to THz frequencies. For example, Yao et al. design a 1D photonic crystal filled with low-temperature plasma for controlling broadband microwave transmission [21]. Sakaguchi et al. given the proposal of bandgaps observed in millimeter and sub-THz ranges in two-dimensional micro-plasma arrays [22]. In all theoretical models discuss here, the plasma has been introduced in terms of the simplest approach available, namely the cold-plasma model. Recently, Qi et al. [15] and King et al. [16] theoretically investigated the propagation of the electromagnetic wave through an extrinsic magnetized plasma multilayer using the same model and suggested that this kind of system have photonic bandgaps in the microwave region. Although extrinsic magnetized plasma multilayer properties have been extensively studied in the literature, the interest has been shifted to how it can work with the quasiperiodic external magnetic field: one of the goals is to enhance the performance of photonic multilayers. In this context, Nayak et al. showed that extrinsic Octonacci magnetized plasma multilayer form four forbidden bands, instead of a single forbidden band while
E-mail address: [email protected]. https://doi.org/10.1016/j.optmat.2020.109653 Received 28 November 2019; Received in revised form 25 December 2019; Accepted 1 January 2020 Available online 16 January 2020 0925-3467/© 2020 Elsevier B.V. All rights reserved.
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shown that two measure bandgaps, one appeared above the plasma frequency and the second appeared above the plasma frequency, are found in the Dodecanacci extrinsic magnetized plasma multilayer structure. Both the aroused bandgaps are dependent on external mag netic field and energy density. It is also found that, distinct from the bandgap appeared below the plasma frequency, the bandgap above the plasma frequency is also insensitive to Dodecanacci quasiperiodic sequence. These interesting results will offer many prospects for mi crowave photonic switches, filters and other photonic devices. The plan of this paper is as follows: in Sec. 2, we present the theory of Dodecanacci extrinsic magnetized plasma quasicrystal, in Subsecs. 2.1 and 2.2, and the method of calculation employed here, which is based on the transfer-matrix approach, in Subsec. 2.3; Sec. 3 is devoted to the discussion of this transmission spectra and dispersion relation for the Dodecanacci multilayer structures; and the conclusions of this work are presented in Sec. 4.
Fig. 1. (Color online) Schematic representation of a one-dimensional extrinsic magnetized plasma photonic crystal. Here we consider N ¼ 4. (For interpre tation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2. Theory of Dodecanacci extrinsic magnetized plasma quasicrystal
considering the periodic structure [8]. In another recent report, Nayak et al. demonstrated that the deterministic rule of the considered quasi periodic sequences specifying the layer count, type, and location could not able to change the bandgap, appeared above the plasma frequency, offered by such extrinsic magnetized plasma multilayer [5]. Since such interesting bandgap properties of the presented photonic extrinsic quasicrystals depend on the substitution rules, obtaining different characteristics is contingent on implementing different possible deter ministic extrinsic photonic multilayer. In the context of experimental investigations of the plasma photonic crystal, recently, the light propagation on plasma photonic systems has been investigated experimentally by Fan and Dong [23], by using the dielectric barrier discharge from two liquid electrodes, and by Zhang and Ouyang [24] that used a series of gas discharge tubes. However, extrinsic plasma photonic crystal is still undiscovered. Today, the gen eration of magnetized plasma is not a huge obstacle but the difficulties may arise to design the spatially varying external magnetic field. However, if the reference photonics structure is implemented then they can add an extra degree of freedom to the device which is expected quite effective in the control of the electromagnetic wave. In view of the impotence of the subject, based on previous works [5, 8] and in a recent report presented by Silva et al. [25], in this study, we consider an extrinsic magnetized plasma multilayer composed of an external magnetic field following Dodecanacci quasicrystal. We have
To study the propagation of light waves through extrinsic Dodeca nacci quasicrystal structure, we have assumed that material of the pro posed multilayered structure to be magnetized plasma and select a particular axis as the x-axis which is along the direction normal to the layers. The complex permittivity profile of the magnetized plasma is given by Ref. [26]. # "
ε�Plasma ðω; Be Þ ¼ 1
� ω2 1
ω2p
i ωγ � ωωl
� ;
(1)
where ωp ¼ ðne e2 =me ε0 Þ1=2 is the plasma frequency, ωl ¼ eBðxÞ=me is the gyromagnetic frequency, and γ is the effective collision frequency, ne being the electron concentration or called as energy density. e, me and Be denote elementary electron charge, electron mass, and external mag netic field intensity, respectively. The - (þ) sign in ωl refer to the positive (negative) magnetic field and is called the right (left)-hand polarization or RHP (LHP). 2.1. Extrinsic magnetized plasma multilayer The extrinsic magnetized plasma multilayer structure is a full plasma photonic crystal, which is created by the bulk plasma embedded into the external spatially alternating static magnetic field. As a first approxi mation, it is assumed that the magnitude of the magnetic field is
Fig. 2. (Color online) The transmittance spectra of the (from bottom to top) periodic and the 3rd, 6th, 9th and 12th generations of the (a) case-I and (b) case-II Dodecanacci extrinsic magnetized plasma multilayer as a function of the frequency for normal incidence with an external magnetic field jBe j ¼ 1 T and energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
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Fig. 3. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from LHP plasma (layer A)- RHP plasma (layer B) as a function of wavelength and an external magnetic field, jBe j for normal incident at energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
homogeneous everywhere in the applied region, but its polarity changes periodically. In Fig. 1 we have a schematic representation of the system where layer A (thickness dA ) is the region under the negative magnetic field of Be , i.e., the region of bulk plasma system which is left-hand polarized. Similarly, layer B (thickness dB ) denotes the region under the positive magnetic field þBe and this region of the bulk plasma system is right-hand polarized. Therefore, according to Eq. (1), the permittiv ities of layers A and B are ε Plasma ðω; Be Þ and εþPlasma ðω;Be Þ, respectively. The spatial period of the extrinsic magnetized plasma photonic crystal is defined as d ¼ dA þ dB , the number of periods is N, and thus the length of the extrinsic magnetized plasma multilayer structure, L ¼ Nd.
2.3. Mathematical modeling of proposed structure The transmittance T of an N layered one-dimensional system is given � � by T ¼ �tj2 , where t ¼ 1=M11 is the transmission coefficient of the sys tem. To calculate the parameter M11 of the random system, we should use the transfer-matrix M of the system, and it is given by Ref. [28]: � � �YN � M11 M12 M¼ Dr Pr Dr 1 D0 ; (2) ¼ D0 1 r¼1 M21 M22 where M11 , M12 , M21 and M22 are the transfer-matrix elements. Pr and Dr stand for propagation matrix and dynamic matrix of the r-th slab having thickness dr and permittivity εr . D0 is the dynamic matrix for vacuum. At normal incidence, Pr and Dr are expressed as (c is the light speed in vacuum) [13]. � pffiffiffiffi . � " # 0 exp iω εr dr c � . � Pr ¼ ; (3) pffiffiffiffi 0 exp iω εr dr c
2.2. Dodecanacci sequence A generalized Dodecanacci sequence [25,27] can be obtained by the deterministic mathematical principle, which is defined as Sg ¼
fASg 2 Sg 1 g2 Sg 1 , where S1 ¼ fA; A; Bg, S2 ¼ fAS1 g2 S1 ¼ fA; A; A; B; A; A;A;B;A;A;Bg, and g � 3 is the generation number. The number of layers N for a particular value of generation number can be calculated by Pg ¼ 4Pg 1 Pg 2 , where P1 ¼ 3 and P2 ¼ 11. Here, A and B are the building blocks representing the materials having permittivities εA and εB , respectively. An alternative way to obtain this sequence is by using the substitution rule A→AAAB and B→AAB. One may obtain the generations Sg of the Dodecanacci sequence, for example: S3 ¼ fA; A; A; B; A; A; A; B; A; A; A; B; A; A; B; A; A; A; B; A; A; A; ⋯;B;A;A;A;B;A;A;B;A;A;A;B;A;A;A; B; A; A; A; B; A; A; Bg. The number of the building blocks for the higher generations of the Dodecanacci sequence are P4 ¼ 153, P5 ¼ 571, P6 ¼ 2131, and so on.
� 1 Dr ¼ pffiffiffiffi
εr
� 1 pffiffiffiffi :
εr
(4)
3. Result analysis In order to investigate the transmission characteristics of the pro posed structure in the microwave frequency region throughout the manuscript, we choose one of the magnetized plasma parameters, i.e., effective collision frequency, as being γ ¼ 4π � 102 GHz. This exclusion is because of the frequency range of the bandgap can not be changed by changing plasma collision frequency [29]. Consequently, the plasma collision frequency does not affect the bandgap. Therefore, in this 3
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Fig. 4. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from RHP plasma (layer A)- LHP plasma (layer B) as a function of wavelength and an external magnetic field, jBe j for normal incident at energy density ne ¼ 8 � 1017 m 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
investigation, we are not introducing the effect of the plasma parameter collision frequency on the transmittance spectrum. The incident field is always assumed to be injected normally. The thickness of each segment of the plasma system is constant, i.e., dA ¼ dB ¼ 15 mm. The considered periodic structure has N ¼ 32, whereas the considered Dodecanacci sequence generations are 3rd, 6th, 9th, and 12th. Here, we focus only on the bandgaps that appear in the frequency range from 0.1 to 40 GHz. It should be noted that the reason for selecting such a wide range of fre quencies is to analyze the behavior of the forbidden regions and the localization of electromagnetic waves, which are expected to be below as well as above the plasma frequency. To have a good analysis of the transmission properties of Dodeca nacci extrinsic magnetized plasma quasicrystal, in the following, two cases have been considered: the case-I exhibits the LHP as layer A, and RHP as layer B; and the case-II is taken in the opposite way, i.e., the layer A represents RHP and layer B represents LHP. Moreover, we know that the constituent material of the proposed system is sensitive to the magnitude external magnetic field jBe j and energy density ne , indicating other possibilities of manipulating the transmission spectra. Therefore, after analyzing our main goal, i.e., the impact of generation number, in Subsec. 3.1, we investigate the effect of transmission spectra of Dodec anacci extrinsic magnetized plasma multilayer as a function of jBe j as well as ne in the Subsecs. 3.2 and 3.3, respectively.
spectra patterns, it is implicit that the bandgap observed above the plasma frequency (labeled as bandgap #2) is not at all affected by the considered quasi-sequence. Recently, the same kind of observation was presented by C. Nayak et al. [5] by considering Fibonacci, Thue-Morse, double-period and Octonacci extrinsic magnetized plasma multilayers. These observations inform the deterministic rule of the quasi-sequences specifying the layer count, type, and location regardless they do not affect this bandgap. Therefore, it may state that such kind of structure has a robust bandgap above the plasma frequency. On the other hand, the major bandgap below the plasma frequency (labeled bandgap #1), which appears for the periodic extrinsic magne tized plasma multilayer, vanishes for almost all the generations of the Dodecanacci sequence investigated. This modification of the forbidden region is because of the sequence under consideration instead of period one. Another notable impact is that with an increase in the generation number from 3rd to 12th, the average transmittance between the spe cific bandgap is decreasing. Since resonant peaks are aroused at a moderate value of generation number, therefore, these structures can be effectively employed as multi-channel filters. Only for the 12th gener ation, the extrinsic magnetized plasma again behave as a perfect reflector. The appearance of such modification in the transmission spectra is due to the long term periodic effect of the considered sequence. Moreover, for the case-I, the results also show that there are two more bandgaps are appeared below to 4.3 GHz [5]. It happens due to the presence of a comparably very higher layer count of the RHP plasma layers than the LHP plasma layers. The only significant change that we observed in these two cases is, as like case-I, case-II Dodecanacci does not have extremely low GHz bandgap. It is clear from Fig. 2 of Ref. [5]. In a general way, the considered cases show the qualitative difference because of the geometry of the structure and the polarization of the bulk plasma region.
3.1. Effect of generation number At first, we study the effect of generation numbers on both the cases discussed in this current section. Fig. 2 illustrates the spectral properties of the periodic and the 3rd, 6th, 9th, and 12th generations of the Dodecanacci quasiperiodic photonic crystals as a function of the fre quency for: (a) case-I and (b) case-II. The physical parameters used here are jBe j ¼ 1 T and ne ¼ 8 � 1017 m 3. By observing both transmittance 4
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Fig. 5. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from LHP plasma (layer A)- RHP plasma (layer B) as a function of wavelength and energy density ne for normal incident at external magnetic field jBe j ¼ 1 T. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
3.2. Effect of external magnetic field
correspond to the transmissions through the case-II Dodecanacci extrinsic magnetized plasma multilayers with the generation number of 3rd, 6th, 9th, and 12th, respectively. From Figs. 3 and 4, we can see a very similar kind of response for the bandgap above the plasma fre quency, like the discussed bandgap is blue-shifted, and the width of the bandgap becomes narrower when the magnitude external field jBe j in creases. Moreover, the bandgap is also not affected by the generation number of the Dodecanacci sequence. On the other hand, the bandgap, which appears below to the plasma frequency, as we expected, the transmission coefficients for the case-II systems are not similar to the case-I systems. The bandgap appeared for the case-I structure is splits by the moderate value of the transmission region. From Fig. 4(b)–(d), we can observe some narrow transmission peaks as the generation number gown up. The number of narrow transmission peaks also decreases with an increase in generation num ber, but the count of narrow transmission peaks is always higher for case-II structures. As a consequence, the average transmission of this frequency region decreases with an increase in generation number. The reason for the same is the consequence of the effective permittivity of magnetized plasma in the considered frequency range.
Now we investigate the dependence of the transmission spectra on the variation of jBe j for the Dodecanacci extrinsic magnetized plasma multilayer. Fig. 3 shows the effect of jBe j on electromagnetic wave transmissivities for the case-I structures with energy density ne ¼ 8� 1017 m 3. Here, the dark and bright areas denote the bandgap and the allowed band, respectively. Fig. 3(a)–(d) correspond to the trans missions through the Dodecanacci extrinsic magnetized plasma multi layers with the generation number of 3rd, 6th, 9th, and 12th, respectively. From them, it is clear that the significant bandgap above the plasma frequency is blue-shifted, and the width of the bandgap be comes narrower. In the intraband transition, when jBe j increases, only electromagnetic waves with higher frequencies can be absorbed by electrons efficiently; therefore, the bandgap moves toward higher fre quencies, proving that the bandgap can be easily tuned by jBe j. It is also clearly seen that the bandgap is insensitive to the generation number, for the entire range of external magnetic fieldsjBe j, i.e., from 0 to 1.25 T. On the other hand, in Fig. 3, as the value of generation number in creases, the bandgap below the plasma frequency becomes broader with some intermediate localized states. The intermediate localized state comes from the fact that the long-range order of the arrangement of the layers in the Dodecanacci quasi-sequence. Again the count of the same localized states decreases with an increase in generation number. Therefore, as stated before, the localization states for a moderate value of Dodecanacci generations may be accepted as an effective narrow band filter design. For a comparison of case-II with case-I type structures, we have calculated the transmission at the same set of quasiperiodic structures. Fig. 4 displays the same as Fig. 3 but for the case-II, i.e., Fig. 4(a)–(d)
3.3. Effect of electron density Let us now examine the change of transmission spectra, for an external magnetic field jBe j ¼ 1 T, due to one of the plasma parameters, i.e., energy density ne . The obtained results for case-I type structures are plotted in Fig ?? Fig. 5(a)-(d) correspond to the transmissions through the case-I Dodecanacci extrinsic magnetized plasma multilayers with the, and 12th, respectively. From these figures, it can observe that with an increase in electron density, the behavior of the transmittance 5
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Optical Materials 100 (2020) 109653
Fig. 6. (Color online) The transmittance spectra of the (a) 3rd, (b) 6th, (c) 9th and (d) 12th generation of the Dodecanacci extrinsic magnetized plasma multilayer comprising from RHP plasma (layer A)- LHP plasma (layer B) as a function of wavelength and energy density ne for normal incident at external magnetic field jBe j ¼ 1 T. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
spectrum indicates an opposite trend to that of Fig. 4. In brief, the aroused bandgaps become wider with an increase in electron density ne . It is also revealed that, as compared to Fig. 3, the shift of bandgap to ward higher frequency is very nominal. This observation is significantly different from our recent studies [8], in terms of the shifting in the frequency axis, where we are practicing an Octonacci magnetized cold plasma quasicrystal. On the other hand, by observing Fig. 5(a)–(d), we can note the same kind of response from 3(a)-(d). In exact, the effective transmission of the observed bandgap range decreases with an increase in the generation number of Dodecanacci extrinsic magnetized plasma multilayers. Be sides, a marginal increase in the width of the bandgap is also noticed. These observations are obvious and happen due to the relevance dimension of the structure under the trail. In Fig. 6, we plotted the same as Fig. 5, but for the case-II structures. It is easy to observe that a similar result occurs when we analyze the behavior of case-II structures. In particular, as the generation index increases, the number of transmission peaks decreases, and they become narrower, which is a clear indication of the reduction in the count of localized states. Iit is significant to note that the effective average transmission of the bandgap regions for case-II drops as compared to case-I.
multilayers simultaneously. For this Dodecanacci structure, it was demonstrated that increasing the magnetic field shrinks the gap widths and change the gap location to higher frequencies. A dissimilar result is observed in increasing the electron density, whereas the gap location is still changed to the higher frequencies. As the generation number in creases, unlike the Bragg photonic band gaps get narrower and narrower [30] here, the bandgap above the plasma frequency remains unchanged. In contrast, the bandgap below the plasma frequency is a border with some intermediate resonant peaks. A very interesting fact is that the count of these spiky transmission spectra is decreased with an increase in generation number. We believe that similar to conventional photonic quasicrystals, this type of extrinsic quasicrystal could have potential applications in designing tunable photonic crystal devices.
4. Conclusions
Chittaranjan Nayak: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Writing - review & editing.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement
In this paper, we have theoretically investigated the tunable reflec tion properties of Dodecanacci extrinsic magnetized plasma multilayer in the presence of a spatially varying external magnetic field. Without spatially varying external magnetic fields, they are bulk plasma systems; when spatially varying external magnetic fields arranged in dodeca naccically, they are Dodecanacci extrinsic magnetized plasma
Acknowledgements The author acknowledges the HOD, Department of Electronics and Communication Engineering, the Director of Engineering and Technol ogy, and the Vice-Chancellor, SRM Institute of Science and Technology, 6
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Optical Materials 100 (2020) 109653
Chennai, for their continuous encouragement. The author also ac �, for knowledges Dr. C. H. Costa, from Federal University of Ceara managing the language of the manuscript.
[14] [15] [16] [17]
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