- Email: [email protected]
Design of output-graded narrow polychromatic filter by using photonic quasicrystals
Journal Pre-proof Design of output-graded narrow polychromatic filter by using photonic quasicrystals
Naim Ben Ali, Vigneswaran Dhasarathan, Haitham Alsaif, Youssef Trabelsi, Truong Khang Nguyen, Y. Bouazzi, Mounir Kanzari PII:
S0921-4526(19)30798-7
DOI:
https://doi.org/10.1016/j.physb.2019.411918
Reference:
PHYSB 411918
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
23 June 2019
Accepted Date:
30 November 2019
Please cite this article as: Naim Ben Ali, Vigneswaran Dhasarathan, Haitham Alsaif, Youssef Trabelsi, Truong Khang Nguyen, Y. Bouazzi, Mounir Kanzari, Design of output-graded narrow polychromatic filter by using photonic quasicrystals, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411918
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof
Design of output-graded narrow polychromatic filter by using photonic quasicrystals
Naim Ben Ali1,5 , Vigneswaran Dhasarathan2,3*, Haitham Alsaif 4, Youssef Trabelsi5,6, Truong Khang Nguyen2,3, Y.Bouazzi1,5, and Mounir Kanzari5,7 1Department 2Division
of Industrial Engineering, College of Engineering, University of Ha’il, 2440, Ha’il City, Saudi Arabia
of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3Faculty
of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi, Minh City Vietnam Email : [email protected]
4Department
of Electrical Engineering, College of Engineering, University of Ha’il, 2440, Ha’il City, Saudi Arabia
5University
of Tunis El Manar, National Engineering School of Tunis, Photovoltaic and Semiconductor Materials Laboratory,1002, Tunis, Tunisia
6King
Khaled university, College of Arts and Sciences in Mahayel Aseer, Physics department , KSA.
7University
of Tunis, Preparatory Engineering Institute of Tunis, Montfleury, 1008, Tunisia
Abstract The properties of one-dimensional photonic quasicrystals built according to one-dimensionalgeneralized Fibonacci (GF) / Generalized Thue-Morse (GTM) were investigated in order to design an output polychromatic filter. This main aperiodic multilayered structure was made up of alternate two dielectrics materials Silica and Bi4Ge3O12(BGO)with higher and lower refractive indices respectively. The Transfer Matrix Method (TMM) was adopted to calculate the photometric response. The transmittance spectrum exhibited a stacking of similar Bragg gaps (BGs) for specific arrangement (m=n) of GF and GTM. We noted that the positions and the number and size of BGs were sensitive to the constituent of the Photonic quasicrystal (PQC) system and lattice parameter of quasiperiodic sequence. Therefore, these configurations of multilayered stacks can be useful as a graded polychromatic filter.
Keywords: Quasiperiodic sequences, Photonic Quasicrystals (PQC), Photonic band gap, GTM sequence, GF sequence, polychromatic filter.
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1. Introduction The area of photonic quasicrystals (PQCs) is a new field of optics with a quasiperiodic modulation in refractive indices of two different material A and B. One-dimensional photonic quasicrystal is the simplest form of PC that can be considered as an intermediate class between random and periodic media. It is an artificial multilayered stack containing two kinds of material with higher and lower refractive indices, respectively. The considered PQC still determined by structural substitution rules of quasiperiodic sequence like Fibonacci [1-2], Thue Morse[3-5], Rudin Shapiro[6], Cantor[7], etc. Besides, these multilayered stacks prohibited the propagation of the electromagnetic wave in a specific range of frequency that creates forbidden bands called photonic band gaps (PBG). Thus, the ability of light manipulating is strongly useful for the design of potential applications in photonic technology and optical communications. Numerous applications based on one-dimensional and two-dimensional photonic quasicrystals (1D PQCs) are achieved and successfully used for the development of optoelectronic devices like a high reflector, optical filters [8-11, 22, 23]. Various 1D GF and 1D GTM systems have been used to enhance the omnidirectional Bragg gaps (OBG)of the conventional photonic crystals. In order to enhance the OBG, certain researchers have applied the deformation through the main quasiperiodic structure [12]. They proved that the proprieties of main PBG were considerably improved. In the last decade, the stop band filter based on 1D-PQC has attracted much attention due to its interesting properties and abilities to localize the wave in specific channel frequency. It was found that a succession of discrete PBGs covers a large zone of the optical frequency domain. It may be used to develop new narrowband optical filters, in which channels are adjusted by the lattice parameters of the quasiperiodic sequence. This considered filter can be also used in Dense wavelength division multiplexing (DWDM)[13-14]. In this case, a certain researcher [15], proposed design of multiband optical filter using Fibonacci quasiperiodic class for suitable selection parameters. They found that the characteristic of multi-band for this quasiperiodic structure is adjusted by the basic elements and parameters of the Fibonacci class. Then, an efficient relationship between the Fibonacci class and superimposed Bragg Gratings has been established [16]. In addition, this design was extended to other mathematical models based on quasiperiodic structures that suggest a new arrangement of 2
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basic PC elements like the Morse, Cantor, Rudin Shapiro. Indeed, all these aperiodic systems exhibit a typical distribution of layers in which periodicity is broken for long-range order and permits to create a localized mode within the main PBG. This defect mode leads to create multiple peaks in the transmission spectrum. In this work, we investigate the filtering properties through regular GF and GTM quasiperiodic sequences. This Work aims to study the optical properties of structural Bragg gaps (BGs) with the appropriate arrangement of layers. Also, tenable channels with zero transmission that inhibit the propagation of EM wave in a specific zone of the microwave frequency range are determined. Besides, a comparative study of two quasiperiodic configurations is useful to identify the perfect muli-channeled filter at precise frequencies range. The transmittance and reflectance are used to examine the effects the thickness of BGO layer, the distributed Bragg gaps (BGs) and transmission peak for different GTM (m=n) sequences. 2. Problem formulation In this section, we used the transfert matrix method (TMM) to determine the reflectance R and the transmittance T through the one-dimensional Fibonacci and Thue-Morse class quasicrystals. Based on this approach, the input amplitude fields E 0+ reflected fields E -0 , and output transmitted field E +m+1 after m layers are related by the sequential multiplication complex transfer matrix C j that can be given by [17] as:
E 0+ m C j - = E 0 j=1 t j
E +m+1 - E m+1
1
is expressed as [17] in the following formula: C j For both TM and TE modes,
exp iφ j-1 rjexp -iφ j-1 Cj= rjexp iφ j-1 exp -iφ j-1
2
The phase shift of the wave between the jth and (j-1)th boundaries is defined by [17] in the following expression:
φ j-1 =
2π nˆ j-1d j-1cosθ j-1 λ 3
3 ,
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The Fresnel coefficients t j and r j can be expressed as follows by using the complex refractive index nˆ j and the complex refractive angle j . For parallel P- polarization (TM mode) and perpendicular S- polarization (TE mode) the Fresnel coefficients are [17]: ; rjp =
nˆ j-1cosθ j -nˆ jcosθ j-1 nˆ j-1cosθ j +nˆ jcosθ j-1
4
t jp =
2nˆ j-1cosθ j-1 nˆ j-1cosθ j +nˆ jcosθ j-1
5
; rjs =
nˆ j-1cosθ j-1 -nˆ jcosθ j nˆ j-1cosθ j-1 +nˆ jcosθ j
6
t js =
2nˆ j-1cosθ j-1 nˆ j-1cosθ j-1 +nˆ jcosθ j
7
For both polarisations S and P the transmittance energy T are reduced as [17]:
nˆ cosθ m+1 2 Trs =Re m+1 tS nˆ 0 cosθ 0
8
nˆ m+1cosθ m+1 tP nˆ 0 cosθ 0
; Trp =Re
2
9 .
3. Numerical Results and quasiperiodic models The studied aperiodic structures in this paper are the Generalized Fibonacci (GF) and the Generalized Thue-Morse (GTM) sequence. We chose to study only these two aperiodic structures because, according to several previous scientific studies and among all the aperiodic structures, only these two photonic structures make it possible to obtain polychromatic filters [111]. 3.1. GTM(m, n) quasiperiodic structure A one-dimensional
GTM quasiperiodic multilayered stack is an aperiodic structure
including two types of materials with lower and higher refractive indices. Besides, the two different elements L and H that form the 1D PC are organized following the inflation rule:
σ GTM (H,L):H H m Ln , L Lm H n ([18]). Therefore, it should be generated by the corresponding recursive rule: Sk+1 =Smk Skn , where Skn is the conjugated sequence of Smk , where the set (m, n) and k are the two parameters and the order of GTM sequence, respectively. Table 1 shows the structural generic form of organized multilayers stacks through the GTM(m, n) sequence for (m=n=2) and k=1, 2,3 with S0=H. 4
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GTM
Distributed { H,L }chain of GTM(2, 2) sequence
order 1
H2L2 , with S0=H
2
H2L2H2L2L2 H2L2H2
3
H2L2H2L2L2H2L2H2H2L2H2L2L2H2L2H2L2H2L2H2H2L2H2L2L2H2L2H2H2L2H2L2
Table1: Organized blocks (H, L) follow GTM(2,2) sequence satisfying the substitution rule σ GTM (H,L) .
Figure 1 shows the geometry of an example of1D multilayered stack organized according to the second level of GTM(2,2) sequence.
Figure 1: Schematic representation showing the organization of (H, L) chain of 1D GTM(m, n) quasiperiodic photonic structure with m=n=2 and k=3. 3.2. GF(m, n) quasiperiodic sequence 1D GF(m, n) quasiperiodic multilayered stacks including the two elements H and L were built according to the Fibonacci inflation rule[18]: σ GF (H,L):H H m Ln , L H . Here, H and L denote the material with the higher and lower refractive indices, respectively. Then, the series of n
GF(m, n) structure can be generated by the following sequence: Sk+1 =Smk Sk-1 . Here m and n are the two parameters of GF(m, n). We recall that the quasicrystals based on GF class show multi Bragg peaks. In table 2, we give the first four organized (H, L) elements according to GF(m=n, 3) sequence. order k
Distributed { H,L }chain of GF(2, 2) sequence
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1
H2L2 , with S0 =L and S1 =H
2
H2L2H2L2H2
3
H2L2H2L2H2H2L2H2L2H2H2L2H2L2
Table 2: Organized blocks (H, L) follow GF(2, 2) sequence satisfying the substitution rule σ GF (H,L) .
Figure 2 shows an example of the 1D multilayered stack including the elements H and L organized according to the second level of GF(2, 2) class. Here represents the incidence angle of the wave and using the transfert matrix method (TMM) we can determine the transmitted wave spectra after 28 layers.
Figure 2: Schematic representation showing the organization of(H, L)chain of 1D GF(m,n) quasiperiodic photonic structure with m=n=2 and k=3. For the of numerical calculation, we considered that the refractive indices of two limited media of 1D multilayered stacks are equal to 1 and all regions of the considered PC are linear, homogeneous, and transparent. The materials that form the PC are the Bi4Ge3O12 (H) and the silica (L). The refractive index of silica can be calculated as [19, 20]: 𝑛𝐿(𝜆) = 1 +
0.6961663𝜆2 𝜆2 ― 0.06840432
+
0.4079426𝜆2 𝜆2 ― 0.11624142
+
0.8974794𝜆2 𝜆2 ― 9.8961612
The refractive index of the Bi4Ge3O12 (BGO) material can be calculated as [21]: 𝑛𝐻(𝜆) = 1 + Where 𝝺 is the wavelength in micrometers. 6
3.1218393𝜆2 𝜆2 ― 0.18072
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Via literature and previous works, we can assume that the silica [20] and Bi4Ge3O12 [24] materials are transparent in the studied region. The thicknesses dL, dH of the two elements of PC were taken to satisfy the Bragg law:
n Ld L n Hd H
0 , where 0 is the central wavelength. The TMM approach was adopted to 4
exhibit the transmittance and characteristics of the photonic band gaps (PBGs). 3.3. Effect of lattice parameters of GF(m, n) and GTM(m, n) quasiperiodic sequences Figure 3 shows the transmission spectrum for different lattice parameters m=n=p. The GF(m, n) exhibited similar band gaps that inhibited the propagation of EM wave in a certain range of frequencies. The bandwidth of each photonic band gap (PBG) can be divided for an increase of p due to the number of cavities open by the local defect of periodicity inside the crystals. In addition, It appeared in the same figure an oscillation beam between adjacent PBGs that increased for an increase of parameter p. As a result, a polychromatic filter was obtainable for regular GF(p) class.
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Figure 3: Transmittance spectrum of regular GF(p) multilayered stacks versus frequency for p set to be 1,2,3, and 4. Figure 4 shows the transmission spectra of GF(m, n) for m=n=2 that exhibited two relative band . Thus, It is clear that from this n gaps and can be adjusted for the change of the contrast index figure the dip transmission augmented progressively and reached an appreciable PBG for high . n
100
n=1 n=2 n=3 n=4
90 80
Transmission[%]
70 60 50 40 30 20 10 0 1
2
10
3
Frequency[10 Hz]
4
5
Figure 4: Transmittance spectrum of regular GF(2) multilayered stacks versus frequency for n set to be 1,2,3, and 4. Figure 5 shows the transmission spectrum of aperiodic GTM multilayered stacks. A typical multichannel filter without PBG beam in inter-band is clear from this figure. Similarly to GF class, the number of PBGs increased progressively for an increase of parameters p=m=n. Furthermore, the increase of the structural Bragg gaps series can be used in multi-mode fiber cables. Then, the quality Q factor and the filtering properties are improved. Thus, the Bragg peaks were due to the confinement of light inside the PC and were sensitive to parameter p. Indeed, a local defect can be created by the repetition of auto-similar distribution of alternating (HL) within the GTM heterolayers that depends on the given parameter p. Consequently, the presented GTM structure led to the design of an executive microwave switching device (TOFF=0) and (TON=1) for the appropriately optimized parameter p. As seen in figure 5, the number of 8
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Bragg peaks (Npeak) was approximately proportional to the given parameter p. The transmission peaks became narrow with the addition of repetitive (LH) layers within the PC and led to an increase of the value of quality factor Q. The improvement of the Q can be explained by the decrease of the Full width at half maximum (FWHM) of Bragg peaks. So with p increasing, the number of transmission peaks also is increased, however, the values of transmission peaks are relatively small because the increase of number of layers in front of the incident wave allows the attenuation of the transmitted wave, and a part of the incident wave will be reflected.
Figure 5: Transmittance spectrum of aperiodic GTM(p) multilayered stacks versus frequency at given GTM parameters p set to be 8and 16. Figure 6 shows the occupied position of Bragg spectrum of aperiodic GTM multilayered stacks for different thicknesses dH of corresponding materials with the high refractive index. It should be appreciated that the position of Npeak shifted toward the lower frequency for an increase of thicknesses dH that improved the quality factor, where all peaks were similar to bandwidth.
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100 dH=0.75mm
90
dH=0.81mm dH=0.85mm
Transmission[%]
80 70 60 50 40 30 20 10 0 2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
10
Frequency[10 Hz]
Figure 6: Evaluation of transmission peaks of GTM(5) multilayered stacks versus frequency at given thicknesses dH of material with a higher refractive index with dH (mm) set to be 0.75,0.81 and 0.85. Table 3 summarizes the corresponding evolution of Q for the set value of thicknesses dH. We choose to compute the transmission spectrum for p=5. Table 3: Evolution of Q-factor for corresponding transmission peaks appearing at appropriate thicknesses dH(mm). Thicknesses dH(mm) f peak(1010Hz) fpeak(1010Hz) Q
0.75
0.81
0.85
2.735 0.5 5.47
2.5 0.5 5
2.295 0.5 4.59
3.4. Effect of incident angle
Figure 7 shows the present plotted of 3D reflectance spectrum versus incident angle and frequency at the given parameter p. All spectra exhibited a multitude of Bragg peaks opened at different frequencies and sensitive to lattice parameters of GTM in which bandwidth decreased progressively for augmentation of p. From figure 7 it is clear, that the stacking of typical 10
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channels covers the maximum of frequency range and enhances the total reflection's band. Similarly, we plotted at the same time the disposition of PBGs for different variations of θ(rad) . Where, the yellow areas presented the forbidden reflections’ band with zero transmission. Whereas, the blue insert zones that separate all PBGs corresponded to the peak transmission. We noted the presence of similar transmission peaks that separate all multichannel with zero transmission due to the symmetrical defects inserted in the main PC that derived from the inflation rule of GTM sequence. From figure 7, it is clear that for both p values and from the incidence angle =0.75 rad, the PBGs show a small shift towards the higher frequency region, and begin to be wider. The difference in PBGs positions is due to the changes in the wave propagation path inside the layers, which result from the changes in the incidence angle. Also for the both p values and from =1.435 rad, it is noticed that the intensity of the transmission peaks become too small. Therefore, the structure can be considered as a wave-reflector, for the studied frequency range. Finally we can conclude that the PBGs can be also tuned by changing the wave incidence angle to be near 1.5 rad. The physical explanation of this effect was first predicted by Belyakov and Dmitrienko and confirmed in experiments [25]. They approved that for an oblique incidence, waves become diffractive and for the large angles of incidence, there arises a region of total diffractive reflection [25].
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Figure 7: 3D Reflectance spectrums of 1D GTM multilayered stack as a function of frequency (Hz) and incident angle with the corresponding view of PBGs areas in yellow separated by fine Bragg peaks in blue for different incident angle with p set to be 5 and 10. 5.Conclusion A multichannel filter based on regular GF and aperiodic GTM sequence can be achieved at microwave frequency range. All transmission and reflection properties were investigated by the TMM approach. Therefore, we showed for the two quasiperiodic configurations, a stacking of forbidden bands in which bandwidths were adjusted by the given lattice parameter m=n of quasiperiodic sequences. Also, a series of transmission peaks were shown within the present PBGs representing the number of filtering channels due to the insert defect inside the quasiperiodic generation. In addition, a broad discrete PBG can be achieved through the GTM heterolayers for the different incident angle. Acknowledgments The first and the third author would like to acknowledge the research deanship of University of Ha’il, KSA for funding the project ’’RG-191250’’. References [1]
Y. Zhang, Z. Wu, Y. CAO, H. Zhang, Optical properties of one-dimensional Fibonacci quasi-periodic graphene photonic crystal, Optics Communications 338 (2015) 168–173.
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[2]
B.K Singh, K.B Thapa, P.C Pandey, Optical reflectance and omnidirectional bandgaps in Fibonacci quasicrystals type 1-D multilayer structures containing exponentially graded material, Optics Communications 297 (2013) pp. 65–73.
[3]
S.Z. Gharamaleki, Narrowband optical filter design for DWDM communication applications
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on
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Morse
structures,
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S. Sahel, R. Amri, D. Gamra, M. Lejeune, M. Benlahsen, K. Zellama and H. Bouchriha, Effect of sequence built on photonic band gap properties of one-dimensional quasiperiodic photonic crystals: Application to Thue-Morse and Double-period structures, Superlattices and Microstructures 111 (2017) 1-9.
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H. Rahimi , Analysis of photonic spectra in Thue-Morse, double-period and RudinShapiro quasiregular structures made of high temperature superconductors in visible range, Optical Materials 57 (2016) 264-271.
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Y.Trabelsi, Y. Bouazzi, N. Ben Ali, M. Kanzari, Narrow stop band optical filter using one-dimensional regular Fibonacci/Rudin Shapiro photonic quasicrystals, Optical and Quantum Electronics 48 (2016) 1-9.
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S. Sahel, R. Amri, L. Bouaziz, D. Gamra, M. Lejeune, M. Benlahsen, and H. Bouchriha, Optical filters using Cantor quasi-periodic one dimensional photonic crystal based on Si/SiO2, Superlattices and Microstructures 97(2016) 429-438.
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Bipin K.Singh, Praveen C. Pandey , Influence of graded index materials on the photonic localization in one-dimensional quasiperiodic (Thue–Mosre and Double-Periodic) photonic crystals, Optics Communications 333 (2014) 84-91.
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M. KANZARI, A.BOUZIDI and B.REZIG, Interferential polychromatic filters, The European Physical Journal B 36 (2003) 431–443.
[10] S. Roshan Entezar, Photonic crystal wedge as a tunable multichannel filter, Superlattices and Microstructures 82 (2015) 33–39. [11] NB Ali, J Zaghdoudi, M Kanzari and R Kuszelewicz,The slowing of light in onedimensional hybrid periodic and non-periodic photonic crystals, Journal of Optics 12 (2010), 045402. [12] N. Ben Ali and M. Kanzari, Omni-directional high reflectors using one-dimensional deformed quasi-periodic Cantor band gap structure at optical telecommunication 13
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wavelength band, The Mediterranean Journal of Electronics and Communications, 6 (2010) 1-6. [13] M. Vasconcelos, P. W. Mauriz and E. L. Albuquerque, Optical filters based in quasiperiodic photonic crystal, Microelectronics Journal 40 (2009) 851-853. [14] S. Golmohammadi, M. K. Moravvej-Farshi, A. Rostami, and A. Zarifkar,Narrowband DWDM filters based on Fibonacci-class quasi-periodic structure, Optics Express 15 (2007) 10520-10532. [15] N. Ben Ali, Optical Fabry–Perot filters using hybrid Periodic, Fibonacci and Cantor photonic structures, Nano Communication Networks 13 (2017) 34-42. [16] A. Rostami, Generalized Fibonacci quasi photonic crystals and generation of superimposed Bragg Gratings for opticalcommunication, Microelectronics Journal 37 (2006) 897-903. [17] N. Ben Ali, Narrow stop band microwave filters by using hybrid generalized quasiperiodic photonic crystals, Chinese journal of physics 55 (2017), 2384-2392. [18] Y. Trabelsi, N. Ben Ali, Y. Bouazzi and M.Kanzari, Microwave transmission through onedimensional hybrid quasi-regular (Fibonacci and Thue-Morse)/periodic structures, Photonic Sensors 3 (2013) 246–255. [19] I. H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am. 55 (1965), 1205 [20] M. Naftalya and R. E. Miles, Terahertz time-domain spectroscopy of silicate glasses and the relationship to material properties, Journal of Applied Physics 102 (2007) 043517. [21] Arezou Rashidi and Abdolrahman Namdar, Tunability of temperature-dependent absorption in a graphene-based hybrid nanostructure cavity, Eur. Phys. J. B 91: 68 (2018) 2-7. [22] Luigi Moretti and Vito Mocella, Two-dimensional photonic aperiodic crystals based on Thue-Morse sequence, Optics Express 15 (2007) 15314-23 [23] K Ben Abdelaziz, Yassine Bouazzi and Kanzari Mounir, Study of deformed quasi-periodic Fibonacci two dimensional photonic crystals, Journal of Physics Conference Series 633 (2015) 012007.
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[24] Yali Wang, Bihui Hou, Haiyan, Guozhong Zhao, Bingxin Yang, The THz spectra and analysis of Bi4Ge3O12 crystal, Published in: 35th International Conference on Infrared, Millimeter, and Terahertz Waves, IEEE. [25] A. H. Gevorgyan, Effects of angle of incidence and polarization in the chiral photonic crystals, Opt. Spectrosc. 105 (2008) 624-632. https://doi.org/10.1134/S0030400X08100172.
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AUTHOR CONSENT FORM The following authors present in the paper entitled on “Design of output-graded narrow polychromatic filter by using photonic quasicrystals”, has assured that there is no conflict of interest to propose this work. The author also agrees that the contribution was given in mutual manner for this successful completion of application. 1. N. Ben Ali 2. Vigneswaran Dhasarathan 3. Y.Trabelsi, 4. Truong Khang Nguyen 5. Y.Bouazzi 6. M. Kanzari
Naim Ben Ali, Vigneswaran Dhasarathan, Haitham Alsaif, Youssef Trabelsi, Truong Khang Nguyen, Y. Bouazzi, Mounir Kanzari PII:
S0921-4526(19)30798-7
DOI:
https://doi.org/10.1016/j.physb.2019.411918
Reference:
PHYSB 411918
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
23 June 2019
Accepted Date:
30 November 2019
Please cite this article as: Naim Ben Ali, Vigneswaran Dhasarathan, Haitham Alsaif, Youssef Trabelsi, Truong Khang Nguyen, Y. Bouazzi, Mounir Kanzari, Design of output-graded narrow polychromatic filter by using photonic quasicrystals, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411918
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof
Design of output-graded narrow polychromatic filter by using photonic quasicrystals
Naim Ben Ali1,5 , Vigneswaran Dhasarathan2,3*, Haitham Alsaif 4, Youssef Trabelsi5,6, Truong Khang Nguyen2,3, Y.Bouazzi1,5, and Mounir Kanzari5,7 1Department 2Division
of Industrial Engineering, College of Engineering, University of Ha’il, 2440, Ha’il City, Saudi Arabia
of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3Faculty
of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi, Minh City Vietnam Email : [email protected]
4Department
of Electrical Engineering, College of Engineering, University of Ha’il, 2440, Ha’il City, Saudi Arabia
5University
of Tunis El Manar, National Engineering School of Tunis, Photovoltaic and Semiconductor Materials Laboratory,1002, Tunis, Tunisia
6King
Khaled university, College of Arts and Sciences in Mahayel Aseer, Physics department , KSA.
7University
of Tunis, Preparatory Engineering Institute of Tunis, Montfleury, 1008, Tunisia
Abstract The properties of one-dimensional photonic quasicrystals built according to one-dimensionalgeneralized Fibonacci (GF) / Generalized Thue-Morse (GTM) were investigated in order to design an output polychromatic filter. This main aperiodic multilayered structure was made up of alternate two dielectrics materials Silica and Bi4Ge3O12(BGO)with higher and lower refractive indices respectively. The Transfer Matrix Method (TMM) was adopted to calculate the photometric response. The transmittance spectrum exhibited a stacking of similar Bragg gaps (BGs) for specific arrangement (m=n) of GF and GTM. We noted that the positions and the number and size of BGs were sensitive to the constituent of the Photonic quasicrystal (PQC) system and lattice parameter of quasiperiodic sequence. Therefore, these configurations of multilayered stacks can be useful as a graded polychromatic filter.
Keywords: Quasiperiodic sequences, Photonic Quasicrystals (PQC), Photonic band gap, GTM sequence, GF sequence, polychromatic filter.
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1. Introduction The area of photonic quasicrystals (PQCs) is a new field of optics with a quasiperiodic modulation in refractive indices of two different material A and B. One-dimensional photonic quasicrystal is the simplest form of PC that can be considered as an intermediate class between random and periodic media. It is an artificial multilayered stack containing two kinds of material with higher and lower refractive indices, respectively. The considered PQC still determined by structural substitution rules of quasiperiodic sequence like Fibonacci [1-2], Thue Morse[3-5], Rudin Shapiro[6], Cantor[7], etc. Besides, these multilayered stacks prohibited the propagation of the electromagnetic wave in a specific range of frequency that creates forbidden bands called photonic band gaps (PBG). Thus, the ability of light manipulating is strongly useful for the design of potential applications in photonic technology and optical communications. Numerous applications based on one-dimensional and two-dimensional photonic quasicrystals (1D PQCs) are achieved and successfully used for the development of optoelectronic devices like a high reflector, optical filters [8-11, 22, 23]. Various 1D GF and 1D GTM systems have been used to enhance the omnidirectional Bragg gaps (OBG)of the conventional photonic crystals. In order to enhance the OBG, certain researchers have applied the deformation through the main quasiperiodic structure [12]. They proved that the proprieties of main PBG were considerably improved. In the last decade, the stop band filter based on 1D-PQC has attracted much attention due to its interesting properties and abilities to localize the wave in specific channel frequency. It was found that a succession of discrete PBGs covers a large zone of the optical frequency domain. It may be used to develop new narrowband optical filters, in which channels are adjusted by the lattice parameters of the quasiperiodic sequence. This considered filter can be also used in Dense wavelength division multiplexing (DWDM)[13-14]. In this case, a certain researcher [15], proposed design of multiband optical filter using Fibonacci quasiperiodic class for suitable selection parameters. They found that the characteristic of multi-band for this quasiperiodic structure is adjusted by the basic elements and parameters of the Fibonacci class. Then, an efficient relationship between the Fibonacci class and superimposed Bragg Gratings has been established [16]. In addition, this design was extended to other mathematical models based on quasiperiodic structures that suggest a new arrangement of 2
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basic PC elements like the Morse, Cantor, Rudin Shapiro. Indeed, all these aperiodic systems exhibit a typical distribution of layers in which periodicity is broken for long-range order and permits to create a localized mode within the main PBG. This defect mode leads to create multiple peaks in the transmission spectrum. In this work, we investigate the filtering properties through regular GF and GTM quasiperiodic sequences. This Work aims to study the optical properties of structural Bragg gaps (BGs) with the appropriate arrangement of layers. Also, tenable channels with zero transmission that inhibit the propagation of EM wave in a specific zone of the microwave frequency range are determined. Besides, a comparative study of two quasiperiodic configurations is useful to identify the perfect muli-channeled filter at precise frequencies range. The transmittance and reflectance are used to examine the effects the thickness of BGO layer, the distributed Bragg gaps (BGs) and transmission peak for different GTM (m=n) sequences. 2. Problem formulation In this section, we used the transfert matrix method (TMM) to determine the reflectance R and the transmittance T through the one-dimensional Fibonacci and Thue-Morse class quasicrystals. Based on this approach, the input amplitude fields E 0+ reflected fields E -0 , and output transmitted field E +m+1 after m layers are related by the sequential multiplication complex transfer matrix C j that can be given by [17] as:
E 0+ m C j - = E 0 j=1 t j
E +m+1 - E m+1
1
is expressed as [17] in the following formula: C j For both TM and TE modes,
exp iφ j-1 rjexp -iφ j-1 Cj= rjexp iφ j-1 exp -iφ j-1
2
The phase shift of the wave between the jth and (j-1)th boundaries is defined by [17] in the following expression:
φ j-1 =
2π nˆ j-1d j-1cosθ j-1 λ 3
3 ,
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The Fresnel coefficients t j and r j can be expressed as follows by using the complex refractive index nˆ j and the complex refractive angle j . For parallel P- polarization (TM mode) and perpendicular S- polarization (TE mode) the Fresnel coefficients are [17]: ; rjp =
nˆ j-1cosθ j -nˆ jcosθ j-1 nˆ j-1cosθ j +nˆ jcosθ j-1
4
t jp =
2nˆ j-1cosθ j-1 nˆ j-1cosθ j +nˆ jcosθ j-1
5
; rjs =
nˆ j-1cosθ j-1 -nˆ jcosθ j nˆ j-1cosθ j-1 +nˆ jcosθ j
6
t js =
2nˆ j-1cosθ j-1 nˆ j-1cosθ j-1 +nˆ jcosθ j
7
For both polarisations S and P the transmittance energy T are reduced as [17]:
nˆ cosθ m+1 2 Trs =Re m+1 tS nˆ 0 cosθ 0
8
nˆ m+1cosθ m+1 tP nˆ 0 cosθ 0
; Trp =Re
2
9 .
3. Numerical Results and quasiperiodic models The studied aperiodic structures in this paper are the Generalized Fibonacci (GF) and the Generalized Thue-Morse (GTM) sequence. We chose to study only these two aperiodic structures because, according to several previous scientific studies and among all the aperiodic structures, only these two photonic structures make it possible to obtain polychromatic filters [111]. 3.1. GTM(m, n) quasiperiodic structure A one-dimensional
GTM quasiperiodic multilayered stack is an aperiodic structure
including two types of materials with lower and higher refractive indices. Besides, the two different elements L and H that form the 1D PC are organized following the inflation rule:
σ GTM (H,L):H H m Ln , L Lm H n ([18]). Therefore, it should be generated by the corresponding recursive rule: Sk+1 =Smk Skn , where Skn is the conjugated sequence of Smk , where the set (m, n) and k are the two parameters and the order of GTM sequence, respectively. Table 1 shows the structural generic form of organized multilayers stacks through the GTM(m, n) sequence for (m=n=2) and k=1, 2,3 with S0=H. 4
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GTM
Distributed { H,L }chain of GTM(2, 2) sequence
order 1
H2L2 , with S0=H
2
H2L2H2L2L2 H2L2H2
3
H2L2H2L2L2H2L2H2H2L2H2L2L2H2L2H2L2H2L2H2H2L2H2L2L2H2L2H2H2L2H2L2
Table1: Organized blocks (H, L) follow GTM(2,2) sequence satisfying the substitution rule σ GTM (H,L) .
Figure 1 shows the geometry of an example of1D multilayered stack organized according to the second level of GTM(2,2) sequence.
Figure 1: Schematic representation showing the organization of (H, L) chain of 1D GTM(m, n) quasiperiodic photonic structure with m=n=2 and k=3. 3.2. GF(m, n) quasiperiodic sequence 1D GF(m, n) quasiperiodic multilayered stacks including the two elements H and L were built according to the Fibonacci inflation rule[18]: σ GF (H,L):H H m Ln , L H . Here, H and L denote the material with the higher and lower refractive indices, respectively. Then, the series of n
GF(m, n) structure can be generated by the following sequence: Sk+1 =Smk Sk-1 . Here m and n are the two parameters of GF(m, n). We recall that the quasicrystals based on GF class show multi Bragg peaks. In table 2, we give the first four organized (H, L) elements according to GF(m=n, 3) sequence. order k
Distributed { H,L }chain of GF(2, 2) sequence
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1
H2L2 , with S0 =L and S1 =H
2
H2L2H2L2H2
3
H2L2H2L2H2H2L2H2L2H2H2L2H2L2
Table 2: Organized blocks (H, L) follow GF(2, 2) sequence satisfying the substitution rule σ GF (H,L) .
Figure 2 shows an example of the 1D multilayered stack including the elements H and L organized according to the second level of GF(2, 2) class. Here represents the incidence angle of the wave and using the transfert matrix method (TMM) we can determine the transmitted wave spectra after 28 layers.
Figure 2: Schematic representation showing the organization of(H, L)chain of 1D GF(m,n) quasiperiodic photonic structure with m=n=2 and k=3. For the of numerical calculation, we considered that the refractive indices of two limited media of 1D multilayered stacks are equal to 1 and all regions of the considered PC are linear, homogeneous, and transparent. The materials that form the PC are the Bi4Ge3O12 (H) and the silica (L). The refractive index of silica can be calculated as [19, 20]: 𝑛𝐿(𝜆) = 1 +
0.6961663𝜆2 𝜆2 ― 0.06840432
+
0.4079426𝜆2 𝜆2 ― 0.11624142
+
0.8974794𝜆2 𝜆2 ― 9.8961612
The refractive index of the Bi4Ge3O12 (BGO) material can be calculated as [21]: 𝑛𝐻(𝜆) = 1 + Where 𝝺 is the wavelength in micrometers. 6
3.1218393𝜆2 𝜆2 ― 0.18072
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Via literature and previous works, we can assume that the silica [20] and Bi4Ge3O12 [24] materials are transparent in the studied region. The thicknesses dL, dH of the two elements of PC were taken to satisfy the Bragg law:
n Ld L n Hd H
0 , where 0 is the central wavelength. The TMM approach was adopted to 4
exhibit the transmittance and characteristics of the photonic band gaps (PBGs). 3.3. Effect of lattice parameters of GF(m, n) and GTM(m, n) quasiperiodic sequences Figure 3 shows the transmission spectrum for different lattice parameters m=n=p. The GF(m, n) exhibited similar band gaps that inhibited the propagation of EM wave in a certain range of frequencies. The bandwidth of each photonic band gap (PBG) can be divided for an increase of p due to the number of cavities open by the local defect of periodicity inside the crystals. In addition, It appeared in the same figure an oscillation beam between adjacent PBGs that increased for an increase of parameter p. As a result, a polychromatic filter was obtainable for regular GF(p) class.
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Figure 3: Transmittance spectrum of regular GF(p) multilayered stacks versus frequency for p set to be 1,2,3, and 4. Figure 4 shows the transmission spectra of GF(m, n) for m=n=2 that exhibited two relative band . Thus, It is clear that from this n gaps and can be adjusted for the change of the contrast index figure the dip transmission augmented progressively and reached an appreciable PBG for high . n
100
n=1 n=2 n=3 n=4
90 80
Transmission[%]
70 60 50 40 30 20 10 0 1
2
10
3
Frequency[10 Hz]
4
5
Figure 4: Transmittance spectrum of regular GF(2) multilayered stacks versus frequency for n set to be 1,2,3, and 4. Figure 5 shows the transmission spectrum of aperiodic GTM multilayered stacks. A typical multichannel filter without PBG beam in inter-band is clear from this figure. Similarly to GF class, the number of PBGs increased progressively for an increase of parameters p=m=n. Furthermore, the increase of the structural Bragg gaps series can be used in multi-mode fiber cables. Then, the quality Q factor and the filtering properties are improved. Thus, the Bragg peaks were due to the confinement of light inside the PC and were sensitive to parameter p. Indeed, a local defect can be created by the repetition of auto-similar distribution of alternating (HL) within the GTM heterolayers that depends on the given parameter p. Consequently, the presented GTM structure led to the design of an executive microwave switching device (TOFF=0) and (TON=1) for the appropriately optimized parameter p. As seen in figure 5, the number of 8
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Bragg peaks (Npeak) was approximately proportional to the given parameter p. The transmission peaks became narrow with the addition of repetitive (LH) layers within the PC and led to an increase of the value of quality factor Q. The improvement of the Q can be explained by the decrease of the Full width at half maximum (FWHM) of Bragg peaks. So with p increasing, the number of transmission peaks also is increased, however, the values of transmission peaks are relatively small because the increase of number of layers in front of the incident wave allows the attenuation of the transmitted wave, and a part of the incident wave will be reflected.
Figure 5: Transmittance spectrum of aperiodic GTM(p) multilayered stacks versus frequency at given GTM parameters p set to be 8and 16. Figure 6 shows the occupied position of Bragg spectrum of aperiodic GTM multilayered stacks for different thicknesses dH of corresponding materials with the high refractive index. It should be appreciated that the position of Npeak shifted toward the lower frequency for an increase of thicknesses dH that improved the quality factor, where all peaks were similar to bandwidth.
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100 dH=0.75mm
90
dH=0.81mm dH=0.85mm
Transmission[%]
80 70 60 50 40 30 20 10 0 2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
10
Frequency[10 Hz]
Figure 6: Evaluation of transmission peaks of GTM(5) multilayered stacks versus frequency at given thicknesses dH of material with a higher refractive index with dH (mm) set to be 0.75,0.81 and 0.85. Table 3 summarizes the corresponding evolution of Q for the set value of thicknesses dH. We choose to compute the transmission spectrum for p=5. Table 3: Evolution of Q-factor for corresponding transmission peaks appearing at appropriate thicknesses dH(mm). Thicknesses dH(mm) f peak(1010Hz) fpeak(1010Hz) Q
0.75
0.81
0.85
2.735 0.5 5.47
2.5 0.5 5
2.295 0.5 4.59
3.4. Effect of incident angle
Figure 7 shows the present plotted of 3D reflectance spectrum versus incident angle and frequency at the given parameter p. All spectra exhibited a multitude of Bragg peaks opened at different frequencies and sensitive to lattice parameters of GTM in which bandwidth decreased progressively for augmentation of p. From figure 7 it is clear, that the stacking of typical 10
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channels covers the maximum of frequency range and enhances the total reflection's band. Similarly, we plotted at the same time the disposition of PBGs for different variations of θ(rad) . Where, the yellow areas presented the forbidden reflections’ band with zero transmission. Whereas, the blue insert zones that separate all PBGs corresponded to the peak transmission. We noted the presence of similar transmission peaks that separate all multichannel with zero transmission due to the symmetrical defects inserted in the main PC that derived from the inflation rule of GTM sequence. From figure 7, it is clear that for both p values and from the incidence angle =0.75 rad, the PBGs show a small shift towards the higher frequency region, and begin to be wider. The difference in PBGs positions is due to the changes in the wave propagation path inside the layers, which result from the changes in the incidence angle. Also for the both p values and from =1.435 rad, it is noticed that the intensity of the transmission peaks become too small. Therefore, the structure can be considered as a wave-reflector, for the studied frequency range. Finally we can conclude that the PBGs can be also tuned by changing the wave incidence angle to be near 1.5 rad. The physical explanation of this effect was first predicted by Belyakov and Dmitrienko and confirmed in experiments [25]. They approved that for an oblique incidence, waves become diffractive and for the large angles of incidence, there arises a region of total diffractive reflection [25].
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Figure 7: 3D Reflectance spectrums of 1D GTM multilayered stack as a function of frequency (Hz) and incident angle with the corresponding view of PBGs areas in yellow separated by fine Bragg peaks in blue for different incident angle with p set to be 5 and 10. 5.Conclusion A multichannel filter based on regular GF and aperiodic GTM sequence can be achieved at microwave frequency range. All transmission and reflection properties were investigated by the TMM approach. Therefore, we showed for the two quasiperiodic configurations, a stacking of forbidden bands in which bandwidths were adjusted by the given lattice parameter m=n of quasiperiodic sequences. Also, a series of transmission peaks were shown within the present PBGs representing the number of filtering channels due to the insert defect inside the quasiperiodic generation. In addition, a broad discrete PBG can be achieved through the GTM heterolayers for the different incident angle. Acknowledgments The first and the third author would like to acknowledge the research deanship of University of Ha’il, KSA for funding the project ’’RG-191250’’. References [1]
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[24] Yali Wang, Bihui Hou, Haiyan, Guozhong Zhao, Bingxin Yang, The THz spectra and analysis of Bi4Ge3O12 crystal, Published in: 35th International Conference on Infrared, Millimeter, and Terahertz Waves, IEEE. [25] A. H. Gevorgyan, Effects of angle of incidence and polarization in the chiral photonic crystals, Opt. Spectrosc. 105 (2008) 624-632. https://doi.org/10.1134/S0030400X08100172.
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AUTHOR CONSENT FORM The following authors present in the paper entitled on “Design of output-graded narrow polychromatic filter by using photonic quasicrystals”, has assured that there is no conflict of interest to propose this work. The author also agrees that the contribution was given in mutual manner for this successful completion of application. 1. N. Ben Ali 2. Vigneswaran Dhasarathan 3. Y.Trabelsi, 4. Truong Khang Nguyen 5. Y.Bouazzi 6. M. Kanzari