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Properties of defect modes in cylindrical photonic crystals
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Properties of defect modes in cylindrical photonic crystals Sahar A. El-Naggar
T
Department of Engineering Math and Physics, Faculty of Engineering, Cairo University, 12613, Giza, Egypt
ARTICLE INFO
ABSTRACT
Keywords: Cylindrical photonic crystals Defect mode Refractive index sensor
In this research, we study the properties of the defect modes that arise in the photonic band gap of a cylindrical photonic crystal (CPC) by using the transfer matrix method in the cylindrical coordinates. We consider two defective CPCs stacked in symmetric and asymmetric geometries. We examine the number of modes and their properties in the two CPCs structures. The dependencies of the modes on the azimuthal number and the inner radius of the CPC are also discussed. Numerical results show that the defect mode in the asymmetric CPCs has higher Q factor than those in the symmetric CPCs. Beside many potential optical communications and filtering applications, the structure may be a candidate for refractive index sensing with average sensitivity of 200 nm/RIU in the refractive index range from 1 to 1.6.
1. Introduction Photonic crystals (PCs) are known as regular arrays of materials with different refractive indices. The motion of photons in the PCs is analogues to that of electrons in ordinary crystals composed of a regular array of atoms. Therefore, photonic bandgaps (PBGs), which are frequency range where electromagnetic are forbidden to propagate though the PCs, are similar to electronic bandgaps that arises in ordinary crystals [1]. When a lattice defect is introduced in the PCs, the lattice defect will support an electromagnetic wave mode with a certain resonant frequency mode that exists in the PBGs. PCs are classified to one-dimensional (1D) PCs, two-dimensional and three dimensional structures. The main advantage of one-dimensional (1D) PCs is their simple structure that can be easily fabricated. Cylindrical PCs (CPCs) are type of 1D PCs with a periodic cylindrical multilayer structure. These CPCs have also received attention in recent years [2–8]. However, to the best of our knowledge, there has been little focus [9,10] on filtering properties based on CPCs. In this work, we study transverse wave (TE) propagation in defective CPCs. We use the cylindrical wave transfer matrix that has been developed in [2] to calculate the transmittance of the CPC. We consider two defective CPCs that have been arranged in symmetric and asymmetric geometries. We examine the number of modes and their properties in the two CPCs structures. We first focus our attention on the properties of the defect modes that arises in these CPCs. We observe that multiple defect modes with frequencies lying in the PBG arise. Therefore, we evaluate the quality factor (Q) and the full width at half maximum (FWHM) for these modes. We then show that these modes can be dependent on the azimuthal mode number and the starting radius. Finally, we test the possibility of using the suggested structure as a refractive index sensor. Our work may help many optical communications and filtering applications because our structure works at wavelength of 1550 nm This paper is organized as follows; in Section 2, we describe the two CPCs structure that have been addressed and we summarize the method has been used. In Section 3, we present the numerical results. Finally, the conclusions are presented in Section 4.
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163447 Received 17 June 2019; Accepted 18 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 1. A cross-sectional view in the x–y plane of a defective CPC in (a) Asymmetric (HL)N/D/(HL)N and (b) Symmetric (HL)N/D/(LH)N geometries. The defective CPC is bounded by two mediums, the inner and the outer regions of refractive indices ni and no respectively. The letter D refers to the defect layer shown in blue. The dimension in z direction is assumed to be much larger than the dimensions in x and y. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
2. Basic model and numerical methods We consider TE waves that are propagating in a defective CPCs structure. We consider two geometries shown in Fig. 1. The asymmetric geometry Inner/(HL)N/D/(HL)N/Outer, and the symmetric geometry Inner/(HL)N/D/(LH)N/Outer. In Fig. 1, layer 1 is H, layer 2 is L, layer 3 is H and so on until defect layer is introduced. For example, at N = 2, the asymmetric CPCs structure is HLHL/D/ HLHL, and the symmetric one is HLHL/D/LHLH. H and L are used to denote the materials with high refractive index and low refractive index, respectively. Each defective CPCs structure contains a defect layer D that is introduced at the middle of the structure. All layers are arranged to form the cylindrical coaxial structure shown in Fig. 1. The thicknesses of layers H, L, and D are assumed to be dH, dL, and dD, respectively. Layers H, L, D and the outer region have refractive indices nH, nL, nD and no respectively. The inner medium has a refractive index of ni and a starting radius of ρi. The electromagnetic cylindrical waves are assumed to diverge from the axis of symmetry and the fields should obey the source-free Maxwell's equations as follows, ∇×E = - jωH,
(1)
∇×H = jωE.
(2)
It is known that in the TE waves, the electric field E is along the + z direction and the three non-zero components are Ez, Hφ, and Hρ. The solution for the field Ez can be expressed as: Ez (ρ,ϕ) = [AJm(kρ)+ BYm(kρ)] exp(jmϕ),
(3)
where A and B are constants, Jm is a Bessel function, Ym is a Neumann function. k is the wave number in the medium and m is the azimuthal mode number. The azimuthal part of the magnetic field is given by, Hϕ (ρ,ϕ) = - j γ [AJ`m(kρ)+ BY`m(kρ)] exp(jmϕ),
(4)
with γ =√(ε/μ), where ε and μ are the permittivity and permeability of the medium. Note that the sign of the derivative denotes differentiation with respect to the entire argument of the function, and not just with respect to ρ. At each cylindrical surface between two different mediums, we apply the continuity condition of the tangential fields components Ez, Hφ. Therefore, a total transfer matrix can be constructed. The transfer matrix relates the fields components at the inner and outer boundaries of the CPCs. The final 2
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
form of the equations is written to define the total transfer matrix Mt as follows,
Ez H
=Mt = i
Ez H
,
(5)
= o
where ρo is the outermost radius. Note that the elements of the transfer matrix are functions of the radii unlike those are used in the planner PCs. To be able to calculate the transmittance and the reflectance, we use Hankel functions to express the fields components in the inner region and in the outer region as a sum of waves traveling in opposite directions, as follows
Ez H
= = i,o
(1)
H(1) m (k
i,o)
-j H` (1) m (k
H (2) m (k
i,o)
(2) i,o) -j H` m (k
i,o)
Ai,o , Bi,o
(6)
(2) m
where H m and H are Hankel functions of the first and the second kinds. A and B are the amplitudes of the waves. The subscripts “i” and “o” are used to refer to the inner and the outer regions respectively. We rewrite Eqs. (5) and (6) as follows,
Ai T T = 11 12 Bi T21 T22
Ao Bo
(7)
The amplitudes of the electric fields at ρ=ρi and those at ρ=ρo can be written in terms of the amplitude reflection rd and amplitude transmission td as follows, Ez )
ρ=ρi
=1+rd, Ez)
ρ=ρo
=td.
(8)
In terms of rd and td, the associated reflectance and transmittance are R = |rd|2, T = (no/ni) |td|2.
(9)
In the next section, we focus our results on the transmittance T through the CPCs. 3. Numerical results and discussion The characteristic of the defect modes in CPCs is studied in near infrared wavelength region having design wavelength λo of 1550 nm, the common wavelength in optical communications. We consider two geometries shown in Fig. 1. The asymmetric geometry Inner/(HL)N/D/(HL)N/Outer, and the symmetric geometry Inner/(HL)N/D/(LH)N/Outer. The layer with high refractive index H is titanium dioxide TiO2 (nH = 2.32), the layer with low refractive index L is aluminum oxide Al2O3 (nL = 1.63) [6]. These materials are selected because this CPC has been experimentally made [6]. All layers are quarter wavelength, therefore H, L and D have respective thickness dH =λo/4nH,dL =λo/4nL and dD =λo/4nD. In addition, both the inner and the outer regions are air. The starting radius and the defect layer D are initially taken as ρi = 100a where a = dH + dL, and air respectively. Then the effect of changing the refractive index of the defect layer and varying the starting radius are addressed. First, we investigate the number of defect modes arising in the asymmetric CPCs and the symmetric CPCs at lowest azimuthal mode (m = 0). In Fig. 2, we plot the calculated transmittance for the (HL)N/D/(HL)N and that for (HL)N/D/(LH)N at N = 12. We note that one defect mode arises in the asymmetric CPCs at the design wavelength with peak height of 0.8. We call this defect mode D0. While, the symmetric CPCs has two defect modes D1 and D2 within the PBG as shown in Fig. 2(b) at 1431.7 nm and 1689.6 nm respectively. We denote the wavelengths of the defect modes for the D0, D1 and D2 by λD0, λD1 and λD2, respectively. The transmission peaks for both D1 and D2 are 0.8. It is worth noting that the transmittance of CPCs does not reach unity like that of planner PCs. This is consistent with conservation of energy, because energy density on a cylindrical interface is inversely proportional to the radius. Therefore, the transmittance reaches a maximum value of ρi/ρo. In Fig. 2(c), the defect mode D0 is plotted for three different values of N = 8, 10 and 12. It is known that increasing the number of periods N leads to an enhancement of the quality factor Q (Δλ/
Fig. 2. Transmittance through the asymmetric CPCs, (HL)12/D/(HL)12, and through the symmetric CPCs, (HL)12/D/(LH)12, in (a) and (b) respectively. The defect mode D0 in (c) at various values of N. 3
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Table 1 Variation in FWHM, Quality Q and transmittance of CPCs with number of periods (N). D0 in Asymmetric CPC
D1 in symmetric CPC
D2 in symmetric CPC
N
T
Q
FWHM (nm)
Q
FWHM (nm)
Q
FWHM (nm)
8 10 12 14
0.855 0.827 0.8 0.775
777 3099 12289 48413
1.994 0.5002 0.1261 0.0320
519 1369 3490 8728
2.7543 1.0457 0.4101 0.1641
439 1159 2946 7451
3.8546 1.4588 0.5735 0.2267
λ) due to fields’ confinement within the defect layer. Unfortunately, in the CPC increasing N tends to scarify the peak height of the defect mode. In Table 1, the variation in FWHM, Q and transmittance peak of the three defect modes D0, D1 and D2 at different values of N are listed. We observe that the peak height of the transmittance for the three modes decreases from 0.855 to 0.775 by increasing N from 8 to 14, respectively. In addition, increasing N leads to a decrease in FWHM for the three modes. Therefore, the Q factor increases for the three defect modes D0, D1 and D2 from 777 to 48413, from 519 to 8728 and from 439 to 7451 by increasing N from 8 to 14, respectively. We conclude that D0, arising in the asymmetric CPCs, has higher Q and narrower FWHM than the two defects arising in the symmetric CPCs. Next, we study the behavior of the electric field intensity inside the CPCs to understand the reason behind the presence of two defect modes in the symmetric CPCs and investigate the differences between the three defect modes. We plot the electric field |Ez | along the radius, ρ, of the CPCs at λD0, λD1 and λD2 as shown in Fig. 3. We observe that the field is highly confined in the defect layer for D0 compared to that for D1 and D2 which explains higher Q associated with D0. In addition, comparison between the field distributions of D1 and that of D2 reveals that the symmetrical geometry gives rise to two possible field’s distributions in the defect layer at resonance. One of these distributions is like sine function which corresponds to a state in the PBG with high energy (appears at low wavelength). The other distribution is like cosine function which corresponds to a state with low energy (appears at high wavelength). 3.1. Effect of azimuthal mode number m and the starting radius ρi In this subsection, we start by studying the effect of the azimuthal mode number (m) on the defect modes arising in the asymmetric CPCs and the symmetric one. In the analysis that follows, we consider N = 12 to guarantee high Q without scarifying the peak height of transmittance. We plot the transmittance in the vicinity of λD0, λD1 and λD2 in Fig. 4(a)–(c) respectively at different four values of m = 1, 5, 9 and 13. We note that the defect mode shifts to lower wavelength by increasing m. The shift in the wavelength is up to 1.5 nm when m increases to 13. This wavelength shift originates because the field solutions of cylindrical waves depend on m. In addition, we get the expression of the radial component Hρ = −(mcEz)/(ωμρ). At m = 0, the radial component of the field equals zero. We observe that m is in the numerator which indicates that the radial component increases by increasing m. Therefore, increasing m is analogues to increasing the angle of incidence in the planner PCs as stated in [8]. In Table 2, the variation in FWHM, Q and transmittance peak of the three defect modes D0, D1 and D2 at different values of m are listed. We observe that the FWHM and Q are almost invariant by changing m. Next, we investigate the effect of the starting radius on the defect modes in CPCs at nonzero azimuthal mode number (m = 5). We plot the transmittance around λD0, λD1 and λD2 in Fig. 5(a)–(c) respectively at different four values of ρi = 200a, 75a, 50a and 25a. We note that by decreasing the starting radius, the wavelength of the defect is blue shifted. In addition, a decrease in the peak transmittance value can be observed by increasing ρ. From Table 3, which lists the properties of the modes, we note that for the three
Fig. 3. The electric field intensity distribution inside the CPCs for D0, D1 and D2 at λD0 = 1550 nm, λD1 = 1431.7 nm, and λD2 = 1689.6 nm in (a), (b) and (c) respectively. The vertical blue dotted lines indicate the boundaries of the layers. The defect layer is bounded by two solid red vertical lines. N = 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). 4
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 4. The defect modes D0, D1 and D2 at different azimuthal modes in (a), (b) and (c) respectively. Table 2 Variation in FWHM, Quality Q and transmittance of CPCs with azimuthal mode (m). D0 in Asymmetric CPCs
D1 in Symmetric CPCs
D2 in Symmetric CPCs
m
λD0 (nm)
T
Q
FWHM (nm)
λD1 (nm)
T
Q
FWHM (nm)
λD2 (nm)
T
Q
FWHM (nm)
1 5 9 13
1550 1549.8 1549.3 1548.5
0.8 0.8 0.79 0.79
12289 12394 12390 12484
0.1261 0.1250 0.1250 0.1240
1431.7 1431.6 1431.2 1430.6
0.8 0.8 0.79 0.79
3490.8 3490.4 3498.1 3505.3
0.4101 0.4101 0.4091 0.4081
1689.6 1689.3 1688.8 1687.9
0.8 0.8 0.79 0.79
2963.2 2971.5 2988.0 3031.1
0.5702 0.5685 0.5652 0.5569
Fig. 5. The defect modes D0, D1 and D2 at different starting radii in (a), (b) and (c) respectively. Table 3 Variation in FWHM, Quality Q and transmittance of CPCs with inner radius (ρi). Asymmetric CPCs
Symmetric CPCs
D0
D1
D2
ρi
λD0 (nm)
T
Q
FWHM (nm)
λD1 (nm)
T
Q
FWHM (nm)
λD2 (nm)
T
Q
FWHM (nm)
200a 75a 50a 25a
1549.9 1549.6 1549.3 1548.0
0.89 0.75 0.67 0.49
12298 12296 12391 12582
0.1260 0.1260 0.1250 0.1230
1431.7 1431.4 1431.2 1430.3
0.89 0.75 0.67 0.49
3490.7 3498.7 3498.1 3539.1
0.4101 0.4091 0.4091 0.4041
1689.5 1689.2 1688.8 1687.3
0.89 0.75 0.67 0.49
2954.5 2971.2 2996.9 3048.4
0.5719 0.5685 0.5635 0.5535
modes, the peak value decreases from 0.89 to 0.49 by decreasing the inner radius from 200a to 25a respectively. The decrease in the value of the transmittance occurs because the ratio ρi/ρo decreases by decreasing ρi. The shift in the wavelength is up to 2 nm when ρi decreases to 25a. The shift in the wavelength can be explained by the radial component of the field discussed in the previous paragraph. By observing the expression of the radial component, it is inverse proportional to the radius. Therefore, at fixed m, the decrease in the radius is associated with an increase in the radial component of the field. 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 6. The defect mode D0 at different values of nD. Table 4 Variation in FWHM, Quality Q, peak transmittance of asymmetric CPC, the wavelength of defect D0 (λD0), and the wavelength sensitivity S with various values of (nD). nD
λD0 (nm)
T
Q
FWHM (nm)
S (nm/RIU)
1.1 1.2 1.3 1.4 1.5 1.6
1570.1 1591.4 1613.3 1635 1655.8 1675
0.7932 0.7766 0.7575 0.7409 0.7301 0.7261
11293 9642.6 7680.6 5745.5 4094.8 2829.9
0.139 0.165 0.2101 0.2846 0.4044 0.5919
201 213 219 217 208 192
3.2. Refractive index sensing In this subsection, we test the feasibility of designing refractive index sensor based on CPCs structure. We focus our attention on studying the effect of the refractive index of the defect layer on the defect mode arising in the asymmetric CPCs, D0, because we found out that this defect has higher Q than those arise in the symmetric CPCs. We set the starting radius ρi = 100a and dD=λo/4. To investigate the performance of using our structure as a sensor, we calculate the wavelength sensitivity Sλ as follows, Sλ (nm/RIU)= Δλpeak/Δn,
(10)
where Δλpeak is the change in the wavelength where peak resonance occurs and Δn is the change in the refractive index. In Fig. 6, we plot the transmittance of the asymmetric CPCs at different values of nD. We observe that the defect’s wavelength increases from 1570.1 nm to 1675 nm by increasing nD from 1.1 to 1.6 respectively. Increasing nD leads to an increase in the resonance wavelength because the resonance wavelength depends on the effective refractive index which increases when the refractive index of the defect layer increases. In addition, the peak height is lowered down from 0.793 to 0.726 as nD increases from 1.1 to 1.6 respectively. The Q factor decreases due to an increase in the FWHM. However, an average wavelength sensitivity Sλ of 200 nm/RIU is calculated which is comparable to the earlier photonic crystal cavity lasers. The variation in FWHM, Q, peak transmittance, and the wavelength sensitivity with various values of nD are listed in Table 4. Our analysis on the defect modes provides useful information for the design of a transmission filter with high Q based on the CPCs. 4. Conclusion By using the cylindrical wave transfer matrix method, the properties of the defect modes for the asymmetric and symmetric structures of CPCs have been theoretically investigated. Numerical results show that there is a single defect mode inside the PBG in cylindrical photonic crystals with asymmetric geometry. On the other hand, we have found two defect modes in a symmetric one. Unlike the planner PCs, the peak heights of the defect modes are dependent on the stack number N. The fields’ distributions inside the CPCs structures for the three defects have been discussed. The mode defect in the asymmetric CPC has higher Q factor than those arises in the symmetric one. The effect of increasing the azimuthal mode number and the decrease of the inner radius is to shift the defect modes’ positions to shorter wavelengths. Our analysis on the defect modes can help designing of a narrowband transmission filter based on the CPCs. Moreover, by monitoring the wavelength of the defect arising in the asymmetric structure, the structure is candidate to work as a refractive index sensor. References [1] Kazuaki Sakoda, Optical Properties of Photonic Crystals, 1st ed., Springer-Verlag, Berlin Heidelberg, NY, 2005, pp. 1–6.
6
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar [2] [3] [4] [5] [6] [7] [8] [9] [10]
M.A. Kaliteevski, R.A. Abram, V.V. Nikolaev, G.S. Sokolovski, Bragg reflectors for cylindrical waves, J. of Modern Optics 46 (5) (1999) 875–890. J. Scheuer, A. Yariv, Optical annular resonators based on radial Bragg and photonic crystal reflectors, Opt. Express 11 (21) (2003) 2736–2746. M.-S. Chen, C.-J. Wu, T.-J. Yang, Optical properties of a superconducting annular periodic multilayer structure, Solid State Commun. 149 (2009) 1888–1893. C.-A. Hu, C.-J. Wu, T.-J. Yang, S.-L. Yang, Analysis of optical properties in cylindrical dielectric photonic crystal, Opt. Commun. 291 (2013) 424–434. M. Roussey, E. Descrovi, M. Häyrinen, A. Angelini, M. Kuittinen, S. Honkanen, One-dimensional photonic crystals with cylindrical geometry, Opt. Express 22 (22) (2014) 27236–27241. S.A. El-Naggar, Optical guidance in cylindrical photonic crystals, Optik 130 (2017) 584–588. S.A. El-Naggar, Photonic gaps in one dimensional cylindrical photonic crystal that incorporates single negative materials, Eur. Phys. J. 71 (1) (2017) 11. J. Scheuer, A. Yariv, Annular Bragg defect mode resonators, J. Opt. Soc. Am. B 20 (11) (2003) 2285–2291. M.-S. Chen, C.-J. Wu, T.-J. Yang, Narrowband reflection-and-transmission filter in an annular defective photonic crystal containing an ultrathin metallic film, Opt. Commun. 285 (2012) 3143–3149.
7
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Properties of defect modes in cylindrical photonic crystals Sahar A. El-Naggar
T
Department of Engineering Math and Physics, Faculty of Engineering, Cairo University, 12613, Giza, Egypt
ARTICLE INFO
ABSTRACT
Keywords: Cylindrical photonic crystals Defect mode Refractive index sensor
In this research, we study the properties of the defect modes that arise in the photonic band gap of a cylindrical photonic crystal (CPC) by using the transfer matrix method in the cylindrical coordinates. We consider two defective CPCs stacked in symmetric and asymmetric geometries. We examine the number of modes and their properties in the two CPCs structures. The dependencies of the modes on the azimuthal number and the inner radius of the CPC are also discussed. Numerical results show that the defect mode in the asymmetric CPCs has higher Q factor than those in the symmetric CPCs. Beside many potential optical communications and filtering applications, the structure may be a candidate for refractive index sensing with average sensitivity of 200 nm/RIU in the refractive index range from 1 to 1.6.
1. Introduction Photonic crystals (PCs) are known as regular arrays of materials with different refractive indices. The motion of photons in the PCs is analogues to that of electrons in ordinary crystals composed of a regular array of atoms. Therefore, photonic bandgaps (PBGs), which are frequency range where electromagnetic are forbidden to propagate though the PCs, are similar to electronic bandgaps that arises in ordinary crystals [1]. When a lattice defect is introduced in the PCs, the lattice defect will support an electromagnetic wave mode with a certain resonant frequency mode that exists in the PBGs. PCs are classified to one-dimensional (1D) PCs, two-dimensional and three dimensional structures. The main advantage of one-dimensional (1D) PCs is their simple structure that can be easily fabricated. Cylindrical PCs (CPCs) are type of 1D PCs with a periodic cylindrical multilayer structure. These CPCs have also received attention in recent years [2–8]. However, to the best of our knowledge, there has been little focus [9,10] on filtering properties based on CPCs. In this work, we study transverse wave (TE) propagation in defective CPCs. We use the cylindrical wave transfer matrix that has been developed in [2] to calculate the transmittance of the CPC. We consider two defective CPCs that have been arranged in symmetric and asymmetric geometries. We examine the number of modes and their properties in the two CPCs structures. We first focus our attention on the properties of the defect modes that arises in these CPCs. We observe that multiple defect modes with frequencies lying in the PBG arise. Therefore, we evaluate the quality factor (Q) and the full width at half maximum (FWHM) for these modes. We then show that these modes can be dependent on the azimuthal mode number and the starting radius. Finally, we test the possibility of using the suggested structure as a refractive index sensor. Our work may help many optical communications and filtering applications because our structure works at wavelength of 1550 nm This paper is organized as follows; in Section 2, we describe the two CPCs structure that have been addressed and we summarize the method has been used. In Section 3, we present the numerical results. Finally, the conclusions are presented in Section 4.
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163447 Received 17 June 2019; Accepted 18 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 1. A cross-sectional view in the x–y plane of a defective CPC in (a) Asymmetric (HL)N/D/(HL)N and (b) Symmetric (HL)N/D/(LH)N geometries. The defective CPC is bounded by two mediums, the inner and the outer regions of refractive indices ni and no respectively. The letter D refers to the defect layer shown in blue. The dimension in z direction is assumed to be much larger than the dimensions in x and y. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
2. Basic model and numerical methods We consider TE waves that are propagating in a defective CPCs structure. We consider two geometries shown in Fig. 1. The asymmetric geometry Inner/(HL)N/D/(HL)N/Outer, and the symmetric geometry Inner/(HL)N/D/(LH)N/Outer. In Fig. 1, layer 1 is H, layer 2 is L, layer 3 is H and so on until defect layer is introduced. For example, at N = 2, the asymmetric CPCs structure is HLHL/D/ HLHL, and the symmetric one is HLHL/D/LHLH. H and L are used to denote the materials with high refractive index and low refractive index, respectively. Each defective CPCs structure contains a defect layer D that is introduced at the middle of the structure. All layers are arranged to form the cylindrical coaxial structure shown in Fig. 1. The thicknesses of layers H, L, and D are assumed to be dH, dL, and dD, respectively. Layers H, L, D and the outer region have refractive indices nH, nL, nD and no respectively. The inner medium has a refractive index of ni and a starting radius of ρi. The electromagnetic cylindrical waves are assumed to diverge from the axis of symmetry and the fields should obey the source-free Maxwell's equations as follows, ∇×E = - jωH,
(1)
∇×H = jωE.
(2)
It is known that in the TE waves, the electric field E is along the + z direction and the three non-zero components are Ez, Hφ, and Hρ. The solution for the field Ez can be expressed as: Ez (ρ,ϕ) = [AJm(kρ)+ BYm(kρ)] exp(jmϕ),
(3)
where A and B are constants, Jm is a Bessel function, Ym is a Neumann function. k is the wave number in the medium and m is the azimuthal mode number. The azimuthal part of the magnetic field is given by, Hϕ (ρ,ϕ) = - j γ [AJ`m(kρ)+ BY`m(kρ)] exp(jmϕ),
(4)
with γ =√(ε/μ), where ε and μ are the permittivity and permeability of the medium. Note that the sign of the derivative denotes differentiation with respect to the entire argument of the function, and not just with respect to ρ. At each cylindrical surface between two different mediums, we apply the continuity condition of the tangential fields components Ez, Hφ. Therefore, a total transfer matrix can be constructed. The transfer matrix relates the fields components at the inner and outer boundaries of the CPCs. The final 2
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
form of the equations is written to define the total transfer matrix Mt as follows,
Ez H
=Mt = i
Ez H
,
(5)
= o
where ρo is the outermost radius. Note that the elements of the transfer matrix are functions of the radii unlike those are used in the planner PCs. To be able to calculate the transmittance and the reflectance, we use Hankel functions to express the fields components in the inner region and in the outer region as a sum of waves traveling in opposite directions, as follows
Ez H
= = i,o
(1)
H(1) m (k
i,o)
-j H` (1) m (k
H (2) m (k
i,o)
(2) i,o) -j H` m (k
i,o)
Ai,o , Bi,o
(6)
(2) m
where H m and H are Hankel functions of the first and the second kinds. A and B are the amplitudes of the waves. The subscripts “i” and “o” are used to refer to the inner and the outer regions respectively. We rewrite Eqs. (5) and (6) as follows,
Ai T T = 11 12 Bi T21 T22
Ao Bo
(7)
The amplitudes of the electric fields at ρ=ρi and those at ρ=ρo can be written in terms of the amplitude reflection rd and amplitude transmission td as follows, Ez )
ρ=ρi
=1+rd, Ez)
ρ=ρo
=td.
(8)
In terms of rd and td, the associated reflectance and transmittance are R = |rd|2, T = (no/ni) |td|2.
(9)
In the next section, we focus our results on the transmittance T through the CPCs. 3. Numerical results and discussion The characteristic of the defect modes in CPCs is studied in near infrared wavelength region having design wavelength λo of 1550 nm, the common wavelength in optical communications. We consider two geometries shown in Fig. 1. The asymmetric geometry Inner/(HL)N/D/(HL)N/Outer, and the symmetric geometry Inner/(HL)N/D/(LH)N/Outer. The layer with high refractive index H is titanium dioxide TiO2 (nH = 2.32), the layer with low refractive index L is aluminum oxide Al2O3 (nL = 1.63) [6]. These materials are selected because this CPC has been experimentally made [6]. All layers are quarter wavelength, therefore H, L and D have respective thickness dH =λo/4nH,dL =λo/4nL and dD =λo/4nD. In addition, both the inner and the outer regions are air. The starting radius and the defect layer D are initially taken as ρi = 100a where a = dH + dL, and air respectively. Then the effect of changing the refractive index of the defect layer and varying the starting radius are addressed. First, we investigate the number of defect modes arising in the asymmetric CPCs and the symmetric CPCs at lowest azimuthal mode (m = 0). In Fig. 2, we plot the calculated transmittance for the (HL)N/D/(HL)N and that for (HL)N/D/(LH)N at N = 12. We note that one defect mode arises in the asymmetric CPCs at the design wavelength with peak height of 0.8. We call this defect mode D0. While, the symmetric CPCs has two defect modes D1 and D2 within the PBG as shown in Fig. 2(b) at 1431.7 nm and 1689.6 nm respectively. We denote the wavelengths of the defect modes for the D0, D1 and D2 by λD0, λD1 and λD2, respectively. The transmission peaks for both D1 and D2 are 0.8. It is worth noting that the transmittance of CPCs does not reach unity like that of planner PCs. This is consistent with conservation of energy, because energy density on a cylindrical interface is inversely proportional to the radius. Therefore, the transmittance reaches a maximum value of ρi/ρo. In Fig. 2(c), the defect mode D0 is plotted for three different values of N = 8, 10 and 12. It is known that increasing the number of periods N leads to an enhancement of the quality factor Q (Δλ/
Fig. 2. Transmittance through the asymmetric CPCs, (HL)12/D/(HL)12, and through the symmetric CPCs, (HL)12/D/(LH)12, in (a) and (b) respectively. The defect mode D0 in (c) at various values of N. 3
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Table 1 Variation in FWHM, Quality Q and transmittance of CPCs with number of periods (N). D0 in Asymmetric CPC
D1 in symmetric CPC
D2 in symmetric CPC
N
T
Q
FWHM (nm)
Q
FWHM (nm)
Q
FWHM (nm)
8 10 12 14
0.855 0.827 0.8 0.775
777 3099 12289 48413
1.994 0.5002 0.1261 0.0320
519 1369 3490 8728
2.7543 1.0457 0.4101 0.1641
439 1159 2946 7451
3.8546 1.4588 0.5735 0.2267
λ) due to fields’ confinement within the defect layer. Unfortunately, in the CPC increasing N tends to scarify the peak height of the defect mode. In Table 1, the variation in FWHM, Q and transmittance peak of the three defect modes D0, D1 and D2 at different values of N are listed. We observe that the peak height of the transmittance for the three modes decreases from 0.855 to 0.775 by increasing N from 8 to 14, respectively. In addition, increasing N leads to a decrease in FWHM for the three modes. Therefore, the Q factor increases for the three defect modes D0, D1 and D2 from 777 to 48413, from 519 to 8728 and from 439 to 7451 by increasing N from 8 to 14, respectively. We conclude that D0, arising in the asymmetric CPCs, has higher Q and narrower FWHM than the two defects arising in the symmetric CPCs. Next, we study the behavior of the electric field intensity inside the CPCs to understand the reason behind the presence of two defect modes in the symmetric CPCs and investigate the differences between the three defect modes. We plot the electric field |Ez | along the radius, ρ, of the CPCs at λD0, λD1 and λD2 as shown in Fig. 3. We observe that the field is highly confined in the defect layer for D0 compared to that for D1 and D2 which explains higher Q associated with D0. In addition, comparison between the field distributions of D1 and that of D2 reveals that the symmetrical geometry gives rise to two possible field’s distributions in the defect layer at resonance. One of these distributions is like sine function which corresponds to a state in the PBG with high energy (appears at low wavelength). The other distribution is like cosine function which corresponds to a state with low energy (appears at high wavelength). 3.1. Effect of azimuthal mode number m and the starting radius ρi In this subsection, we start by studying the effect of the azimuthal mode number (m) on the defect modes arising in the asymmetric CPCs and the symmetric one. In the analysis that follows, we consider N = 12 to guarantee high Q without scarifying the peak height of transmittance. We plot the transmittance in the vicinity of λD0, λD1 and λD2 in Fig. 4(a)–(c) respectively at different four values of m = 1, 5, 9 and 13. We note that the defect mode shifts to lower wavelength by increasing m. The shift in the wavelength is up to 1.5 nm when m increases to 13. This wavelength shift originates because the field solutions of cylindrical waves depend on m. In addition, we get the expression of the radial component Hρ = −(mcEz)/(ωμρ). At m = 0, the radial component of the field equals zero. We observe that m is in the numerator which indicates that the radial component increases by increasing m. Therefore, increasing m is analogues to increasing the angle of incidence in the planner PCs as stated in [8]. In Table 2, the variation in FWHM, Q and transmittance peak of the three defect modes D0, D1 and D2 at different values of m are listed. We observe that the FWHM and Q are almost invariant by changing m. Next, we investigate the effect of the starting radius on the defect modes in CPCs at nonzero azimuthal mode number (m = 5). We plot the transmittance around λD0, λD1 and λD2 in Fig. 5(a)–(c) respectively at different four values of ρi = 200a, 75a, 50a and 25a. We note that by decreasing the starting radius, the wavelength of the defect is blue shifted. In addition, a decrease in the peak transmittance value can be observed by increasing ρ. From Table 3, which lists the properties of the modes, we note that for the three
Fig. 3. The electric field intensity distribution inside the CPCs for D0, D1 and D2 at λD0 = 1550 nm, λD1 = 1431.7 nm, and λD2 = 1689.6 nm in (a), (b) and (c) respectively. The vertical blue dotted lines indicate the boundaries of the layers. The defect layer is bounded by two solid red vertical lines. N = 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). 4
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 4. The defect modes D0, D1 and D2 at different azimuthal modes in (a), (b) and (c) respectively. Table 2 Variation in FWHM, Quality Q and transmittance of CPCs with azimuthal mode (m). D0 in Asymmetric CPCs
D1 in Symmetric CPCs
D2 in Symmetric CPCs
m
λD0 (nm)
T
Q
FWHM (nm)
λD1 (nm)
T
Q
FWHM (nm)
λD2 (nm)
T
Q
FWHM (nm)
1 5 9 13
1550 1549.8 1549.3 1548.5
0.8 0.8 0.79 0.79
12289 12394 12390 12484
0.1261 0.1250 0.1250 0.1240
1431.7 1431.6 1431.2 1430.6
0.8 0.8 0.79 0.79
3490.8 3490.4 3498.1 3505.3
0.4101 0.4101 0.4091 0.4081
1689.6 1689.3 1688.8 1687.9
0.8 0.8 0.79 0.79
2963.2 2971.5 2988.0 3031.1
0.5702 0.5685 0.5652 0.5569
Fig. 5. The defect modes D0, D1 and D2 at different starting radii in (a), (b) and (c) respectively. Table 3 Variation in FWHM, Quality Q and transmittance of CPCs with inner radius (ρi). Asymmetric CPCs
Symmetric CPCs
D0
D1
D2
ρi
λD0 (nm)
T
Q
FWHM (nm)
λD1 (nm)
T
Q
FWHM (nm)
λD2 (nm)
T
Q
FWHM (nm)
200a 75a 50a 25a
1549.9 1549.6 1549.3 1548.0
0.89 0.75 0.67 0.49
12298 12296 12391 12582
0.1260 0.1260 0.1250 0.1230
1431.7 1431.4 1431.2 1430.3
0.89 0.75 0.67 0.49
3490.7 3498.7 3498.1 3539.1
0.4101 0.4091 0.4091 0.4041
1689.5 1689.2 1688.8 1687.3
0.89 0.75 0.67 0.49
2954.5 2971.2 2996.9 3048.4
0.5719 0.5685 0.5635 0.5535
modes, the peak value decreases from 0.89 to 0.49 by decreasing the inner radius from 200a to 25a respectively. The decrease in the value of the transmittance occurs because the ratio ρi/ρo decreases by decreasing ρi. The shift in the wavelength is up to 2 nm when ρi decreases to 25a. The shift in the wavelength can be explained by the radial component of the field discussed in the previous paragraph. By observing the expression of the radial component, it is inverse proportional to the radius. Therefore, at fixed m, the decrease in the radius is associated with an increase in the radial component of the field. 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar
Fig. 6. The defect mode D0 at different values of nD. Table 4 Variation in FWHM, Quality Q, peak transmittance of asymmetric CPC, the wavelength of defect D0 (λD0), and the wavelength sensitivity S with various values of (nD). nD
λD0 (nm)
T
Q
FWHM (nm)
S (nm/RIU)
1.1 1.2 1.3 1.4 1.5 1.6
1570.1 1591.4 1613.3 1635 1655.8 1675
0.7932 0.7766 0.7575 0.7409 0.7301 0.7261
11293 9642.6 7680.6 5745.5 4094.8 2829.9
0.139 0.165 0.2101 0.2846 0.4044 0.5919
201 213 219 217 208 192
3.2. Refractive index sensing In this subsection, we test the feasibility of designing refractive index sensor based on CPCs structure. We focus our attention on studying the effect of the refractive index of the defect layer on the defect mode arising in the asymmetric CPCs, D0, because we found out that this defect has higher Q than those arise in the symmetric CPCs. We set the starting radius ρi = 100a and dD=λo/4. To investigate the performance of using our structure as a sensor, we calculate the wavelength sensitivity Sλ as follows, Sλ (nm/RIU)= Δλpeak/Δn,
(10)
where Δλpeak is the change in the wavelength where peak resonance occurs and Δn is the change in the refractive index. In Fig. 6, we plot the transmittance of the asymmetric CPCs at different values of nD. We observe that the defect’s wavelength increases from 1570.1 nm to 1675 nm by increasing nD from 1.1 to 1.6 respectively. Increasing nD leads to an increase in the resonance wavelength because the resonance wavelength depends on the effective refractive index which increases when the refractive index of the defect layer increases. In addition, the peak height is lowered down from 0.793 to 0.726 as nD increases from 1.1 to 1.6 respectively. The Q factor decreases due to an increase in the FWHM. However, an average wavelength sensitivity Sλ of 200 nm/RIU is calculated which is comparable to the earlier photonic crystal cavity lasers. The variation in FWHM, Q, peak transmittance, and the wavelength sensitivity with various values of nD are listed in Table 4. Our analysis on the defect modes provides useful information for the design of a transmission filter with high Q based on the CPCs. 4. Conclusion By using the cylindrical wave transfer matrix method, the properties of the defect modes for the asymmetric and symmetric structures of CPCs have been theoretically investigated. Numerical results show that there is a single defect mode inside the PBG in cylindrical photonic crystals with asymmetric geometry. On the other hand, we have found two defect modes in a symmetric one. Unlike the planner PCs, the peak heights of the defect modes are dependent on the stack number N. The fields’ distributions inside the CPCs structures for the three defects have been discussed. The mode defect in the asymmetric CPC has higher Q factor than those arises in the symmetric one. The effect of increasing the azimuthal mode number and the decrease of the inner radius is to shift the defect modes’ positions to shorter wavelengths. Our analysis on the defect modes can help designing of a narrowband transmission filter based on the CPCs. Moreover, by monitoring the wavelength of the defect arising in the asymmetric structure, the structure is candidate to work as a refractive index sensor. References [1] Kazuaki Sakoda, Optical Properties of Photonic Crystals, 1st ed., Springer-Verlag, Berlin Heidelberg, NY, 2005, pp. 1–6.
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Optik - International Journal for Light and Electron Optics 200 (2020) 163447
S.A. El-Naggar [2] [3] [4] [5] [6] [7] [8] [9] [10]
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