- Email: [email protected]
Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial
Superlattices and Microstructures xxx (2017) 1e8
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial Behnam Kazempour a, *, Kazem Jamshidi-Ghaleh b, Majid Shabzendeh a a b
Department of Physics, Ahar Branch, Islamic Azad University, Ahar, Iran Department of Physics, Azarbaijan Shahid Madani University, Tabriz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 March 2017 Received in revised form 22 May 2017 Accepted 22 May 2017 Available online xxx
The influence of the variation in the optical axis and the incidence angle a the P and the S polarizations on the transmittance properties of the tunable filter in a 1D photonic crystal (1DPC) with an anisotropic metamaterial defect layer is theoretically investigated. By using the 4 4 transfer matrix method, the pronounced contrast in the behavior of the P-and the S-polarized modes were demonstrated. The results indicate that, the intensity and the peak wavelength of the defect mode under the P epolarized wave can be tuned by a variation of the optical axis of the anisotropic layer, whereas the intensity and the peak wavelength of the defect mode under the S polarized wave remains unchanged. In addition, we show that the peak wavelength and the intensity of the tunable filter can be tailored by varying the incidence angle of light. The results, therefore, illustrate that the peak wavelength of the defect mode shifts towards blue as the angle of incidence increases for both polarizations. These results lead to certain new findings concerning the design of new types of tunable narrowband filters. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Photonic crystal Anisotropic metamaterial Tunable Optical axis
1. Introduction As a type of artificial material, one-dimensional photonic crystals (1DPCs) with a periodic arrangement of refractive indexes have attracted considerable attention for the past few years because of their capability to create a range of forbidden frequencies known as the photonic band gap (PBG) [1]. PBGs have many interesting and attractive applications in optical filters [2], reflectors [3], and nonlinear diodes [4], as well as in fabricating PC waveguides [5,6]. By breaking the periodicity of a usual PC structure, we have a defective crystal. It is possible to generate a transmission peak within the PBG in the frequency (or the wavelength) range; such a peak is called a defect mode, which is similar to the defect state in semiconductor solids. With the creation of a transmission peak, a defective PC can thus be designed as an optical filter [7,8]. Recently, tunable filters have been of much interest to the community because of their potential use in optical electronics and communications. A tunable filter is a filter in which the position of the transmission peak and the intensity of the defect modes can be tuned by the means of certain external agents or physical conditions such as the temperature, the electric field, the magnetic field, and so on; this changes result of the variation in the electric permittivity, the magnetic permeability, or the physical properties of the PC structure [9e11]. In addition to the conventional all-dielectric PCs and metamaterial PCs, during the past decade, anisotropic metamaterial photonic crystal have received much attention by researchers owing to their significant scientific
* Corresponding author. E-mail address: [email protected] (B. Kazempour). http://dx.doi.org/10.1016/j.spmi.2017.05.062 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
2
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
and high optical sensitivity to the parameters such as the optical axis and their dependence on the direction of propagation as well as their polarization, which, contains more desired characteristics than conventional PCs [12e17]. In the past, much attention has been paid to the properties of electromagnetic wave propagation in single-negative (SNG) materials. SNG materials, which are materials with negative permittivity or negative permeability, can possess zero-feff (zero effective phase) photonic gap [18e20]. In recent years, there have been many reports on the metamaterials that have been introduced into PCs to create multichannel optical filters that, can be tuned by the structure period or the layer thickness [21,22]. These investigations indicate that it is possible to manipulate these filters by replacing the physical or the optical parameters of the PC structure in order to fabricate the modulating devices [23,24]. Although several reports on filters have been studied using the anisotropic metamaterials, the creation of a polarization-dependent anisotropic metamaterial, the effect of the optical axis and the incidence angle on the shift trend, and the intensity of the defect mode have not been made available thus far. In this study, we are theoretically interested in investigating the transmittance properties of the tunable filter in 1DPC, which is composed of anisotropic metamaterial as a defect layer. To further this aim, we apply the 4 4 transfer matrix method for both, the S and the P polarization states of this structure and study the influence of the incidence angle and the optical axis of the anisotropic metamaterial on the tunability of the optical filter. Our numerical results show that the shift trend and the controlling intensity of the tunable filter in the proposed structure are strongly sensitive to the optical axis of the anisotropic metamaterial under the P-polarized wave; by varying the optical axis, red-shift is observed in the defect mode, which originates from the fact that the dielectric permittivity in the metamaterials can be changed by controlling the orientation of the optical axis. On the contrary, for S polarization, the wavelength position and the intensity of the tunable filter is unchanged as a function of the optical axis. In addition, the effects of the incident angle on the wavelength position and the intensity of the tunable filter will be specifically explored for both polarizations. The results suggest that an anisotropic metamaterial can be used to design the tunable filter and the shifting feature is dependent on the material. The paper is organized as the following. In Sec. 2, 1D symmetric anisotropic metamaterial PC structure and the characteristic 4 4 transfer matrix method is presented to calculate the transmittance properties of tunable filter. The numerical results and the effect of the optical axis and the incidence angle on tuning of the tunable filter are illustrated in Sec. 3. The conclusion is presented in Sec. 4.
2. Model and theory Consider a 1DPC structure consisting of anisotropic metamaterial as a defect is sandwiched between two Bragg mirrors formed by alternating two dielectric layers having arrangement of ðBAÞ5 CðBAÞ5 , as shown in Fig. 1. Here, the PC layers, B and A are taken to be SiO2 and Si .The middle C denotes the anisotropic metamaterial with arbitrary optical axis as a defect layer. Here, layers A and B with corresponding thicknesses of d1 and d2 are dielectric materials with constant optical parameters. We propose that the optic axis lies in the x z plane and makes angle 4 with the periodicity direction (z). In this case, the permittivity tensors of the anisotropic metamaterial layer is given by Refs. [25,26]:
0
ε cos2 4 þ εjj sin2 4 0 B ⊥ εC ¼ B 0 ε ⊥ @ ε⊥ εjj cos4sin4 0
1 ε⊥ εjj cos4sin4 C C 0 A ε⊥ sin2 4 þ εjj cos2 4
(1)
here, ε⊥ and εjj are the principle elements of the permittivity tensors of the layer C along the optical axis and perpendicular to the optical axis, respectively, and 4 is the angle between the optical axis and the z-axis. Consider an electromagnetic wave with frequency of u, electric and magnetic fields of E and H, respectively, incident to the structure with angle q with respect to the z-axis. The fundamental equations for an electromagnetic wave are given by the following Maxwell equations:
Fig. 1. Schematic of proposed 1DPC structure, A and B are the dielectric materials with corresponding thicknesses of d1 and d2, and C stands for defect, composed of the anisotropic metamaterial with thicknesses of d3.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
!! ! !! V E ð r ; tÞ ¼ ium0 H ð r ; tÞ !! ! !! V H ð r ; tÞ ¼ iuε0 ! ε E ð r ; tÞ
3
(2)
where ! ε is the relative permittivity tensor, which, for anisotropic metamaterial with arbitrary optical axis is described Eq. (1). In this work, the 4 4 transfer matrix method is introduced to investigate the propagation of monochromatic plane waves in birefringent layered materials and magnetic materials. All materials are assumed to be non-magnetic, therefore, in the numerical calculations; the permeability of the layers is set to unity. After solving the Maxwell equations, it will be shown that the equations for tangential components of the electric and magnetic fields has this form [26e30]:
vJ ¼ ik0 DJ vz
(3)
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi m0 Hy ; ε0 Ey ; m0 Hx Þ, and D is a coefficient matrix with where k0 ¼ uc is the vacuum wave vector, j is vector ð ε0 Ex ; elements that depend on the optical properties of the medium [7,27]. Solving Eq. (2) for anisotropic metamaterial layer (C), where the principal axes coincide with the x; y and z axes, the coefficient matrix D is given as:
0
Nx εxz εzz
1
Nx2 εzz
B B B B ε ε Nx εxz DC ¼ B B εxx xz zx B ε εzz zz B B @ 0 0 0 0
1 C C C C 0 0 C C; C C C 0 1A 0 0
εyy Nx2
(4)
0
here, Nx ¼ n0 sin q, where n0 and q are the refractive index of the incident medium and incident angle, respectively. It has been shown that the elements of the transfer matrix depend on the eigenvalues of matrix DC . After some algebraic calculation, we can find eigenvalues of DC :
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
az1;3 ¼ ± D12 D21 þ D11 D22 and az2;4 ¼ ± D43 :
(5)
The eigenvectors corresponding to the eigenvalues aiz are the elements of DC matrix. Solving Eq. (3) using the eigenvalues and eigenvectors of matrix DC shows that the tangential components of the fields through the anisotropic metamaterial layer can be transferred as follows:
Jðz þ d3 Þ ¼ MC ,JðzÞ;
(6)
For our case of having the optic axis in the x z plane, the elements of the 4 4 transfer matrix of the anisotropic layer ðMC Þ are given by:
M11 ¼
ðD11 az1 Þexpðik0 d3 az3 Þ ðD11 az3 Þexpðik0 d3 az1 Þ ; az3 az1
M12 ¼
D12 ðexpðik0 d3 az3 Þ expðik0 d3 az1 ÞÞ ; az3 az1
M21 ¼
D21 ðexpðik0 d3 az3 Þ expðik0 d3 az1 ÞÞ ; az3 az1
M22 ¼
ðD11 az1 Þexpðik0 d3 az3 Þ ðD11 az3 Þexpðik0 d3 az1 Þ ; az3 az1
M33 ¼ cosðik0 d3 az2 Þ; M34 ¼ i sinðik0 d3 az2 Þ=a ; z2 M43 ¼ i az2 sinðik0 d3 az2 Þ; M44 ¼ M33 ;
(7)
Also, the element of the transfer matrix M for A and B layers can be calculated by the same method as follows [7]: Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
4
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
M11 ¼ M22 ¼ M33 ¼ M44 ¼ cos k0 dj azj ; M12 ¼ iazj sin k0 dj azj ε ; j M21 ¼ iεj sin k0 dj azj a ; zj M34 ¼ M21 ε ; M43 ¼ εj M12 ; j
(8)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here, j ¼ A and B , azj ¼ εj N2x and dj is the thickness of the corresponding layer. The total transfer matrix representing the entire structure is given as:
2
M11 6 M21 5 5 M ¼ ðMB MA Þ MC ðMB MA Þ ¼ 4 M31 M41
M12 M22 M32 M41
M13 M23 M33 M41
3 M14 M24 7 : M34 5 M41
(9)
After a series of algebraic calculations applying the boundary conditions to the interfaces of the layers, the relation between the incident, reflection, and transmission fields can be expressed as [29]:
0
1 2 EPi B Er C 6 B P C¼4 @ Ei A S ESr
M11 M21 M31 M41
M12 M22 M32 M41
M13 M23 M33 M41
30 t M14 EP M24 7B 0 B M34 5@ ESr M41 0
1 C C; A
(10)
where EPi , EPr , ESi , ESr , EPt and ESr denote the complex amplitudes of the S and P polarization of the incident, reflected, and transmitted waves, respectively. For example, the corresponding transmittances are the transmission coefficients of the S and P polarized waves are:
tP ¼
tS ¼
EPt
!
EPi ESt ESi
¼
M33 M11 M33 M13 M31
¼
M11 : M11 M33 M13 M31
ESi ¼0
! EPi ¼0
(11)
3. Results and discussions In the following calculations, the refractive indexes of isotropic dielectric layers A and B is assumed to be nA ¼ 3:7, nB ¼ 1:5 and the corresponding thickness are dA ¼ 90nm, dB ¼ 180nm; respectively. The layers A and B refer to SiO2 and Si, respectively [31]. However, the anisotropic metamaterial discussed here has been experimentally reported from GHz to optical frequencies [32,33]. For the anisotropic metamaterial, we assume that has losses factor and used εjj ¼ 2 þ 0:001i, ε⊥ ¼
Fig. 2. Transmittance spectra of proposed 1DPC structure at five different orientation optical axis, for the normal incidence of the wave under P polarized wave.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
5
2 þ 0:01i in the anisotropic metamaterials [25]. At first, we discussed the transmission properties of the1DPCs with an anisotropic metamaterial as a defect layer for five different values of the optical axis of the anisotropic layer under the P polarized wave and for the normal angle of incidence, as shown in Fig. 2. It is evident from Fig. 2 that, in the case of q ¼ 0+ , there is a peak of transmission on the defect mode at l ¼ 1:223mm. As observable in the figure that the tunable filters are shifted toward the higher wavelengths (red shift) as the optical axis (4) increases. Also, we can conclude that owing to increase in the optical axis in our considered structure, the intensity transmittance of the tunable filter increases. In addition it is clear from this figure that there is a good enhancement for the intensity transmittance with an increase in the optical axis and the transmittance at 4 ¼ 60 is about 100 for wavelength l ¼ 1:357mm. The controlling of the tunable filter is strongly related to the intrinsically anisotropic properties in metamaterials, whose orientation of the optical axis gives rise to a variation in the effective permittivity, which leads to a direct change in the optical path. Subsequently, the wavelength position of the defect mode will be eventually changed with the optical axis. A key factor producing tunability is the optical axis; the lower (upper) limit of the red shift is determined by the optical axis at 4 ¼ 0 (4 ¼ 90 ). The dependence of the peak wavelength of the defect mode on the optical axis is further illustrated in Fig. 3. With a variation of the optical axis from 4 ¼ 0 to 4 ¼ 90 the peak wavelength of the defect mode is displaced from l ¼ 1:223mm to l ¼ 1:427mm , and 204nm wide shifts can be obtained. As observable, the peaks wavelength of the defect mode shift slightly to higher wavelength (red shift) as the orientation of the optical axis increases. Next, we change the incidence angle under the P polarized wave and observe how the tunable filter and the PBG structure can be changed. The result is shown in Fig. 4 for different incidence angles q ¼ 0 , 45 and 60 . The other parameters are same as those in Fig. 4 for a specified value of the optical axis 4 ¼ 60 . As illustrated, by increasing the incidence angle, the wavelength of the defect mode and the PBG are shifted towards the lower wavelengths (blue shift); under this circumstance. We see that in addition to the shift in the PBG, the width of the PBG is compressed as the incidence angle increases. At larger values of q, we can see that there is a further difference between the wavelengths of transmittance peaks, at smaller angles of incidence, the structure has high transmission. The dependence of the peak wavelength on the angle of incidence for P polarizations is further illustrated in Fig. 5. As observable, by increasing the angle of incidence, the peak wavelength of the defect modes is blue-shifted. Moreover, the degree of shift in the peak wavelength has no sensible change at smaller angles (less than 20 ) whereas the trend blue shift in the peak wavelength of the defect mode is shifted considerably by varying the higher angle of incidence. The results indicate that the optical filter can also be tuned by varying the angle of incidence. Interestingly, a tuning range up to 122nm could still be achieved when the angle of incidence changed from 0 to 85 . Under the same conditions, we would like to investigate how the tuning of the defect mode is affected by the optical axis under the S polarized wave. In Fig. 6, we show the transmittance for the structure in the S polarized wave at four different values of the optical axis: 4 ¼ 0 ; 30 ; 45 and 60 , respectively. The results show that the wavelength position, the intensity of the defect mode and the PBGs remain unchanged as the 4-value changes; these results are attributed to the nature of the dielectric tensor in anisotropic metamaterials. It is clear that the optical parameters of the component tensor metamaterial depend on the parameter 4 such as εxx ; εxz and εzz , while εyy remains unchanged as the optical axis is varied. The determined parameters in the anisotropic metamaterials for the P-polarized wave are εxx ; εxz and εzz , which leads to the corresponding tunability. However, the component εyy is responsible for the optical properties for S-polarized wave, i.e., the independence on the change of the optical axis. Finally, to show the effect of the incidence angle (q) on the tunability of the optical filter and the PBGs of the proposed structure, under the S polarized wave is plotted in Fig. 7, for different incidence angles q ¼ 0 , 30 and 60 . The other parameters are same as those in Fig. 4, at a specified value of the optical axis 4 ¼ 60 . As can be seen in Fig. 7, by increasing the
Fig. 3. Dependence of peak wavelength of the defect mode as a function of the optical axis for the case of normal incident angle, under P polarized wave.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
6
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
Fig. 4. Transmittance spectra of proposed 1DPC structure at three different incident angles, for a specified value of optical axis 4 ¼ 60 , under P polarized wave.
Fig. 5. Dependence of peak wavelengths on the incidence angle under P polarized wave, for 4 ¼ 60 .
Fig. 6. Calculated transmittance spectra of proposed 1DPC structure at four different orientation optical axis, for the normal incidence of the wave under S polarized wave.
incidence angle, the wavelength of the defect mode and the PBG are shifted toward the lower wavelengths (blue shift). It is of notable interest that the intensity of defect mode decreases as the angle of incidence increases. The dependence of the peak wavelength on the angle of incidence under the S polarization is further illustrated in Fig. 8. By increasing the angle of incidence, the peak wavelength of the defect modes is blue-shifted for this polarization. We can, therefore, conclude that the normal incidence of light is a better choice for obtaining high transmission. Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
7
Fig. 7. Transmittance spectra for the tunable filter structure at different angles of incidence, angles q ¼ 0 , 30 and 60 , respectively, under the S polarized wave.
Fig. 8. Dependence of peak wavelengths of the defect mode on the incidence angle under S polarized wave, at 4 ¼ 60 .
4. Conclusions We investigated the optical properties of the tunable filter in 1DPCs containing an anisotropic metamaterial defect layer with an arbitrary optical axis at an oblique incidence of light by using the 4 4 Transfer matrix method. We found that a large shifting as well as a high transmission of the tunable filter can be simultaneously achieved in our proposed structure. The influence of variation in the optical axis and the incidence angle on the transmittance properties of the tunable filter have been investigated. In the P polarized wave, it has been shown that the defect mode is red shifted as a function of the optical axis of the anisotropic metamaterial and the intensity of the defect mode can be enhanced by increasing the optical axis. Also, the results indicate that for S polarized wave, the position wavelength and the intensity of the defect mode remains unchanged as the optical axis changes. In addition, in the oblique incidence, the peak wavelength of the defect mode and the PBGs are blue- shifted as the angle of incidence increases for the P and the S polarized wave, and the intensity of the defect mode and the width of the PBGs decreases and becomes compressed with increase in the incidence angle. The analysis of the properties of the defect mode is informative to the design of a tunable transmission filter, which is of technical use in photonic applications. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Modeling the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [2] T. Hattori, N. Tsurumachi, H. Nakatsuka, Analysis of optical nonlinearity by effect states in one-dimensional photonic crystals, J. Opt. Soc. Am. B 14 (1997) 348e355. [3] D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Observation of total omnidirectional reflection from a one-dimensional dielectric lattice, Appl. Phys. A 68 (1999) 25e28. [4] M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, C.M. Bowden, Thin-film nonlinear optical diode, Appl. Phys. Lett. 66 (1995) 2324e2326.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
8
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
€ sele, V. Sandoghdar, Highly directional emission from photonic crystal [5] P. Kramper, M. Agio, C.M. Soukoulis, A. Birner, F. Müller, R.B. Wehrspohn, U. Go waveguides of subwavelength width, Phys. Rev. Lett. 92 (2004) 113903e113907. [6] Y.-H. Chang, Y.-Y. Jhu, C.-J. Wu, Temperature dependence of defect mode in a defective photonic crystal, Opt. Commun. 285 (2012) 1501e1504. [7] K. Jamshidi- Ghaleh, B. Kazempour, External tunability of optical filter in symmetric one-dimensional photonic crystals containing ferroelectric and ITO material defect, Optik 127 (2016) 10626e10631. [8] K. Jamshidi- Ghaleh, F. Bayat, Generating frequency dependant twisted beam shapes using 1DPC nanostructure with graded-index defect layer, Opt. Lett. 39 (2014) 3802e3805. [9] Q. Zhu, Y. Zhang, Defect modes and wavelength tuning of one-dimensional photonic crystal with lithium Niobate, Optik 120 (2009) 195e198. [10] X. Hu, Q. Gong, S. Feng, B. Cheng, D. Zhang, Tunable multichannel filter in nonlinear ferroelectric photonic crystal, Opt. Commun. 253 (2005) 138e144. [11] F. Moslemi, K. Jamshidi- Ghaleh, Electrically tunable optical bistability based on one-dimensional photonic crystals with nonlinear nanocomposite materials, J. Appl. Phys. 119 (2016) 093101e093106. [12] E. Cojocaru, Birefringence dispersion in Solc-type anisotropic periodic band gap structures, Appl. Opt. 40 (2004) 1089e1097. [13] V.I. Kopp, R. Bose, A.Z. Genack, Transmission through chiral twist defects in anisotropic periodic structures, Opt. Lett. 28 (2003) 349e351. [14] N. Ouchani, D. Bria, B. Djafari-Rouhani, A. Nougaoui, Transverse -electric/transverse -magnetic polarization converter using 1D finite biaxial photonic crystal, J. Opt. Soc. Am. A 24 (2007) 2710e2718. [15] K. Jamshidi- Ghaleh, B. Kazemour, A. Phirouznia, Electrically tunable polarization splitting and conversion based on 1DPC structure with anisotropic defect layer, Superlattices Microstruct. 101 (2017) 109e116. [16] J. Zhang, H. Chen, Y. Luo, Y. Yuan, L. Shen, L. Ran, J. Kong, Wide band backward coupling based on anisotropic left-handed metamaterial, Appl. Phys. Lett. 90 (2007) 043506. [17] T. Pan, G.-D. Xu, T.-C. Zang, L. Gao, Study of a slab waveguide loaded with dispersive anisotropic metamaterials, Appl. Phys. A 95 (2009) 367e372. [18] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat- Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000) 4184e4187. [19] R.A. Shelby, D.R. Smith, S. Schultz, Experimental verification of a negative index of refraction, Science 292 (2001) 77e79. [20] X. Hu, Z. Liu, Q. Gong, Tunable multichannel filter in photonic crystal heterostructure containing permeability-negative materials, Phys. Lett. A 372 (2008) 333e339. [21] X. Hu, Z. Liu, Q. Gong, A multichannel filter in a photonic crystal heterostructure containing single-negative material, J. Opt. A Pure Appl. Opt. 9 (2007) 877e883. [22] B. Thubthimthong, F. Chollet, Design and simulation of a tunable photonic band gap filter, Microelectron. Eng. 85 (2008) 1421e1424. [23] J.W. Liu, T.W. Chang, C.J. Wu, Filtering properties of photonic crystal dual-channel tunable filter containing superconducting defects, J. Super. Conduct. Nov. Mag. 27 (2013) 67e72. [24] D. Park, S. Kim, I. Park, H. Lim, Higher order optical resonant filters based on coupled defect resonators in photonic crystals IEEE, J. Light Wave Tech. 23 (2005) 1923e1928. [25] G. Zheng, Abrupt change of reflectivity from the strongly anisotropic metamaterial, Opt. Commun. 281 (2008) 1941e1944. [26] A. Yariv, P. Yeh, Optical Waves in Crystals Propagation and Control of Laser Radiation, Wiley, 1983. [27] I. Abdulhalim, Analytic propagation matrix method for linear optics of arbitrary biaxial layered media, J. Opt. A 1 (1999) 646e653. [28] T. Pan, G.-D. Xu, T.-C. Zang, L. Gao, Goose H€ anchen shift in one-dimensional photonic crystals containing uniaxial indefinite medium, Phys. Status Solidi B 246 (2009) 1088. [29] R. Srivastava, K.B. Thapa, S. Pati, S.P. Ojha, Negative refraction in 1D photonic crystals, Solid State.Commun. 147 (2008) 157e161. [30] P. Han, C.T. Chan, Z.Q. Zhang, Wave localization in one-dimensional random structures composed of single-negative metamaterials, Phys. Rev. B 77 (2008) 115332. [31] H.-Y. Lee, H. Makino, T. Yao, A. Tanaka, Si-based omnidirectional reflector and transmission filter optimized at a wavelength of 1.55 mm, Appl. Phys. Lett. 81 (2002) 4502e4504. [32] H. Luo, Z. Ren, W. Shu, F. Li, Wave propagation in an anisotropic metamaterial with single sheeted hyperboloid dispersion relation, Appl. Phys. A 87 (2007) 245e249. [33] N.-H. Shen, S. Foteinopoulou, M. Kafesaki, T. Koschny, E. Ozbay, E.N. Economou, C.M. Soukoulis, Compact planar far- fields upper lens based on anisotropic left- handed metamaterials, Phys. Rev. B 80 (2009) 115123.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial Behnam Kazempour a, *, Kazem Jamshidi-Ghaleh b, Majid Shabzendeh a a b
Department of Physics, Ahar Branch, Islamic Azad University, Ahar, Iran Department of Physics, Azarbaijan Shahid Madani University, Tabriz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 March 2017 Received in revised form 22 May 2017 Accepted 22 May 2017 Available online xxx
The influence of the variation in the optical axis and the incidence angle a the P and the S polarizations on the transmittance properties of the tunable filter in a 1D photonic crystal (1DPC) with an anisotropic metamaterial defect layer is theoretically investigated. By using the 4 4 transfer matrix method, the pronounced contrast in the behavior of the P-and the S-polarized modes were demonstrated. The results indicate that, the intensity and the peak wavelength of the defect mode under the P epolarized wave can be tuned by a variation of the optical axis of the anisotropic layer, whereas the intensity and the peak wavelength of the defect mode under the S polarized wave remains unchanged. In addition, we show that the peak wavelength and the intensity of the tunable filter can be tailored by varying the incidence angle of light. The results, therefore, illustrate that the peak wavelength of the defect mode shifts towards blue as the angle of incidence increases for both polarizations. These results lead to certain new findings concerning the design of new types of tunable narrowband filters. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Photonic crystal Anisotropic metamaterial Tunable Optical axis
1. Introduction As a type of artificial material, one-dimensional photonic crystals (1DPCs) with a periodic arrangement of refractive indexes have attracted considerable attention for the past few years because of their capability to create a range of forbidden frequencies known as the photonic band gap (PBG) [1]. PBGs have many interesting and attractive applications in optical filters [2], reflectors [3], and nonlinear diodes [4], as well as in fabricating PC waveguides [5,6]. By breaking the periodicity of a usual PC structure, we have a defective crystal. It is possible to generate a transmission peak within the PBG in the frequency (or the wavelength) range; such a peak is called a defect mode, which is similar to the defect state in semiconductor solids. With the creation of a transmission peak, a defective PC can thus be designed as an optical filter [7,8]. Recently, tunable filters have been of much interest to the community because of their potential use in optical electronics and communications. A tunable filter is a filter in which the position of the transmission peak and the intensity of the defect modes can be tuned by the means of certain external agents or physical conditions such as the temperature, the electric field, the magnetic field, and so on; this changes result of the variation in the electric permittivity, the magnetic permeability, or the physical properties of the PC structure [9e11]. In addition to the conventional all-dielectric PCs and metamaterial PCs, during the past decade, anisotropic metamaterial photonic crystal have received much attention by researchers owing to their significant scientific
* Corresponding author. E-mail address: [email protected] (B. Kazempour). http://dx.doi.org/10.1016/j.spmi.2017.05.062 0749-6036/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
2
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
and high optical sensitivity to the parameters such as the optical axis and their dependence on the direction of propagation as well as their polarization, which, contains more desired characteristics than conventional PCs [12e17]. In the past, much attention has been paid to the properties of electromagnetic wave propagation in single-negative (SNG) materials. SNG materials, which are materials with negative permittivity or negative permeability, can possess zero-feff (zero effective phase) photonic gap [18e20]. In recent years, there have been many reports on the metamaterials that have been introduced into PCs to create multichannel optical filters that, can be tuned by the structure period or the layer thickness [21,22]. These investigations indicate that it is possible to manipulate these filters by replacing the physical or the optical parameters of the PC structure in order to fabricate the modulating devices [23,24]. Although several reports on filters have been studied using the anisotropic metamaterials, the creation of a polarization-dependent anisotropic metamaterial, the effect of the optical axis and the incidence angle on the shift trend, and the intensity of the defect mode have not been made available thus far. In this study, we are theoretically interested in investigating the transmittance properties of the tunable filter in 1DPC, which is composed of anisotropic metamaterial as a defect layer. To further this aim, we apply the 4 4 transfer matrix method for both, the S and the P polarization states of this structure and study the influence of the incidence angle and the optical axis of the anisotropic metamaterial on the tunability of the optical filter. Our numerical results show that the shift trend and the controlling intensity of the tunable filter in the proposed structure are strongly sensitive to the optical axis of the anisotropic metamaterial under the P-polarized wave; by varying the optical axis, red-shift is observed in the defect mode, which originates from the fact that the dielectric permittivity in the metamaterials can be changed by controlling the orientation of the optical axis. On the contrary, for S polarization, the wavelength position and the intensity of the tunable filter is unchanged as a function of the optical axis. In addition, the effects of the incident angle on the wavelength position and the intensity of the tunable filter will be specifically explored for both polarizations. The results suggest that an anisotropic metamaterial can be used to design the tunable filter and the shifting feature is dependent on the material. The paper is organized as the following. In Sec. 2, 1D symmetric anisotropic metamaterial PC structure and the characteristic 4 4 transfer matrix method is presented to calculate the transmittance properties of tunable filter. The numerical results and the effect of the optical axis and the incidence angle on tuning of the tunable filter are illustrated in Sec. 3. The conclusion is presented in Sec. 4.
2. Model and theory Consider a 1DPC structure consisting of anisotropic metamaterial as a defect is sandwiched between two Bragg mirrors formed by alternating two dielectric layers having arrangement of ðBAÞ5 CðBAÞ5 , as shown in Fig. 1. Here, the PC layers, B and A are taken to be SiO2 and Si .The middle C denotes the anisotropic metamaterial with arbitrary optical axis as a defect layer. Here, layers A and B with corresponding thicknesses of d1 and d2 are dielectric materials with constant optical parameters. We propose that the optic axis lies in the x z plane and makes angle 4 with the periodicity direction (z). In this case, the permittivity tensors of the anisotropic metamaterial layer is given by Refs. [25,26]:
0
ε cos2 4 þ εjj sin2 4 0 B ⊥ εC ¼ B 0 ε ⊥ @ ε⊥ εjj cos4sin4 0
1 ε⊥ εjj cos4sin4 C C 0 A ε⊥ sin2 4 þ εjj cos2 4
(1)
here, ε⊥ and εjj are the principle elements of the permittivity tensors of the layer C along the optical axis and perpendicular to the optical axis, respectively, and 4 is the angle between the optical axis and the z-axis. Consider an electromagnetic wave with frequency of u, electric and magnetic fields of E and H, respectively, incident to the structure with angle q with respect to the z-axis. The fundamental equations for an electromagnetic wave are given by the following Maxwell equations:
Fig. 1. Schematic of proposed 1DPC structure, A and B are the dielectric materials with corresponding thicknesses of d1 and d2, and C stands for defect, composed of the anisotropic metamaterial with thicknesses of d3.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
!! ! !! V E ð r ; tÞ ¼ ium0 H ð r ; tÞ !! ! !! V H ð r ; tÞ ¼ iuε0 ! ε E ð r ; tÞ
3
(2)
where ! ε is the relative permittivity tensor, which, for anisotropic metamaterial with arbitrary optical axis is described Eq. (1). In this work, the 4 4 transfer matrix method is introduced to investigate the propagation of monochromatic plane waves in birefringent layered materials and magnetic materials. All materials are assumed to be non-magnetic, therefore, in the numerical calculations; the permeability of the layers is set to unity. After solving the Maxwell equations, it will be shown that the equations for tangential components of the electric and magnetic fields has this form [26e30]:
vJ ¼ ik0 DJ vz
(3)
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi m0 Hy ; ε0 Ey ; m0 Hx Þ, and D is a coefficient matrix with where k0 ¼ uc is the vacuum wave vector, j is vector ð ε0 Ex ; elements that depend on the optical properties of the medium [7,27]. Solving Eq. (2) for anisotropic metamaterial layer (C), where the principal axes coincide with the x; y and z axes, the coefficient matrix D is given as:
0
Nx εxz εzz
1
Nx2 εzz
B B B B ε ε Nx εxz DC ¼ B B εxx xz zx B ε εzz zz B B @ 0 0 0 0
1 C C C C 0 0 C C; C C C 0 1A 0 0
εyy Nx2
(4)
0
here, Nx ¼ n0 sin q, where n0 and q are the refractive index of the incident medium and incident angle, respectively. It has been shown that the elements of the transfer matrix depend on the eigenvalues of matrix DC . After some algebraic calculation, we can find eigenvalues of DC :
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
az1;3 ¼ ± D12 D21 þ D11 D22 and az2;4 ¼ ± D43 :
(5)
The eigenvectors corresponding to the eigenvalues aiz are the elements of DC matrix. Solving Eq. (3) using the eigenvalues and eigenvectors of matrix DC shows that the tangential components of the fields through the anisotropic metamaterial layer can be transferred as follows:
Jðz þ d3 Þ ¼ MC ,JðzÞ;
(6)
For our case of having the optic axis in the x z plane, the elements of the 4 4 transfer matrix of the anisotropic layer ðMC Þ are given by:
M11 ¼
ðD11 az1 Þexpðik0 d3 az3 Þ ðD11 az3 Þexpðik0 d3 az1 Þ ; az3 az1
M12 ¼
D12 ðexpðik0 d3 az3 Þ expðik0 d3 az1 ÞÞ ; az3 az1
M21 ¼
D21 ðexpðik0 d3 az3 Þ expðik0 d3 az1 ÞÞ ; az3 az1
M22 ¼
ðD11 az1 Þexpðik0 d3 az3 Þ ðD11 az3 Þexpðik0 d3 az1 Þ ; az3 az1
M33 ¼ cosðik0 d3 az2 Þ; M34 ¼ i sinðik0 d3 az2 Þ=a ; z2 M43 ¼ i az2 sinðik0 d3 az2 Þ; M44 ¼ M33 ;
(7)
Also, the element of the transfer matrix M for A and B layers can be calculated by the same method as follows [7]: Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
4
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
M11 ¼ M22 ¼ M33 ¼ M44 ¼ cos k0 dj azj ; M12 ¼ iazj sin k0 dj azj ε ; j M21 ¼ iεj sin k0 dj azj a ; zj M34 ¼ M21 ε ; M43 ¼ εj M12 ; j
(8)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here, j ¼ A and B , azj ¼ εj N2x and dj is the thickness of the corresponding layer. The total transfer matrix representing the entire structure is given as:
2
M11 6 M21 5 5 M ¼ ðMB MA Þ MC ðMB MA Þ ¼ 4 M31 M41
M12 M22 M32 M41
M13 M23 M33 M41
3 M14 M24 7 : M34 5 M41
(9)
After a series of algebraic calculations applying the boundary conditions to the interfaces of the layers, the relation between the incident, reflection, and transmission fields can be expressed as [29]:
0
1 2 EPi B Er C 6 B P C¼4 @ Ei A S ESr
M11 M21 M31 M41
M12 M22 M32 M41
M13 M23 M33 M41
30 t M14 EP M24 7B 0 B M34 5@ ESr M41 0
1 C C; A
(10)
where EPi , EPr , ESi , ESr , EPt and ESr denote the complex amplitudes of the S and P polarization of the incident, reflected, and transmitted waves, respectively. For example, the corresponding transmittances are the transmission coefficients of the S and P polarized waves are:
tP ¼
tS ¼
EPt
!
EPi ESt ESi
¼
M33 M11 M33 M13 M31
¼
M11 : M11 M33 M13 M31
ESi ¼0
! EPi ¼0
(11)
3. Results and discussions In the following calculations, the refractive indexes of isotropic dielectric layers A and B is assumed to be nA ¼ 3:7, nB ¼ 1:5 and the corresponding thickness are dA ¼ 90nm, dB ¼ 180nm; respectively. The layers A and B refer to SiO2 and Si, respectively [31]. However, the anisotropic metamaterial discussed here has been experimentally reported from GHz to optical frequencies [32,33]. For the anisotropic metamaterial, we assume that has losses factor and used εjj ¼ 2 þ 0:001i, ε⊥ ¼
Fig. 2. Transmittance spectra of proposed 1DPC structure at five different orientation optical axis, for the normal incidence of the wave under P polarized wave.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
5
2 þ 0:01i in the anisotropic metamaterials [25]. At first, we discussed the transmission properties of the1DPCs with an anisotropic metamaterial as a defect layer for five different values of the optical axis of the anisotropic layer under the P polarized wave and for the normal angle of incidence, as shown in Fig. 2. It is evident from Fig. 2 that, in the case of q ¼ 0+ , there is a peak of transmission on the defect mode at l ¼ 1:223mm. As observable in the figure that the tunable filters are shifted toward the higher wavelengths (red shift) as the optical axis (4) increases. Also, we can conclude that owing to increase in the optical axis in our considered structure, the intensity transmittance of the tunable filter increases. In addition it is clear from this figure that there is a good enhancement for the intensity transmittance with an increase in the optical axis and the transmittance at 4 ¼ 60 is about 100 for wavelength l ¼ 1:357mm. The controlling of the tunable filter is strongly related to the intrinsically anisotropic properties in metamaterials, whose orientation of the optical axis gives rise to a variation in the effective permittivity, which leads to a direct change in the optical path. Subsequently, the wavelength position of the defect mode will be eventually changed with the optical axis. A key factor producing tunability is the optical axis; the lower (upper) limit of the red shift is determined by the optical axis at 4 ¼ 0 (4 ¼ 90 ). The dependence of the peak wavelength of the defect mode on the optical axis is further illustrated in Fig. 3. With a variation of the optical axis from 4 ¼ 0 to 4 ¼ 90 the peak wavelength of the defect mode is displaced from l ¼ 1:223mm to l ¼ 1:427mm , and 204nm wide shifts can be obtained. As observable, the peaks wavelength of the defect mode shift slightly to higher wavelength (red shift) as the orientation of the optical axis increases. Next, we change the incidence angle under the P polarized wave and observe how the tunable filter and the PBG structure can be changed. The result is shown in Fig. 4 for different incidence angles q ¼ 0 , 45 and 60 . The other parameters are same as those in Fig. 4 for a specified value of the optical axis 4 ¼ 60 . As illustrated, by increasing the incidence angle, the wavelength of the defect mode and the PBG are shifted towards the lower wavelengths (blue shift); under this circumstance. We see that in addition to the shift in the PBG, the width of the PBG is compressed as the incidence angle increases. At larger values of q, we can see that there is a further difference between the wavelengths of transmittance peaks, at smaller angles of incidence, the structure has high transmission. The dependence of the peak wavelength on the angle of incidence for P polarizations is further illustrated in Fig. 5. As observable, by increasing the angle of incidence, the peak wavelength of the defect modes is blue-shifted. Moreover, the degree of shift in the peak wavelength has no sensible change at smaller angles (less than 20 ) whereas the trend blue shift in the peak wavelength of the defect mode is shifted considerably by varying the higher angle of incidence. The results indicate that the optical filter can also be tuned by varying the angle of incidence. Interestingly, a tuning range up to 122nm could still be achieved when the angle of incidence changed from 0 to 85 . Under the same conditions, we would like to investigate how the tuning of the defect mode is affected by the optical axis under the S polarized wave. In Fig. 6, we show the transmittance for the structure in the S polarized wave at four different values of the optical axis: 4 ¼ 0 ; 30 ; 45 and 60 , respectively. The results show that the wavelength position, the intensity of the defect mode and the PBGs remain unchanged as the 4-value changes; these results are attributed to the nature of the dielectric tensor in anisotropic metamaterials. It is clear that the optical parameters of the component tensor metamaterial depend on the parameter 4 such as εxx ; εxz and εzz , while εyy remains unchanged as the optical axis is varied. The determined parameters in the anisotropic metamaterials for the P-polarized wave are εxx ; εxz and εzz , which leads to the corresponding tunability. However, the component εyy is responsible for the optical properties for S-polarized wave, i.e., the independence on the change of the optical axis. Finally, to show the effect of the incidence angle (q) on the tunability of the optical filter and the PBGs of the proposed structure, under the S polarized wave is plotted in Fig. 7, for different incidence angles q ¼ 0 , 30 and 60 . The other parameters are same as those in Fig. 4, at a specified value of the optical axis 4 ¼ 60 . As can be seen in Fig. 7, by increasing the
Fig. 3. Dependence of peak wavelength of the defect mode as a function of the optical axis for the case of normal incident angle, under P polarized wave.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
6
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
Fig. 4. Transmittance spectra of proposed 1DPC structure at three different incident angles, for a specified value of optical axis 4 ¼ 60 , under P polarized wave.
Fig. 5. Dependence of peak wavelengths on the incidence angle under P polarized wave, for 4 ¼ 60 .
Fig. 6. Calculated transmittance spectra of proposed 1DPC structure at four different orientation optical axis, for the normal incidence of the wave under S polarized wave.
incidence angle, the wavelength of the defect mode and the PBG are shifted toward the lower wavelengths (blue shift). It is of notable interest that the intensity of defect mode decreases as the angle of incidence increases. The dependence of the peak wavelength on the angle of incidence under the S polarization is further illustrated in Fig. 8. By increasing the angle of incidence, the peak wavelength of the defect modes is blue-shifted for this polarization. We can, therefore, conclude that the normal incidence of light is a better choice for obtaining high transmission. Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
7
Fig. 7. Transmittance spectra for the tunable filter structure at different angles of incidence, angles q ¼ 0 , 30 and 60 , respectively, under the S polarized wave.
Fig. 8. Dependence of peak wavelengths of the defect mode on the incidence angle under S polarized wave, at 4 ¼ 60 .
4. Conclusions We investigated the optical properties of the tunable filter in 1DPCs containing an anisotropic metamaterial defect layer with an arbitrary optical axis at an oblique incidence of light by using the 4 4 Transfer matrix method. We found that a large shifting as well as a high transmission of the tunable filter can be simultaneously achieved in our proposed structure. The influence of variation in the optical axis and the incidence angle on the transmittance properties of the tunable filter have been investigated. In the P polarized wave, it has been shown that the defect mode is red shifted as a function of the optical axis of the anisotropic metamaterial and the intensity of the defect mode can be enhanced by increasing the optical axis. Also, the results indicate that for S polarized wave, the position wavelength and the intensity of the defect mode remains unchanged as the optical axis changes. In addition, in the oblique incidence, the peak wavelength of the defect mode and the PBGs are blue- shifted as the angle of incidence increases for the P and the S polarized wave, and the intensity of the defect mode and the width of the PBGs decreases and becomes compressed with increase in the incidence angle. The analysis of the properties of the defect mode is informative to the design of a tunable transmission filter, which is of technical use in photonic applications. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Modeling the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [2] T. Hattori, N. Tsurumachi, H. Nakatsuka, Analysis of optical nonlinearity by effect states in one-dimensional photonic crystals, J. Opt. Soc. Am. B 14 (1997) 348e355. [3] D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Observation of total omnidirectional reflection from a one-dimensional dielectric lattice, Appl. Phys. A 68 (1999) 25e28. [4] M.D. Tocci, M.J. Bloemer, M. Scalora, J.P. Dowling, C.M. Bowden, Thin-film nonlinear optical diode, Appl. Phys. Lett. 66 (1995) 2324e2326.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062
8
B. Kazempour et al. / Superlattices and Microstructures xxx (2017) 1e8
€ sele, V. Sandoghdar, Highly directional emission from photonic crystal [5] P. Kramper, M. Agio, C.M. Soukoulis, A. Birner, F. Müller, R.B. Wehrspohn, U. Go waveguides of subwavelength width, Phys. Rev. Lett. 92 (2004) 113903e113907. [6] Y.-H. Chang, Y.-Y. Jhu, C.-J. Wu, Temperature dependence of defect mode in a defective photonic crystal, Opt. Commun. 285 (2012) 1501e1504. [7] K. Jamshidi- Ghaleh, B. Kazempour, External tunability of optical filter in symmetric one-dimensional photonic crystals containing ferroelectric and ITO material defect, Optik 127 (2016) 10626e10631. [8] K. Jamshidi- Ghaleh, F. Bayat, Generating frequency dependant twisted beam shapes using 1DPC nanostructure with graded-index defect layer, Opt. Lett. 39 (2014) 3802e3805. [9] Q. Zhu, Y. Zhang, Defect modes and wavelength tuning of one-dimensional photonic crystal with lithium Niobate, Optik 120 (2009) 195e198. [10] X. Hu, Q. Gong, S. Feng, B. Cheng, D. Zhang, Tunable multichannel filter in nonlinear ferroelectric photonic crystal, Opt. Commun. 253 (2005) 138e144. [11] F. Moslemi, K. Jamshidi- Ghaleh, Electrically tunable optical bistability based on one-dimensional photonic crystals with nonlinear nanocomposite materials, J. Appl. Phys. 119 (2016) 093101e093106. [12] E. Cojocaru, Birefringence dispersion in Solc-type anisotropic periodic band gap structures, Appl. Opt. 40 (2004) 1089e1097. [13] V.I. Kopp, R. Bose, A.Z. Genack, Transmission through chiral twist defects in anisotropic periodic structures, Opt. Lett. 28 (2003) 349e351. [14] N. Ouchani, D. Bria, B. Djafari-Rouhani, A. Nougaoui, Transverse -electric/transverse -magnetic polarization converter using 1D finite biaxial photonic crystal, J. Opt. Soc. Am. A 24 (2007) 2710e2718. [15] K. Jamshidi- Ghaleh, B. Kazemour, A. Phirouznia, Electrically tunable polarization splitting and conversion based on 1DPC structure with anisotropic defect layer, Superlattices Microstruct. 101 (2017) 109e116. [16] J. Zhang, H. Chen, Y. Luo, Y. Yuan, L. Shen, L. Ran, J. Kong, Wide band backward coupling based on anisotropic left-handed metamaterial, Appl. Phys. Lett. 90 (2007) 043506. [17] T. Pan, G.-D. Xu, T.-C. Zang, L. Gao, Study of a slab waveguide loaded with dispersive anisotropic metamaterials, Appl. Phys. A 95 (2009) 367e372. [18] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat- Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000) 4184e4187. [19] R.A. Shelby, D.R. Smith, S. Schultz, Experimental verification of a negative index of refraction, Science 292 (2001) 77e79. [20] X. Hu, Z. Liu, Q. Gong, Tunable multichannel filter in photonic crystal heterostructure containing permeability-negative materials, Phys. Lett. A 372 (2008) 333e339. [21] X. Hu, Z. Liu, Q. Gong, A multichannel filter in a photonic crystal heterostructure containing single-negative material, J. Opt. A Pure Appl. Opt. 9 (2007) 877e883. [22] B. Thubthimthong, F. Chollet, Design and simulation of a tunable photonic band gap filter, Microelectron. Eng. 85 (2008) 1421e1424. [23] J.W. Liu, T.W. Chang, C.J. Wu, Filtering properties of photonic crystal dual-channel tunable filter containing superconducting defects, J. Super. Conduct. Nov. Mag. 27 (2013) 67e72. [24] D. Park, S. Kim, I. Park, H. Lim, Higher order optical resonant filters based on coupled defect resonators in photonic crystals IEEE, J. Light Wave Tech. 23 (2005) 1923e1928. [25] G. Zheng, Abrupt change of reflectivity from the strongly anisotropic metamaterial, Opt. Commun. 281 (2008) 1941e1944. [26] A. Yariv, P. Yeh, Optical Waves in Crystals Propagation and Control of Laser Radiation, Wiley, 1983. [27] I. Abdulhalim, Analytic propagation matrix method for linear optics of arbitrary biaxial layered media, J. Opt. A 1 (1999) 646e653. [28] T. Pan, G.-D. Xu, T.-C. Zang, L. Gao, Goose H€ anchen shift in one-dimensional photonic crystals containing uniaxial indefinite medium, Phys. Status Solidi B 246 (2009) 1088. [29] R. Srivastava, K.B. Thapa, S. Pati, S.P. Ojha, Negative refraction in 1D photonic crystals, Solid State.Commun. 147 (2008) 157e161. [30] P. Han, C.T. Chan, Z.Q. Zhang, Wave localization in one-dimensional random structures composed of single-negative metamaterials, Phys. Rev. B 77 (2008) 115332. [31] H.-Y. Lee, H. Makino, T. Yao, A. Tanaka, Si-based omnidirectional reflector and transmission filter optimized at a wavelength of 1.55 mm, Appl. Phys. Lett. 81 (2002) 4502e4504. [32] H. Luo, Z. Ren, W. Shu, F. Li, Wave propagation in an anisotropic metamaterial with single sheeted hyperboloid dispersion relation, Appl. Phys. A 87 (2007) 245e249. [33] N.-H. Shen, S. Foteinopoulou, M. Kafesaki, T. Koschny, E. Ozbay, E.N. Economou, C.M. Soukoulis, Compact planar far- fields upper lens based on anisotropic left- handed metamaterials, Phys. Rev. B 80 (2009) 115123.
Please cite this article in press as: B. Kazempour et al., Transmittance properties of tunable filter in a 1D photonic crystal doped by an anisotropic metamaterial, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.05.062