- Email: [email protected]

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Tunable complete photonic band gap in anisotropic photonic crystal slabs with non-circular air holes using liquid crystals T. Fathollahi Khalkhali n, A. Bananej Laser and Optics Research School, NSTRI, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 8 December 2015 Received in revised form 11 January 2016 Accepted 16 February 2016

In this study, we analyze the tunability of complete photonic band gap of square and triangular photonic crystal slabs composed of square and hexagonal air holes in anisotropic tellurium background with SiO2 as cladding material. The non-circular holes are inﬁltrated with liquid crystal. Using the supercell method based on plane wave expansion, we study the variation of complete band gap by changing the optical axis orientation of liquid crystal. Our numerical results show that noticeable tunability of complete photonic band gap can be obtained in both square and triangular structures with non-circular holes. & 2016 Elsevier B.V. All rights reserved.

Keywords: Photonic crystal slabs Tunable complete photonic band gap Anisotropic tellurium material Liquid crystals Non-circular scatterers

1. Introduction Photonic crystals (PCs) have been one of the major subjects of research in physics and engineering, due to their special and applicable properties. Some of these features are, controlling and manipulating light ﬂow, designing of optical devices with wavelength and subwavelength dimensions and many other peculiar phenomena interacted with light [1–5]. The most prominent property of PCs is photonic band gap (PBG), a region of frequency spectrum where propagating modes are forbidden [6]. PBG giving rise to physical phenomena such as inhibited spontaneous emission [7] and light localization [8]. PCs are mainly studied as photonic crystal slabs,due to the ease manufacturing technology compared to three-dimensional (3D) PCs and the ability to control light propagation in three dimensions [9,10]. PC slabs are 2D periodic structure of ﬁnite thickness that light is conﬁned by a PBG in-plane, and by total internal reﬂection (TIR) in the vertical or out-plane direction. In PC slabs the modes are not purely TE (transverse electric) or TM (transverse magnetic) modes, but according to the symmetry in PC slabs, the propagation modes can be classiﬁed in two types of even modes (TE-like) and odd modes (TM-like) [11–13]. An complete PBG exist for PC slabs only when PBG in both polarization modes are present and they overlap each other in frequency [14]. There are several reasons for which a n

Corresponding author. E-mail addresses: [email protected], [email protected] (T. Fathollahi Khalkhali). http://dx.doi.org/10.1016/j.optcom.2016.02.039 0030-4018/& 2016 Elsevier B.V. All rights reserved.

complete PBG would be a desirable feature. For instant, coupling between modes of opposite symmetry is possible in real structures, due to fabrication intrinsic imperfections, or it can be useful to support localized modes for both polarizations of light and the waveguides based on such defects can guide light of both polarizations as well. Thus, existence of a complete PBG, independent of mode symmetry, is very important for practical application. [15,16] Fathollahi khalkhali et al. [17] have recently demonstrated that square and triangular-lattice PC slabs, created by square and hexagonal air holes in anisotropic Tellurium (Te) background surrounded by SiO2 as cladding material, represents complete wide PBG. In recent years, there has been much interest on tuning the optical properties of PBG structures in order to design switchable or dynamical devices. In 1999, Bush and John proposed that by inﬁltrating three dimensional (3D) PCs with liquid crystal (LC), an applied electric ﬁeld would tune the PBG [18]. Following this publication, some investigations of band gap tunability have been done by utilizing LCs in one-dimensional (1D) [19–22], two-dimensional (2D) [23–29], and 3D [30–32] PCs. Therefore, in this letter, we consider PC slabs with square and triangular lattice composed of air hole with different geometrical shape (square and hexagonal) in anisotropic Te background where the regions above and below the slabs are occupied by SiO2. Then, we inﬁltrate the non-circular air holes with LC and study the tuning of complete photonic band gap by changing the director of LC.

80

T. Fathollahi Khalkhali, A. Bananej / Optics Communications 369 (2016) 79–83

2. Structures and computational methods In this study, we use square and triangular lattice created by square and hexagonal air holes in anisotropic Te background, surrounded by SiO2 (nSiO2 = 1.45) as cladding material, considering that the air holes are inﬁltrated with LC. The structure under consideration are shown schematically in Fig. 1. The orientation of non-circular air holes relative to the lattice axis is deﬁned by angle Θ , which typically shown in Fig. 2 for square lattice of hexagonal air holes. The anisotropic Te has two different principle refractive indices as ordinary-refractive index noTe = 4.8 and extraordinaryrefractive index neTe = 6.2 over the wavelength range of 4.50 − 6.25 μm with an absorption coefﬁcient of α E1 cm 1 [17]. We assume that the periodicity of the PC slab is in the X − Y plane and the extraordinary axis of Te is considered parallel to the Z -axis. Generally, LCs possesses two kinds of dielectric constants known as ordinary εo and extraordinary εe dielectric constants. The light waves with electric ﬁeld perpendicular and parallel to the director of LC experience ordinary and extraordinary dielectric constants, respectively. When the director of LC rotates, the components of the dielectric tensor can be represented as [33]:

→ εxx ( r ) = εo + (εe − εo) sin2 (θ ) cos2 (ϕ)

(1)

→ εyy ( r ) = εo + (εe − εo) sin2 (θ ) sin2 (ϕ)

(2)

→ → εxy ( r ) = εyx ( r ) = (εe − εo) sin2 (θ ) sin (ϕ) cos (ϕ)

(3)

→ → εxz ( r ) = εzx ( r ) = (εe − εo) sin (θ ) cos (θ ) cos (ϕ)

Fig. 2. The rotated hexagonal air holes in a square lattice with angle Θ , which is deﬁned as the angle between axis of the hexagonal air holes cross section and the lattice axis.

→ → εyz ( r ) = εzy ( r ) = (εe − εo) sin (θ ) cos (θ ) sin (ϕ)

(5)

(4)

Fig. 1. Schematic representation of PC slab structures with (a and b) square lattice and (d and e) triangular lattice of LC-inﬁltrated air holes in anisotropic Te background surrounded by SiO2.

T. Fathollahi Khalkhali, A. Bananej / Optics Communications 369 (2016) 79–83

Fig. 3. Schematic representation of rotation angles for LCs director.

→ εzz ( r ) = εo + (εe − εo) sin2 (θ )

(6)

81

square lattice PC slab of square air holes in Te background with SiO2 as cladding material [17]. The numerical results have been shown that this structure represents a complete PBG with normalized width of Δωmax = 0.0 410 (2πc /a) and gap-midgap ratio of ωr = 13.19% at optimum parameters r = 0.38a (half-side length of square air holes), h = 0.51a (slab thickness) and Θ = 30∘ (rotation angle of square air holes). The gap-midgap ratio is deﬁned as ωr ≡ Δωmax /ωg , where ωg is the center frequency of the PBG. This band gap lies between ω1 = 0.2888 and ω2 = 0.3298 frequencies in units of 2πc /a . With a comparison can be found that, in square lattice PC slabs with air holes composed of isotropic material, the gap-midgap ratio is about 9.0% [15] while the gap-midgap ratio of same lattice with anisotropic material is about 13.1% , which shows good improvement compared to conventional PC slabs. Now, we consider that the square air holes in mentioned structure are ﬁlled with LC as shown in Fig. 1(a). In this case, we study the variation of complete band gap. To obtain the maximum tunability of complete band gap, we change the director of LC for different values of half-side length of hole (r ), rotation angle of square holes (Θ ) and slab thickness (h) by varying the θ and ϕ angels from 0∘ to 90∘ and from 0° to 180°, respectively. Our investigation shows that the maximum tunability of 0.0 405 (2πc /a) is obtained for complete band gap at r = 0.39a , h = 0.51a , Θ = 30∘ and ϕ = 0∘ when θ changes from 0∘ to 90∘ . The gap-midgap ratio of complete band gaps also have been studied. Fig. 4(a) and (b) shows the variation of the complete band gap width and gap-midgap ratio as a function of θ and depicts the band spectrum of the structure, corresponding to mentioned

where θ is the zenith angle of the LC director (i.e. the angle between the LC director and the Z-axis), ϕ is the azimuth angle between the projection of the LC director on the X − Y plane and the → X-axis, and n is the director of the LC, as shown in Fig. 3. In the mention structures we use LCs with ordinary refractive index of noLC = 1.590 and extraordinary refractive index of neLC = 2.223. This LC corresponds to the phenylacetylene type LC [34] In this paper, the projected band structures of PC slabs are calculated by plane wave expansion (PWE) method using the MIT Photonic-Bands (MPB) package [35]. We use the super cell approach based on PWE for such calculations, with the assumption that the periodicity of the PC slabs are in the X − Y plane. Since there is no periodicity in the Z -direction, we add sufﬁcient amount of cladding material in the Z -direction to original ﬁnite height cell to reduce the effect of boundaries on the results (supercell is 1 1 4, 4 lattice periods in the vertical direction).

3. Results and discussions In this section we study the mentioned structures (Fig. 1(a)– (d)). Our main goal here is to investigate the modiﬁcation of complete band gap spectrum by adjusting the geometrical parameters of the mentioned PC slabs ( r , h and Θ ) and changing the orientation of LC optical axis. We choose LC as an anisotropic material for the convenient change of anisotropy by simply changing the orientation of LC molecules using an externally applied static electric ﬁeld. The variation of complete PBG of these two structure (square and triangular lattice) will be separately analyzed in the next two subsections. 3.1. Square lattice In this subsection, the results for the square lattice are presented. Recently, Fathollahi khalkhali et al. have investigated the

Fig. 4. (a) Gap width and gap-midgap ratio as a function of θ (b) band structure of even modes (dashed line) and odd modes (solid line) at θ = 0°, in square lattice slab with LC-inﬁltrated square air holes, for r = 0.39a , h = 0.51a , Θ = 30∘ and ϕ = 0∘ .

82

T. Fathollahi Khalkhali, A. Bananej / Optics Communications 369 (2016) 79–83

Fig. 5. (a) Gap width and gap-midgap ratio as a function of θ (b) band structure of even modes (dashed line) and odd modes (solid line) at θ = 0° , in square lattice slab with LC-inﬁltrated hexagonal air holes, for h = 0.55a , r = 0.46a , Θ = 15∘ and ϕ = 0∘ .

parameters, at θ = 0°. It has been reported that the square lattice PC slab of hexagonal air holes in anisotropic Te background, with SiO2 as cladding material, has a complete PBG with normalized width of Δωmax = 0.0 340 (2πc /a) and gap-midgap ratio of ωr = 11.29% at optimum parameters r = 0.45a (side length of hexagonal air holes), h = 0.50a (slab thickness) and Θ = 15∘ (rotation angle of hexagonal air holes). This band gap lies between ω1 = 0.2849 and ω2 = 0.3180 frequencies in units of (2πc /a) [17]. Now similar to the previous case, we study the modiﬁcation of band gap width of anisotropic PC slab with square lattice of hexagonal air holes where the holes are inﬁltrated with LC. The tuning of complete band gap is performed by changing the parameters r , h, Θ and θ , ϕ . Comprehensive investigations show that the maximum tunability of 0.0 159 (2πc /a) is obtained for complete band gap at h = 0.55a , r = 0.46a , Θ = 15∘ and ϕ = 0∘ when θ changes from 0∘ to 90∘ . Fig. 5(a) shows the variation of the complete band gap width and gap-midgap ratio as a function of θ . Also Fig. 5(b) shows the corresponding photonic band structure at θ = 0°. The above numerical results show that when the air holes are inﬁltrated with LC due to the refractive index contrast reduction, the band gaps become narrower compared to their counterparts with unﬁlled air holes. We know that for square lattice the largest complete PBG is produced by the holes with same symmetry (square holes) [36]. Therefore, it is expected that, the square hole lattice shows larger band gap and greater tunability for which our numerical results, prove it. It can be seen that in the case of square holes the

Fig. 6. (a) Gap width and gap-midgap ratio as a function of θ (b) band structure of even modes (dashed line) and odd modes (solid line) at θ = 0° , in triangular lattice slab with LC-inﬁltrated hexagonal air holes, for h = 0.80a , r = 0.52a , Θ = 15∘ and ϕ = 0∘ .

tunability has been increased compared to hexagonal holes structure and previously studied square lattice structures with LCinﬁltrated circular holes [37]. From gap-midgap ratio of square holes, it is understood that the variation of band gaps are appeared in lower frequencies compared to that of hexagonal and circular holes, which is an advantageous in reducing the leakage of light from the slabs in the vertical direction. It should also be noted that in the case of square holes, the complete PBG has closed when θ changes from 0∘ to 90∘ which can be considered as an advantage in switchable devices. 3.2. Triangular lattice In this subsection we discuss the case of triangular lattice. Investigations reveal that, the triangular lattice PC slab of hexagonal air holes in Te background, with SiO2 as cladding material, has a complete PBG with maximum normalized width of Δωmax = 0.0 552 (2πc /a) and gap-midgap ratio of ωr = 16.35% [17]. This PBG is achieved at optimum parameters: r = 0.52a (side length of hexagonal air holes), h = 0.80a (slab thickness) and Θ = 15∘ (rotation angle of hexagonal air holes). This band gap lies between ω1 = 0.3100 and ω2 = 0.3652 frequencies in units of (2πc /a). Now, by ﬁlling the hexagonal air holes with LC, we evaluate the modiﬁcation of complete band gap by changing the director of LC. In this case, ﬁve parameters ( r , h, Θ and θ , ϕ ) are treated as adjustable parameters to obtain the maximum tunability. The

T. Fathollahi Khalkhali, A. Bananej / Optics Communications 369 (2016) 79–83

83

increased. So, the variation of ϕ is important for the case in which the width of PBG is noticeable at θ = 90° [29,37]. In our structures, calculations show that by varying θ from 0∘ to 90∘ the width of band gaps are reduced remarkably and the dependency of band gaps to ϕ are negligible. Thus we limit ϕ to zero in this work.

4. Conclusion Using the supercell method based on PWE, we have studied the variation of complete band gap of square and triangular PC slabs created by LC-inﬁltrated square and hexagonal air holes in anisotropic Te background which is surrounded by SiO2. Numerical results show that, the square lattice represent noticeable tunability of complete band gap for optimum values of non-circular air holes parameters by changing the optical axis orientation of LC. The present results can be useful in designing realistic LC-inﬁltrated photonic crystal ﬁbers and switchable devices.

References

Fig. 7. (a) Gap width and gap-midgap ratio as a function of θ (b) band structure of even modes (dashed line) and odd modes (solid line) at θ = 0° , in triangular lattice slab with LC-inﬁltrated square air holes, for h = 0.80a , r = 0.40a , Θ = 0∘ and ϕ = 0∘ .

numerical results show that the maximum tunability of 0.0 178 (2πc /a) is obtained for complete band gap at h = 0.80a , r = 0.52a , Θ = 15∘ and ϕ = 0∘ when θ changes from 0∘ to 90∘ . The variation of complete PBG and gap-midgap ratio versus θ and dispersion relations of the structure at θ = 0° are shown in Fig. 6. Fig. 1(d) shows the fourth structure to be analyzed; the PC slab with triangular lattice composed of LC-inﬁltrated square air holes in anisotropic Te background where the slab surrounded by SiO2. We ﬁrst consider the triangular lattice PC slab of square air holes (without LC material) in Te background with SiO2 as cladding. It has been reported, the mentioned structure represents a complete PBG with normalized width of Δωmax = 0.0 232 (2πc /a) and gapmidgap ratio of ωr = 9.03% at optimum parameters r = 0.37a (halfside length of square air holes), h = 0.80a (slab thickness) and Θ = 0∘ (rotation angle of square air holes) [17]. Now, the square air holes are inﬁltrated with LC. Similar to previous cases, the maximum tunability of complete band gap is obtained by changing the director of LC. It has been found that, the maximum tunability of 0.0100 (2πc /a) has been obtained for complete band gap at h = 0.80a , r = 0.40a , Θ = 0∘ and ϕ = 0∘ by varying the angle θ , as shown in Fig. 7(a). Also Fig.7(b) shows the corresponding photonic band spectrum at θ = 0°. The numerical results show that, for the triangular lattice, the largest PBG and tunability is produced by the holes with the same symmetry (hexagonal holes). It should be noted that, at θ = 0° PBG of PCs composed of LCinﬁltrated holes in dielectric background is independent of ϕ and by changing θ from 0∘ to 90∘ , the dependency to ϕ is slowly

[1] K. Inoue, K. Ohtaka, Photonic Crystals: Fabrication and Applications, Springer, Berlin, 2004. [2] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nat. (Lond.) 386 (1997) 143. [3] E. Ozbay, I. Bulu, K. Aydin, H. Caglayan, K. Guven, Photonics Nano Fund Appl. 2 (87) (2004). [4] Y.A. Vlasov, M. O’Boyle, H.F. Hamann, S.J. McNab, Nature 438 (2005) 65. [5] P. Kanakis, T. Kamalakis, T. Sphicopoulos, Opt. Lett. 40 (2015) 1041. [6] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nature 386 (1997) 143. [7] D.L. Bullock, C.C. Shih, R.S. Margulies, J. Opt. Soc. Am. B 10 (1993) 399. [8] J.C. Knight, J. Broeng, T.A. Birks, P. St., J. Russell, Science 282 (1998) 1476. [9] S. Fan, P.R. Villeneuve, J.D. Joannopoulos, E.F. Schubert, Phys. Rev. Lett. 78 (1997) 3294. [10] M. Boroditsky, T.F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, E. Yablonovitch, Appl. Phys. Lett. 75 (1999) 1036. [11] S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, L.A. Kolodziejski, Phys. Rev. B 60 (1999) 5751. [12] M. Qiu, Phys. Rev. B 66 (2002) 033103. [13] S. Shi, C. Chen, D.W. Prather, Appl. Phys. Lett. 86 (2005) 043104. [14] S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, S. Noda, Appl. Phys. Lett. 87 (2005) 061107. [15] C.G. Bostan, R.M. de Ridder, J. Optoelectron. Adv. Mater. 4 (2002) 921. [16] F. Wen, S. David, X. Checoury, M. El Kurdi, P. Boucaud, Opt. Express 16 (2008) 12278. [17] T. Fathollahi Khalkhali, B. Rezaei, A. Soltani Vala, M. Kalaﬁ, Appl. Opt. 52 (2013) 3745–3752. [18] K. Busch, S. John, Phys. Rev. Lett. 83 (1999) 967. [19] K.H. Young, Y.-C. Yang, J.-E. Kim, H.Y. Park, C.-S. Kee, H. Lim, J.-C. Lee, Appl. Phys. Lett. 79 (2001) 15. [20] R. Ozaki, T. Matsui, M. Ozaki, K. Yoshino, Appl. Phys. Lett. 82 (2003) 3593. [21] G. Pucker, A. Mezzetti, M. Crivellari, P. Bellutti, A. Lui, N. Daldosso, L. Pavesi, J. Appl. Phys. 95 (2004) 767. [22] G. Alagappan, X.W. Sun, P. Shum, M.B. Yu, M.T. Doan, J. Opt. Soc. Am. B 23 (2006) 159. [23] S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, Phys. Rev. B 61 (2000) R2389. [24] C.-S. Kee, H. Lim, Y.-K. Ha, J.-E. Kim, H.Y. Park, Phys. Rev. B 64 (2001) 085114. [25] T.T. Larsen, A. Bjarklev, D.S. Hermann, J. Broeng, Opt. Express 11 (2003) 2589. [26] C.-Y. Liu, L.-W. Chen, Phys. Rev. B 72 (2005) 045133. [27] J.A. Reyes, J.A. Reyes-Avendano, P. Halevi, Opt. Commun. 281 (2008) 2535. [28] B. Rezaei, M. Kalaﬁ, Opt. Commun. 282 (2009) 1584. [29] B. Rezaei, T. Fathollahi Khalkhali, M. Kalaﬁ, Opt. Commun. 284 (2011) 813. [30] K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, Appl. Phys. Lett. 75 (1999) 932. [31] D. Kang, J.E. Maclennan, N.A. Clark, A.A. Zakhidov, R.H. Baughman, Phys. Rev. Lett. 86 (2001) 4052. [32] H. Takeda, K. Yoshino, J. Appl. Phys. 92 (2002) 5658. [33] I.-C. Khoo, S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals, World Scientiﬁc, Singapore, 1993. [34] H. Takeda, K. Yoshino, Phys. Rev. E 70 (2004) 026601. [35] S.G. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis, Opt. Express 8 (2000) 173–190 〈http://abinitio.mit.edu/mpb/〉. [36] Rongzhou Wang, et al., J. Appl. Phys. 90 (2001) 4307. [37] T. Fathollahi Khalkhali, B. Rezaei, A.H. Ramezani, Opt. Commun. 285 (2012) 5254.

Copyright © 2023 C.COEK.INFO. All rights reserved.