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Tunable out-of-plane band gap of two-dimensional anisotropic photonic crystals infiltrated with liquid crystals
Optics Communications 284 (2011) 813–817
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / o p t c o m
Tunable out-of-plane band gap of two-dimensional anisotropic photonic crystals infiltrated with liquid crystals B. Rezaei a,⁎, T. Fathollahi Khalkhali a,b, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 12 June 2010 Received in revised form 30 August 2010 Accepted 1 October 2010 Keywords: Tunable out-of-plane bandgap Anisotropic tellurium Liquid crystal Plane wave method
a b s t r a c t We analyze the tunability of out-of-plane band gap in two-dimensional photonic crystals created by square and triangular lattices of air holes in anisotropic tellurium background, considering that the rods are infiltrated with liquid crystal. Using the plane wave expansion method, we study the variation of out-of-plane band gap by changing the optical axis orientation of liquid crystal. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PCs) are periodic dielectric materials designed to affect the propagation of electromagnetic (EM) waves in the same way as the periodic potential in semiconductor crystals affects the electron motion by defining allowed and forbidden energy bands. Since first proposed by John [1] and Yablonovitch [2], their potential scientific and technological applications have inspired great interest among researchers. PCs offer an important opportunity to design new optical devices and hold a great potential for many significant applications, such as semiconductor lasers and solar cells, high quality resonator and filters, and optical fibers. Thus, it is quite essential to investigate the propagation of EM waves in PC structures. Most of the studies have been focused on the propagation of EM waves within the plane of periodicity. Recently, however, several studies have been reported about the wave propagation out of this plane [3–10] which is very important in the designing of PC fibers (PCFs) [11]. In the last decade there has been an increasing emphasis on tuning the optical properties of photonic bandgap structures in order to design switchable or dynamical devices. Busch and John [12] predicted the tunability of in-plane band gap in three dimensional (3D) PCs by utilizing liquid crystals (LCs). Following this publication, some investigations of the band gap tunability have been done by utilizing LCs in one-dimensional (1D) [13–16], two-dimensional (2D) [17–23] and 3D [24–26] PCs.
⁎ Corresponding author. Tel.: +98 4113393027; fax: +98 4113347050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.011
Presently, there are some investigations on LC-infiltrated PCFs [27,28]. Recently, the out-of-plane band structure of LC-infiltrated triangular 2D PC made of air holes in isotropic silica background has been reported using the full-vectorial FEM based eigenvalue algorithm [29]. Since, the use of anisotropic tellurium leads to an enlargement of the photonic band gap [30,31], so, in this paper we have considered 2D square and triangular PCs created by air holes in anisotropic tellurium background. Based on well known plane wave expansion method (PWEM), we study the evolution of out-of-plane band gap by changing the directors of LCs that are infiltrated throughout the air holes. 2. Formulation To determine the photonic band gap in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell's equations using the PWEM as illustrated in several papers [32–34]. Following the discussion of Busch and John [12], the Maxwell's equation for the magnetic field in 2D PCs utilizing LCs can be expressed as: 2 h i ω −1 → → ∇ × εˆ r ∇×H r = 2 H → r c
ð1Þ
where, ω is the frequency of light and c is the light velocity. The → → → dielectric tensor εˆ r = εˆ r + R is periodic with respect to the real → space lattice vector R and we can use Bloch's theorem to expand −1 → εˆ r as a sum of plane waves: −1
εˆ i; j
→ → → iG :r → r = ∑ ηi; j G e ; → G
ði; j = x; y; zÞ
ð2Þ
814
B. Rezaei et al. / Optics Communications 284 (2011) 813–817
→ → where η G is the Fourier transform of the εˆ −1 r , and plays a key role in the determination of photonic band structure. Generally LCs possess two dielectric constants known as ordinary dielectric constant εo and extraordinary dielectric constant εe. 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 Ref. [35]: e o o 2 2 → ε xx r = ε + ε −ε sin ðθÞ cos ðφÞ
ð3Þ
e o o 2 2 → ε yy r = ε + ε −ε sin ðθÞ sin ðφÞ
ð4Þ
e o 2 → → ε xy r = ε yx r = ε −ε sin ðθÞ sinðϕÞ cosðφÞ
ð5Þ
e o → → εxz r = εzx r = ε −ε sinðθÞ cosðθÞ cosðφÞ
ð6Þ
e o → → ε yz r = εzy r = ε −ε sinðθÞ cosðθÞ sinðφÞ
ð7Þ
e o o 2 → ε zz r = ε + ε −ε sin ðθÞ
ð8Þ
→ λ=1 G
∑ H→ →′ →′ G
G; G
h→′
G ;1 h→′ G ;2
!
2
ω = 2 c
h→
!
G;1 h→ G; 2
ð10Þ
where
H→
→ G ; G′ =
→ → 2 ′ ′ → → → →′ 6 eˆ 2 ⋅η G − G ⋅ eˆ 2 j k + G j j k + G j4 → → ′ ′ −ˆe1 ⋅η G − G ⋅ eˆ 2
→ → 3 −ˆe2 ⋅η G − G ′ ⋅ eˆ 1′ 7 5 → → eˆ1 ⋅η G − G ′ ⋅ eˆ 1′
ð11Þ To use Eq. (10) for calculating the out-of-plane photonic band and eˆ2;→ to be perpendicular and structure of 2D PCs, we define eˆ1;→ G G → → parallel to the plane made by k + G and Z axis, respectively, as shown in Fig. 2. According to Fig. 2, the components of unit vectors eˆ → and 1; G eˆ → are as below in the XYZ coordinate: 2; G
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 XY plane and the X axis, and n is the director of the LC, as shown in Fig. 1. Using Bloch's theorem, we can expand the magnetic field in terms of plane waves in the same way: → → → 2 i k + G :r → →e ˆ →e H r = ∑ ∑ hG;λ λ;G
following linear matrix equation for the dispersion of electromagnetic waves [5]
ð9Þ
→ where k is the Bloch vector and the first sum runs over a finite subset of lattice vectors of the reciprocal lattice. eˆ λ;→ ðλ = 1; 2Þ are unit G → → vectors of two transverse magnetic waves perpendicular to k + G are the corresponding two transverse components of the and h→ G;λ
eˆ1 = ðsin α; − cos α; 0Þ ;
eˆ2 = ðcos β cos α; cos β sin α; − sin βÞ ð12Þ
→ → where β is the angle between the k + G and the Z axis, α is the angle → → → → between the projection of the k + G on the XY plane (i.e. k + G p and the X axis. → is real and symmetric; thus the dispersion reThe matrix H → G; G ′ lation of electromagnetic waves can be solved by using standard diagonalization techniques for the linear system of Eq. (10). The main numerical problem in obtaining the eigenvalues is the evaluation of the Fourier coefficients of the inverse dielectric tensor in Eq. (11). The best method is to calculate the matrix of Fourier coefficients of real space tensors and then take its inverse to obtain the required Fourier coefficients, which was shown by Ho, Chan, and Soukoulis [36]. Since, the numerical implementation of infinite–dimensional matrix in a computer with a finite memory is impossible, thus the
Þ
magnetic field. Substituting Eqs. (2) and (9) into Eq. (1), we obtain the
Fig. 1. Schematic representation of rotation angles for the LC directors.
Fig. 2. Out-of-plane wave vector and unit vectors of two transverse magnetic waves.
B. Rezaei et al. / Optics Communications 284 (2011) 813–817
815
Fig. 3. Photonic crystal structures and the corresponding first Brillouin zones for LC-infiltrated (a) square and (b) triangular lattices of air holes in anisotropic tellurium background.
→ → matrix ε G − G ′ is first truncated → →at finite dimension and then inverted to obtain the matrix η G− G ′ as an approximate method. Therefore, the obtained spectrum by this approximate method will not be the same in general. To overcome this problem, in the actual numerical calculation of photonic bands, we consider the large number of reciprocal lattice vectors in Eq. (10) to obtain acceptable convergence. The reader who is interested in this problem should refer to Refs. [37,38]. The structures under consideration and the corresponding first Brillouin zones are depicted in Fig. 3. We have considered (a) square and (b) triangular structures of air holes in anisotropic tellurium background, considering that the holes are infiltrated with LC. The anisotropic tellurium has two different principle refractive indices as ordinary refractive index nTe o = 4.8 and extraordinary refractive index nTe e = 6.2 over the wavelength of 4.5–6.25 μm with an absorption coefficient of α ≃ 1 cm− 1 [39], in which the extraordinary one is parallel to the Z axis. We assumed that the LC with ordinary refractive LC index nLC o = 1.59 and the extraordinary refractive index ne = 2.223. This LC corresponds to the phenylacetylene type LC [40]. Parameter ρ denotes the radius of the air hole and a is being the lattice constant.
3. Numerical results For this study, we consider 2D square and triangular PCs of air holes in anisotropic tellurium background, considering that the holes are infiltrated with LC as shown in Fig. 3. We choose the LCs 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 components of the dielectric tensor of the LC are given by Eqs. (3)–(8). The band structure of 2D PCs is obtained numerically by solving Eq. (10). A total of 441 plane waves were employed for both structures in these calculations. This number is known to be sufficient enough to obtain acceptable convergence for characterizing the out-of-plane band structures. Our main goal here is to study the modification of the band gap spectrum and the value of the out-of-plane band gap by varying LCs optical axis orientation as well as when the radius of air hole varies. The out-of-plane band structures of these two kinds of lattices will be separately analyzed in the next two subsections. 3.1. Square lattice We first consider a 2D PC of square lattice composed of LCinfiltrated air holes in anisotropic tellurium background, as illustrated
Fig. 4. Out-of-plane photonic band structure for square lattice of air holes with radius ρ = 0.47a in tellurium background for kz = 1.4(2πc/a).
Fig. 5. Variation of out-of-plane photonic band gap width in terms of kz in square lattice of air holes with radius ρ = 0.47a.
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B. Rezaei et al. / Optics Communications 284 (2011) 813–817
Fig. 6. Variation of out-of-plane band gap in terms of φ for square lattice of LCinfiltrated air holes in tellurium background at ρ = 0.495a, kz = 1(2π/a) and θ = 90°.
Fig. 8. Out-of-plane photonic band structure for triangular lattice of air holes with radius ρ = 0.47a in tellurium background for kz = 0.2(2πc/a).
in Fig. 3(a). The corresponding first Brillouin zone (BZ) is also depicted in Fig. 3(a). For isotropic PCs, the photonic band structure can be → obtained by simply considering all the wave vectors k in the irreducible BZ (IBZ) which is enclosed by points Γ, X, and M as shown in Fig. 3(a). The resulting photonic band structure obtained from this IBZ for isotropic PCs based on the rotation and reflection symmetry of the structure. In a PC made of anisotropic materials, the definition of the IBZ depends on the orientation of the principal axes of anisotropic material [41] and hence it may be difficult to define the IBZ for each optic axis orientation. So, it is reasonable to consider the more symmetry directions in the first BZ for obtaining the complete photonic band structure. First we consider the case when there is no LC (air holes). Numerical results show that, for the square lattice of air holes with radius lower than ρ = 0.43a, the width of out-of-plane photonic band gap is very small. The calculated out-of-plane photonic band structure for the square lattice of air holes with radius ρ = 0.47a is shown in Fig. 4 for the out-of-plane component of wave vector kz = 1.4(2π/a), where kp is the projection of the wave vector on the XY plane, which has been taken along the Γ − X′ − M − X − M′ − X″ − Γ path of the first BZ. In this figure an out-of-plane photonic band gap with normalized width of Δω = 0.0365(2πc/a) is created between the first and second band. We have also studied the photonic band structure for other values of kz. Fig. 5 shows the variation of out-of-plane photonic band gap width in terms of kz in the square lattice of air holes with radius ρ = 0.47a.
It can be seen that the band gap exists for a wide range of kz and increases as kz increases. At next step, the air holes are infiltrated with LC. In this case, we study the variation of out-of-plane band gap by changing the director of LC. To obtain the maximum tunability of out-of-plane band gap, we change the director of LCs for different values of rod radius ρ and kz. In this structure four parameters θ, φ, kz and ρ are treated as adjustable parameters to obtain the maximum tunability of the out-of-plane band gap. Extensive calculations show that the maximum tunability of 0.0230(2πc/a) is obtained for the out-of-plane band gap at ρ = 0.495a, kz = 1(2π/a) and θ = 90∘ by changing the in-plane alignment of the LC directors, φ. Fig. 6 displays the variation of the out-of-plane band gap width as a function of φ for the above mentioned parameters. From this figure one sees that there exists symmetry at φ = 90∘. Numerical results show that the symmetry angle for θ b 90∘ is φ = 180∘, as depicted in Fig. 7 for ρ = 0.495a, kz = 1(2π/a) and θ = 45∘.
Fig. 7. Variation of out-of-plane band gap in terms of φ for square lattice of LCinfiltrated air holes in tellurium background at ρ = 0.495a, kz = 1(π/a), and θ= 45°.
3.2. Triangular lattice Fig. 3(b) shows the second structure to be analyzed; the 2D PC with triangular lattice composed of the LC-infiltrated air holes in anisotropic tellurium background. The corresponding first BZ of the reciprocal lattice is a hexagon, as depicted in Fig. 3(b). The analysis of band structures for triangular lattice is similar to that of the square lattice, as discussed in the last subsection. For a hexagonal BZ, it is necessary to consider the six distinct sub-zones, as marked in Fig. 3
Fig. 9. Variation of out-of-plane photonic band gap width in terms of kz in triangular lattice of air holes with radius ρ = 0.47a.
B. Rezaei et al. / Optics Communications 284 (2011) 813–817
817
out-of-plane photonic band gap have been analyzed as a function of optical axis orientation of anisotropic LC material. Numerical results show that, the tunability of the out-of-plane band gap is larger in the triangular lattice than in the square lattice. The present results can be useful in designing LC-infiltrated photonic crystal fibers. References [1] [2] [3] [4] [5] [6] [7] [8]
Fig. 10. Variation of out-of-plane band gap in terms of φ for triangular lattice of LCinfiltrated air holes in tellurium background at ρ = 0.49a, kz = 0.53(2π/a) and θ = 90°.
(b), for obtaining the complete band structures of PCs made of anisotropic materials. Similar to the square lattice, we first consider the triangular lattice of air holes (without LC material). Thus, the dielectric configuration, as explained in the last subsection, is similar to isotropic materials. Numerical results show that, no out-of-plane photonic band gap appeared in the triangular lattice of air holes with a radius lower than ρ = 0.4a. Fig. 8 shows the out-of-plane photonic band structure of triangular lattice consisting of air holes with radius ρ = 0.47a in tellurium background at kz = 0.2(2π/a). This figure shows two out-of-plane photonic band gaps. The large out-of-plane photonic band gap with normalized width of Δω = 0.0566(2πc/a)is created between the third and fourth bands. The photonic band structure of this lattice for other values of kz have been studied by plotting the band edge diagrams as a function of kz, in which the first band edge diagram is shown in Fig. 9. It can be seen that, contrary to the square lattice, the band gap width decreases as kz increases. Now, the air holes are infiltrated with LC. Similar to the square lattice, the maximum tunability of out-of-plane band gap is obtained by changing the director of LC. It has been found that, the maximum tunability of 0.0303(2πc/a) has been obtained for the out-of-plane band gap at ρ = 0.49a, kz = 0.53(2π/a) and θ = 90∘ by varying the angle φ, as shown in Fig. 10. Again, it can be seen that the symmetry angle for θ = 90∘ takes place at φ = 90∘. Also, the numerical results show that for θ b 90∘ the symmetry angle is equal to φ = 180∘. 4. Conclusion
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
We have studied the out-of-plane propagation of EM waves in the 2D square and triangular PCs created by the LC-infiltrated air holes in anisotropic tellurium background, using the PWEM. The properties of
[40] [41]
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / o p t c o m
Tunable out-of-plane band gap of two-dimensional anisotropic photonic crystals infiltrated with liquid crystals B. Rezaei a,⁎, T. Fathollahi Khalkhali a,b, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 12 June 2010 Received in revised form 30 August 2010 Accepted 1 October 2010 Keywords: Tunable out-of-plane bandgap Anisotropic tellurium Liquid crystal Plane wave method
a b s t r a c t We analyze the tunability of out-of-plane band gap in two-dimensional photonic crystals created by square and triangular lattices of air holes in anisotropic tellurium background, considering that the rods are infiltrated with liquid crystal. Using the plane wave expansion method, we study the variation of out-of-plane band gap by changing the optical axis orientation of liquid crystal. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PCs) are periodic dielectric materials designed to affect the propagation of electromagnetic (EM) waves in the same way as the periodic potential in semiconductor crystals affects the electron motion by defining allowed and forbidden energy bands. Since first proposed by John [1] and Yablonovitch [2], their potential scientific and technological applications have inspired great interest among researchers. PCs offer an important opportunity to design new optical devices and hold a great potential for many significant applications, such as semiconductor lasers and solar cells, high quality resonator and filters, and optical fibers. Thus, it is quite essential to investigate the propagation of EM waves in PC structures. Most of the studies have been focused on the propagation of EM waves within the plane of periodicity. Recently, however, several studies have been reported about the wave propagation out of this plane [3–10] which is very important in the designing of PC fibers (PCFs) [11]. In the last decade there has been an increasing emphasis on tuning the optical properties of photonic bandgap structures in order to design switchable or dynamical devices. Busch and John [12] predicted the tunability of in-plane band gap in three dimensional (3D) PCs by utilizing liquid crystals (LCs). Following this publication, some investigations of the band gap tunability have been done by utilizing LCs in one-dimensional (1D) [13–16], two-dimensional (2D) [17–23] and 3D [24–26] PCs.
⁎ Corresponding author. Tel.: +98 4113393027; fax: +98 4113347050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.011
Presently, there are some investigations on LC-infiltrated PCFs [27,28]. Recently, the out-of-plane band structure of LC-infiltrated triangular 2D PC made of air holes in isotropic silica background has been reported using the full-vectorial FEM based eigenvalue algorithm [29]. Since, the use of anisotropic tellurium leads to an enlargement of the photonic band gap [30,31], so, in this paper we have considered 2D square and triangular PCs created by air holes in anisotropic tellurium background. Based on well known plane wave expansion method (PWEM), we study the evolution of out-of-plane band gap by changing the directors of LCs that are infiltrated throughout the air holes. 2. Formulation To determine the photonic band gap in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell's equations using the PWEM as illustrated in several papers [32–34]. Following the discussion of Busch and John [12], the Maxwell's equation for the magnetic field in 2D PCs utilizing LCs can be expressed as: 2 h i ω −1 → → ∇ × εˆ r ∇×H r = 2 H → r c
ð1Þ
where, ω is the frequency of light and c is the light velocity. The → → → dielectric tensor εˆ r = εˆ r + R is periodic with respect to the real → space lattice vector R and we can use Bloch's theorem to expand −1 → εˆ r as a sum of plane waves: −1
εˆ i; j
→ → → iG :r → r = ∑ ηi; j G e ; → G
ði; j = x; y; zÞ
ð2Þ
814
B. Rezaei et al. / Optics Communications 284 (2011) 813–817
→ → where η G is the Fourier transform of the εˆ −1 r , and plays a key role in the determination of photonic band structure. Generally LCs possess two dielectric constants known as ordinary dielectric constant εo and extraordinary dielectric constant εe. 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 Ref. [35]: e o o 2 2 → ε xx r = ε + ε −ε sin ðθÞ cos ðφÞ
ð3Þ
e o o 2 2 → ε yy r = ε + ε −ε sin ðθÞ sin ðφÞ
ð4Þ
e o 2 → → ε xy r = ε yx r = ε −ε sin ðθÞ sinðϕÞ cosðφÞ
ð5Þ
e o → → εxz r = εzx r = ε −ε sinðθÞ cosðθÞ cosðφÞ
ð6Þ
e o → → ε yz r = εzy r = ε −ε sinðθÞ cosðθÞ sinðφÞ
ð7Þ
e o o 2 → ε zz r = ε + ε −ε sin ðθÞ
ð8Þ
→ λ=1 G
∑ H→ →′ →′ G
G; G
h→′
G ;1 h→′ G ;2
!
2
ω = 2 c
h→
!
G;1 h→ G; 2
ð10Þ
where
H→
→ G ; G′ =
→ → 2 ′ ′ → → → →′ 6 eˆ 2 ⋅η G − G ⋅ eˆ 2 j k + G j j k + G j4 → → ′ ′ −ˆe1 ⋅η G − G ⋅ eˆ 2
→ → 3 −ˆe2 ⋅η G − G ′ ⋅ eˆ 1′ 7 5 → → eˆ1 ⋅η G − G ′ ⋅ eˆ 1′
ð11Þ To use Eq. (10) for calculating the out-of-plane photonic band and eˆ2;→ to be perpendicular and structure of 2D PCs, we define eˆ1;→ G G → → parallel to the plane made by k + G and Z axis, respectively, as shown in Fig. 2. According to Fig. 2, the components of unit vectors eˆ → and 1; G eˆ → are as below in the XYZ coordinate: 2; G
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 XY plane and the X axis, and n is the director of the LC, as shown in Fig. 1. Using Bloch's theorem, we can expand the magnetic field in terms of plane waves in the same way: → → → 2 i k + G :r → →e ˆ →e H r = ∑ ∑ hG;λ λ;G
following linear matrix equation for the dispersion of electromagnetic waves [5]
ð9Þ
→ where k is the Bloch vector and the first sum runs over a finite subset of lattice vectors of the reciprocal lattice. eˆ λ;→ ðλ = 1; 2Þ are unit G → → vectors of two transverse magnetic waves perpendicular to k + G are the corresponding two transverse components of the and h→ G;λ
eˆ1 = ðsin α; − cos α; 0Þ ;
eˆ2 = ðcos β cos α; cos β sin α; − sin βÞ ð12Þ
→ → where β is the angle between the k + G and the Z axis, α is the angle → → → → between the projection of the k + G on the XY plane (i.e. k + G p and the X axis. → is real and symmetric; thus the dispersion reThe matrix H → G; G ′ lation of electromagnetic waves can be solved by using standard diagonalization techniques for the linear system of Eq. (10). The main numerical problem in obtaining the eigenvalues is the evaluation of the Fourier coefficients of the inverse dielectric tensor in Eq. (11). The best method is to calculate the matrix of Fourier coefficients of real space tensors and then take its inverse to obtain the required Fourier coefficients, which was shown by Ho, Chan, and Soukoulis [36]. Since, the numerical implementation of infinite–dimensional matrix in a computer with a finite memory is impossible, thus the
Þ
magnetic field. Substituting Eqs. (2) and (9) into Eq. (1), we obtain the
Fig. 1. Schematic representation of rotation angles for the LC directors.
Fig. 2. Out-of-plane wave vector and unit vectors of two transverse magnetic waves.
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Fig. 3. Photonic crystal structures and the corresponding first Brillouin zones for LC-infiltrated (a) square and (b) triangular lattices of air holes in anisotropic tellurium background.
→ → matrix ε G − G ′ is first truncated → →at finite dimension and then inverted to obtain the matrix η G− G ′ as an approximate method. Therefore, the obtained spectrum by this approximate method will not be the same in general. To overcome this problem, in the actual numerical calculation of photonic bands, we consider the large number of reciprocal lattice vectors in Eq. (10) to obtain acceptable convergence. The reader who is interested in this problem should refer to Refs. [37,38]. The structures under consideration and the corresponding first Brillouin zones are depicted in Fig. 3. We have considered (a) square and (b) triangular structures of air holes in anisotropic tellurium background, considering that the holes are infiltrated with LC. The anisotropic tellurium has two different principle refractive indices as ordinary refractive index nTe o = 4.8 and extraordinary refractive index nTe e = 6.2 over the wavelength of 4.5–6.25 μm with an absorption coefficient of α ≃ 1 cm− 1 [39], in which the extraordinary one is parallel to the Z axis. We assumed that the LC with ordinary refractive LC index nLC o = 1.59 and the extraordinary refractive index ne = 2.223. This LC corresponds to the phenylacetylene type LC [40]. Parameter ρ denotes the radius of the air hole and a is being the lattice constant.
3. Numerical results For this study, we consider 2D square and triangular PCs of air holes in anisotropic tellurium background, considering that the holes are infiltrated with LC as shown in Fig. 3. We choose the LCs 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 components of the dielectric tensor of the LC are given by Eqs. (3)–(8). The band structure of 2D PCs is obtained numerically by solving Eq. (10). A total of 441 plane waves were employed for both structures in these calculations. This number is known to be sufficient enough to obtain acceptable convergence for characterizing the out-of-plane band structures. Our main goal here is to study the modification of the band gap spectrum and the value of the out-of-plane band gap by varying LCs optical axis orientation as well as when the radius of air hole varies. The out-of-plane band structures of these two kinds of lattices will be separately analyzed in the next two subsections. 3.1. Square lattice We first consider a 2D PC of square lattice composed of LCinfiltrated air holes in anisotropic tellurium background, as illustrated
Fig. 4. Out-of-plane photonic band structure for square lattice of air holes with radius ρ = 0.47a in tellurium background for kz = 1.4(2πc/a).
Fig. 5. Variation of out-of-plane photonic band gap width in terms of kz in square lattice of air holes with radius ρ = 0.47a.
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Fig. 6. Variation of out-of-plane band gap in terms of φ for square lattice of LCinfiltrated air holes in tellurium background at ρ = 0.495a, kz = 1(2π/a) and θ = 90°.
Fig. 8. Out-of-plane photonic band structure for triangular lattice of air holes with radius ρ = 0.47a in tellurium background for kz = 0.2(2πc/a).
in Fig. 3(a). The corresponding first Brillouin zone (BZ) is also depicted in Fig. 3(a). For isotropic PCs, the photonic band structure can be → obtained by simply considering all the wave vectors k in the irreducible BZ (IBZ) which is enclosed by points Γ, X, and M as shown in Fig. 3(a). The resulting photonic band structure obtained from this IBZ for isotropic PCs based on the rotation and reflection symmetry of the structure. In a PC made of anisotropic materials, the definition of the IBZ depends on the orientation of the principal axes of anisotropic material [41] and hence it may be difficult to define the IBZ for each optic axis orientation. So, it is reasonable to consider the more symmetry directions in the first BZ for obtaining the complete photonic band structure. First we consider the case when there is no LC (air holes). Numerical results show that, for the square lattice of air holes with radius lower than ρ = 0.43a, the width of out-of-plane photonic band gap is very small. The calculated out-of-plane photonic band structure for the square lattice of air holes with radius ρ = 0.47a is shown in Fig. 4 for the out-of-plane component of wave vector kz = 1.4(2π/a), where kp is the projection of the wave vector on the XY plane, which has been taken along the Γ − X′ − M − X − M′ − X″ − Γ path of the first BZ. In this figure an out-of-plane photonic band gap with normalized width of Δω = 0.0365(2πc/a) is created between the first and second band. We have also studied the photonic band structure for other values of kz. Fig. 5 shows the variation of out-of-plane photonic band gap width in terms of kz in the square lattice of air holes with radius ρ = 0.47a.
It can be seen that the band gap exists for a wide range of kz and increases as kz increases. At next step, the air holes are infiltrated with LC. In this case, we study the variation of out-of-plane band gap by changing the director of LC. To obtain the maximum tunability of out-of-plane band gap, we change the director of LCs for different values of rod radius ρ and kz. In this structure four parameters θ, φ, kz and ρ are treated as adjustable parameters to obtain the maximum tunability of the out-of-plane band gap. Extensive calculations show that the maximum tunability of 0.0230(2πc/a) is obtained for the out-of-plane band gap at ρ = 0.495a, kz = 1(2π/a) and θ = 90∘ by changing the in-plane alignment of the LC directors, φ. Fig. 6 displays the variation of the out-of-plane band gap width as a function of φ for the above mentioned parameters. From this figure one sees that there exists symmetry at φ = 90∘. Numerical results show that the symmetry angle for θ b 90∘ is φ = 180∘, as depicted in Fig. 7 for ρ = 0.495a, kz = 1(2π/a) and θ = 45∘.
Fig. 7. Variation of out-of-plane band gap in terms of φ for square lattice of LCinfiltrated air holes in tellurium background at ρ = 0.495a, kz = 1(π/a), and θ= 45°.
3.2. Triangular lattice Fig. 3(b) shows the second structure to be analyzed; the 2D PC with triangular lattice composed of the LC-infiltrated air holes in anisotropic tellurium background. The corresponding first BZ of the reciprocal lattice is a hexagon, as depicted in Fig. 3(b). The analysis of band structures for triangular lattice is similar to that of the square lattice, as discussed in the last subsection. For a hexagonal BZ, it is necessary to consider the six distinct sub-zones, as marked in Fig. 3
Fig. 9. Variation of out-of-plane photonic band gap width in terms of kz in triangular lattice of air holes with radius ρ = 0.47a.
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out-of-plane photonic band gap have been analyzed as a function of optical axis orientation of anisotropic LC material. Numerical results show that, the tunability of the out-of-plane band gap is larger in the triangular lattice than in the square lattice. The present results can be useful in designing LC-infiltrated photonic crystal fibers. References [1] [2] [3] [4] [5] [6] [7] [8]
Fig. 10. Variation of out-of-plane band gap in terms of φ for triangular lattice of LCinfiltrated air holes in tellurium background at ρ = 0.49a, kz = 0.53(2π/a) and θ = 90°.
(b), for obtaining the complete band structures of PCs made of anisotropic materials. Similar to the square lattice, we first consider the triangular lattice of air holes (without LC material). Thus, the dielectric configuration, as explained in the last subsection, is similar to isotropic materials. Numerical results show that, no out-of-plane photonic band gap appeared in the triangular lattice of air holes with a radius lower than ρ = 0.4a. Fig. 8 shows the out-of-plane photonic band structure of triangular lattice consisting of air holes with radius ρ = 0.47a in tellurium background at kz = 0.2(2π/a). This figure shows two out-of-plane photonic band gaps. The large out-of-plane photonic band gap with normalized width of Δω = 0.0566(2πc/a)is created between the third and fourth bands. The photonic band structure of this lattice for other values of kz have been studied by plotting the band edge diagrams as a function of kz, in which the first band edge diagram is shown in Fig. 9. It can be seen that, contrary to the square lattice, the band gap width decreases as kz increases. Now, the air holes are infiltrated with LC. Similar to the square lattice, the maximum tunability of out-of-plane band gap is obtained by changing the director of LC. It has been found that, the maximum tunability of 0.0303(2πc/a) has been obtained for the out-of-plane band gap at ρ = 0.49a, kz = 0.53(2π/a) and θ = 90∘ by varying the angle φ, as shown in Fig. 10. Again, it can be seen that the symmetry angle for θ = 90∘ takes place at φ = 90∘. Also, the numerical results show that for θ b 90∘ the symmetry angle is equal to φ = 180∘. 4. Conclusion
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]
We have studied the out-of-plane propagation of EM waves in the 2D square and triangular PCs created by the LC-infiltrated air holes in anisotropic tellurium background, using the PWEM. The properties of
[40] [41]
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