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Study on complete band gap of two-dimensional photonic crystal with quadrangular rods
Optik 123 (2012) 2017–2020
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Study on complete band gap of two-dimensional photonic crystal with quadrangular rods Ming-Bao Yan a,∗ , Zhen-Tang Fu a , Hai-Long Wang b a b
College of Science, Air Force Engineering University, Xian 710051, China College of Physics and Engineering, Qufu Normal University, Shandong 273165, China
a r t i c l e
i n f o
Article history: Received 19 May 2011 Accepted 25 September 2011
PACS: 42.70.Qs 41.20.Jb 74.62.Dh Keywords: Photonic crystal Complete band gap Quadrangular rods Plane wave expansion method
a b s t r a c t The plane wave expansion method (PWM) was employed to study the relation between the photonic band gap (PBG) of 2D triangular lattice photonic crystal (PC) and the shapes of rods and dielectric constant. It is shown that the PBG of PC with quadrangular rods is the widest one, compared with the other case with cross section shapes of triangular, circular and hexagon under the same filling ratio, and a peak value appears when the side length ratio of lx /ly is equal to 1.21 approximately to any filling ratio. In the aspect of the effects of dielectric constant, the PBG width does not increase monotonically with the increase permittivity ε2 of the background material to certain permittivity ε1 of the quadrangular rods, but has a peak value instead. However, the larger the permittivity ε1 is, the narrower the band width is and the lower the central frequency is, and the dispersion ε = ε2 − ε1 is larger also. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Many characteristics and applications of photonic crystals (PCs) [1,2] are based on its photonic band-gap (PBG) [3]. The PBG is thus the essential property of PC. In the last few years, a mass of theoretical and applied investigation of PCs was developed. Due to the existence of band-gaps, PCs are ideal material for developing optical devices with small mode volumes and high quality factors. Many different types of devices are needed in applications such as filters, lasers, waveguides, fibers [4–9] and other nonlinear optical devices. Among various types of PCs, the two-dimensional (2D) ones are of particular interest due to their comparative ease of fabrication, thanks to the planar fabrication techniques developed for the semiconductor industry. They are also much easier to integrate with current photonic devices. In addition, the triangular lattice [10] is of a special interest since the structure can possess a large PBG for TE field polarization and can even possess a complete PBG for both TE and TM field polarization [11] for some lattice parameters. In practice, the research of the periodic structure or medium of the PCs was reported extensively. The periodic structure consists of triangular lattice, square lattice [12], super-lattice structure above which is formed by columns, square or elliptical rods, etc. The
∗ Corresponding author. E-mail address: [email protected] (Y. Ming-Bao). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.09.034
dependence of PBG on different alterable factors is achieved. For example, the width of the PBG becomes wider with the increasing filling ratio and dielectric constant difference. However, the PBG width does not increase monotonically with the former. In this paper, we employ the plane-wave expansion method [13–15] to study the PBG of 2D photonic crystal with triangular lattice. Transmission spectra were obtained by theoretical calculations. The PBG width of PC with quadrangular rods structure is wider than the width of other PC with cross section shapes of triangular, circular or hexagon under the same filling ratio, and a peak value appears when the side length ratio of lx /ly is equal to 1.21 approximately to any filling ratio. Moreover, the case with filling ratio is 0.49 and lx /ly is 1.21 was calculated that the peak value ω is equal to 0.196ωe (ωe = 2c/˛, ˛ indicates lattice constant and c is speed of light in vacuum) and the maximal width is decrescent with the increasing of dielectric constant of quadrangular rods. These results suggest a new basis for designing PC devices.
2. The basic theories The PWM has an obvious advantage that it is easier to reliably automate the identification of frequency bands and gaps and can display mode profiles with no additional efforts. The light spread can be described by Maxwell’s equations in PCs. Considering the isotropic, non-loss and non-magnetic materials,
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Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
Fig. 1. Schematic diagram of 2D triangular lattice PC.
the time-dependent Maxwell’s curl equations in PCs can be written as:
∇ × E(r៝ ) = iω0 H(r៝ ) ∇ × H(r៝ ) = −iωε0 ε(r៝ )E(r៝ )
(1)
where r៝ = (x, y) is the 2D position vector, ε(r៝ ) is the positiondependent dielectric constant, ω is the eigen-angular frequency, and 0 , ε0 denote the permeability and permittivity of free space, respectively. With the assumption of the wave vector k៝ in the x–yplane, i.e., k៝ = (kx, ky, 0), we will refer to as E parallel to the plane (H polarization). The wave equations for H polarizations reduce to: ∂Ey (r៝ ) ∂Ex (r៝ ) − = iω0 Hz (r៝ ), ∂x ∂y
ε(r៝ )Ey (r៝ ) ∂Hz (r៝ ) , = iω ∂x c 2 0
ε(r៝ )Ex (r៝ ) = −iω c 2 0
∂Hz (r៝ ) ∂y
∂ ∂x
1 ∂Hz (r៝ ) ε(r៝ ) ∂x
+
∂ ∂y
1 ∂Hz (r៝ ) ε(r៝ ) ∂y
G
៝ exp(i(k៝ + G) ៝ · r៝ ) H(k៝ + G)
G
៝ = 1/A where K(G)
(5)
៝ · r៝ ) dr៝ , G ៝ is the reciprocal(1/ε(r៝ )) exp(−iG ៝ is expansion coefficient. When Eqs. (4) lattice vector, and H(k៝ + G) ˝
and (5) are substituted into Eq. (3), we can obtained the following eigen-value equation:
៝ k៝ + G ៝ )K(G ៝ −G ៝ )H(k៝ + G) ៝ = (k៝ + G)(
ω2 ៝ ៝ H(k + G) c2
(6)
By solving the equation above, the PBG structure and the field distribution of each mode can be obtained. (2) 3. Results and discussion
+
ω2 Hz (r៝ ) = 0 c2
(3)
To solve Eq. (3), we expand the fields Hz (r៝ ) and dielectric function ៝ ε−1 (r) in a series of plane waves for a given wave vector k:
1 ៝ exp(iG ៝ · r៝ ) = K(G) ε(r៝ )
Hz (r៝ ) =
G
The above simple equations can be easily reduced to a quadratic linear equation for Hz (r៝ ) as follows:
Fig. 2. Transmission spectrum with variation of section shape.
(4)
3.1. Theoretical model For simplicity, we examine the 2D PC in which the dielectric structure is uniform in the z-direction and periodic in the x–yplane. The proposed triangular lattice structure, shown in Fig. 1, consists of 16 rows air holes or dielectric rods in x direction and 16 columns in y direction. Here ε1 , ε2 denote the permittivity of dielectric rods and background materials, respectively, and lx , ly indicate the side length of the dielectric rods. The direction of incident wave parallels with the y axis. In what follows of date
Fig. 3. Transmission spectra with variation of side length ratio.
Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
2019
processing, normalized frequency (˛/) and normalized transmission coefficient were adopted in x-axis and y-axis, respectively, to possess the universality. In order to provide the optimal results, the H polarized wave which can present the wider PBG structure opposed to the E polarized wave was preferred. 3.2. Section-shaped selectivity of transmission properties Large numbers of references reported that the PBG width is lie on many different alterable factors such as dielectric rod shapes, falling ratio, etc. And then, the transmission properties of PC with four different section-shapes such as cylindrical, triangular, square, hexagonal-shaped air (ε1 = 1) hole are calculated, respectively. Where the filling ratio remains unchanged, namely f = 0.49, and ε1 , ε2 are equal to 1 and 10.5, respectively. The result is shown in Fig. 2. Among above PBG structures, the square-shaped PC width is the widest one while the triangular structure width is the smallest. So, the maximum PBG can be achieved by adjusting the parameters under the square structure.
Fig. 5. The PBG width versus ε2 for different ε1 .
3.3. Side length selectivity of transmission properties The filling ratio is fixed, changing the side length with lx and ly , and defining the side length ratio b = lx /ly . Considering f = 0.36, 0.41, 0.45, 0.49, the transmission characteristics were obtained. To each f, the value of b is focused over the range of 0.90–1.40, in which the results of b = 1.00, 1.21, 1.36 were selected to depict the difference obviously. The transmission spectra are plotted in Fig. 3. To validate the affection of the side length ratio to the PBG width exactly, the relation between PBG width and side length ratio was laid out in Fig. 4. The width of PBG does not change wide with the increasing of b all the while, but has a peak at b = 1.21 approximately. The maximum values of width are 0.125ωe , 0.152ωe , 0.173ωe and 0.196ωe , respectively, the width becomes wider with the increasing of the f. The changing tendency of corresponding central frequency is similar to that of PBG width. 3.4. Dielectric constant selectivity of transmission properties Fig. 6. Transmission spectra with (a) ε = 10 and (b) ε = 15.
The PBG width with the variation of ε1 = 1–4 and the corresponding ε2 (from 8 to 40) was calculated for filling ratio f = 0.49 and b = 1.21. It is shown in Fig. 5. We can see that the PBG width reaches a maximum of 0.152ωe for ε1 = 1 (air holes) and ε2 augmented to 13. For ε1 = 2–4, and the maximum width were obtained with 0.112ωe , 0.094ωe and 0.08ωe , and the corresponding ε2 reach to 22, 28 and 36, respectively. Also, the width of PC consisting of air holes is widest one among various dielectric rods PCs. The value of ω diminishes with the increasing of ε1 , and the maximal width for ε1 = 4 decreases to a half of that for ε1 = 1. However, the width ω decreases rapidly with the increasing of ε1 for the invitation of ε2 , exceptionally, ω (ε1 = 3) is less than ω (ε1 = 4) when ε2 is
Fig. 4. The PBG width versus b for different f.
above 33.5. The dispersion between ε2 and ε2 (ε = ε2 − ε1 ) alters bigger and bigger when the maximal BPG width appears. Thus it can be seen that the PBG width lies on the dispersion of dielectric constant, and has a peak value. To same ε, the transmission spectra were further calculated as in Fig. 6, where ε is equal to 10 and 15. Though ε is the same, the ω decreases rapidly with the increasing of ε1 and ε2 . It is revealed that the PBG width is determined not only the ε but also the ε1 and ε2 , respectively. 4. Conclusion By using the plane-wave expansion method, the PBG of 2D photonic crystal with triangular lattice was investigated. Transmission spectra were obtained by theoretical calculations. The PBG width of PC structure with quadrangular air holes provided with a peak value (0.196ωe ) at f = 0.49 and b = 1.21. In the aspect of dielectric constant, the PBG does not become wide monotonically with the increasing of ε2 , but possess of a maximum also. To augment the quadrangular rods ε1 , the ε2 increases dramatically when the maximal width appears, and the dispersion ε becomes larger accordantly. However, the maximal width diminishes promptly with the increasing of ε, and the bigger the quadrangular rods dielectric is, the narrower the PBG width is to fixed ε. Thus, people can adjust the side length ratio of dielectric rods section
2020
Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
and dielectric constant dispersion to achieve the wider PBG width, the larger dielectric constant will be not preferred any more. Although the theoretical results are obtained by taking example for a triangular lattice in this paper, it can be substantially applied to any kind of 2D or 3D PC with quadrangular rods structure. References [1] E. Yablonovich, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (20) (1987) 2059–2062. [2] S. John, Strong localization of photons in certain disordered dielectric superlattice, Phys. Rev. Lett. 58 (23) (1987) 2486–2489. [3] J.B. Pendry, Calculating photonic band structure, Phys. Condens. Matter 8 (1996) 1086–1108. [4] T. Baba, D. Mori, Slow light engineering in photonic crystals, Appl. Phys. 40 (2007) 2659–2665. [5] F.S.-S. Chien, J.B. Tu, W.-F. Hsien, Tight-binding theory for coupled photonic crystal waveguides, Phys. Rev. B 75 (2007), 125113/1–125113/8. [6] L. Dai, C. Jiang, Photonic crystal slow light waveguides with large delaybandwidth product, Appl. Phys. B 95 (2009) 101–105. [7] M. Qiu, He Sailing, Numerical method for computing defect modes in twodimensional photonic crystal with dielectric or metallic inclusions, Phys. Rev. B 61 (2000) 12871–12876.
[8] R.E.P. de Oliveira, C.J.S. de Matos, J.G. Hayashi, et al., Pressure sensing based on non-conventional air-guiding transmission windows in hollowcore photonic crystal fibers, J. Lightwave Technol. 27 (11) (2009) 1605– 1609. [9] S. Noda, M. Yokoyama, M. Imada, A. Chutinan, M. Mochizuki, Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design, Science 293 (2001) 1123–1125. [10] H. Kurt, The directional emission sensitivity of photonic crystal waveguides to air hole removal, Appl. Phys. B 95 (2009) 341–344. [11] S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, S. Noda, Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs, Appl. Phys. Lett. 87 (2005), 061107/1–061107/3. [12] Park J. Hwang, Y. Lee, J. Kim, Super-lattice photonic band-gap single-cell operating in the lowest-order whispering gallery mode, Appl. Phys. Lett. 80 (21) (2002) 3883–3885. [13] M. Maldovan, E.L. Thomas, C.W. Carter, Layer-by-layer diamond-like woodpile structure with a large photonic band gap, Appl. Phys. Lett. 84 (3) (2004) 362–364. [14] M. Che, Z.-Y. Li, Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method, Phys. Rev. B 77 (2008), 125138/1–125138/8. [15] S. Xiao, L. Shen, S. He, A plane-wave expansion method based on the effective medium theory for calculating the band structure of a two-dimensional photonic crystal, Phys. Lett. A 312 (2) (2003) 132–138.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Study on complete band gap of two-dimensional photonic crystal with quadrangular rods Ming-Bao Yan a,∗ , Zhen-Tang Fu a , Hai-Long Wang b a b
College of Science, Air Force Engineering University, Xian 710051, China College of Physics and Engineering, Qufu Normal University, Shandong 273165, China
a r t i c l e
i n f o
Article history: Received 19 May 2011 Accepted 25 September 2011
PACS: 42.70.Qs 41.20.Jb 74.62.Dh Keywords: Photonic crystal Complete band gap Quadrangular rods Plane wave expansion method
a b s t r a c t The plane wave expansion method (PWM) was employed to study the relation between the photonic band gap (PBG) of 2D triangular lattice photonic crystal (PC) and the shapes of rods and dielectric constant. It is shown that the PBG of PC with quadrangular rods is the widest one, compared with the other case with cross section shapes of triangular, circular and hexagon under the same filling ratio, and a peak value appears when the side length ratio of lx /ly is equal to 1.21 approximately to any filling ratio. In the aspect of the effects of dielectric constant, the PBG width does not increase monotonically with the increase permittivity ε2 of the background material to certain permittivity ε1 of the quadrangular rods, but has a peak value instead. However, the larger the permittivity ε1 is, the narrower the band width is and the lower the central frequency is, and the dispersion ε = ε2 − ε1 is larger also. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Many characteristics and applications of photonic crystals (PCs) [1,2] are based on its photonic band-gap (PBG) [3]. The PBG is thus the essential property of PC. In the last few years, a mass of theoretical and applied investigation of PCs was developed. Due to the existence of band-gaps, PCs are ideal material for developing optical devices with small mode volumes and high quality factors. Many different types of devices are needed in applications such as filters, lasers, waveguides, fibers [4–9] and other nonlinear optical devices. Among various types of PCs, the two-dimensional (2D) ones are of particular interest due to their comparative ease of fabrication, thanks to the planar fabrication techniques developed for the semiconductor industry. They are also much easier to integrate with current photonic devices. In addition, the triangular lattice [10] is of a special interest since the structure can possess a large PBG for TE field polarization and can even possess a complete PBG for both TE and TM field polarization [11] for some lattice parameters. In practice, the research of the periodic structure or medium of the PCs was reported extensively. The periodic structure consists of triangular lattice, square lattice [12], super-lattice structure above which is formed by columns, square or elliptical rods, etc. The
∗ Corresponding author. E-mail address: [email protected] (Y. Ming-Bao). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.09.034
dependence of PBG on different alterable factors is achieved. For example, the width of the PBG becomes wider with the increasing filling ratio and dielectric constant difference. However, the PBG width does not increase monotonically with the former. In this paper, we employ the plane-wave expansion method [13–15] to study the PBG of 2D photonic crystal with triangular lattice. Transmission spectra were obtained by theoretical calculations. The PBG width of PC with quadrangular rods structure is wider than the width of other PC with cross section shapes of triangular, circular or hexagon under the same filling ratio, and a peak value appears when the side length ratio of lx /ly is equal to 1.21 approximately to any filling ratio. Moreover, the case with filling ratio is 0.49 and lx /ly is 1.21 was calculated that the peak value ω is equal to 0.196ωe (ωe = 2c/˛, ˛ indicates lattice constant and c is speed of light in vacuum) and the maximal width is decrescent with the increasing of dielectric constant of quadrangular rods. These results suggest a new basis for designing PC devices.
2. The basic theories The PWM has an obvious advantage that it is easier to reliably automate the identification of frequency bands and gaps and can display mode profiles with no additional efforts. The light spread can be described by Maxwell’s equations in PCs. Considering the isotropic, non-loss and non-magnetic materials,
2018
Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
Fig. 1. Schematic diagram of 2D triangular lattice PC.
the time-dependent Maxwell’s curl equations in PCs can be written as:
∇ × E(r៝ ) = iω0 H(r៝ ) ∇ × H(r៝ ) = −iωε0 ε(r៝ )E(r៝ )
(1)
where r៝ = (x, y) is the 2D position vector, ε(r៝ ) is the positiondependent dielectric constant, ω is the eigen-angular frequency, and 0 , ε0 denote the permeability and permittivity of free space, respectively. With the assumption of the wave vector k៝ in the x–yplane, i.e., k៝ = (kx, ky, 0), we will refer to as E parallel to the plane (H polarization). The wave equations for H polarizations reduce to: ∂Ey (r៝ ) ∂Ex (r៝ ) − = iω0 Hz (r៝ ), ∂x ∂y
ε(r៝ )Ey (r៝ ) ∂Hz (r៝ ) , = iω ∂x c 2 0
ε(r៝ )Ex (r៝ ) = −iω c 2 0
∂Hz (r៝ ) ∂y
∂ ∂x
1 ∂Hz (r៝ ) ε(r៝ ) ∂x
+
∂ ∂y
1 ∂Hz (r៝ ) ε(r៝ ) ∂y
G
៝ exp(i(k៝ + G) ៝ · r៝ ) H(k៝ + G)
G
៝ = 1/A where K(G)
(5)
៝ · r៝ ) dr៝ , G ៝ is the reciprocal(1/ε(r៝ )) exp(−iG ៝ is expansion coefficient. When Eqs. (4) lattice vector, and H(k៝ + G) ˝
and (5) are substituted into Eq. (3), we can obtained the following eigen-value equation:
៝ k៝ + G ៝ )K(G ៝ −G ៝ )H(k៝ + G) ៝ = (k៝ + G)(
ω2 ៝ ៝ H(k + G) c2
(6)
By solving the equation above, the PBG structure and the field distribution of each mode can be obtained. (2) 3. Results and discussion
+
ω2 Hz (r៝ ) = 0 c2
(3)
To solve Eq. (3), we expand the fields Hz (r៝ ) and dielectric function ៝ ε−1 (r) in a series of plane waves for a given wave vector k:
1 ៝ exp(iG ៝ · r៝ ) = K(G) ε(r៝ )
Hz (r៝ ) =
G
The above simple equations can be easily reduced to a quadratic linear equation for Hz (r៝ ) as follows:
Fig. 2. Transmission spectrum with variation of section shape.
(4)
3.1. Theoretical model For simplicity, we examine the 2D PC in which the dielectric structure is uniform in the z-direction and periodic in the x–yplane. The proposed triangular lattice structure, shown in Fig. 1, consists of 16 rows air holes or dielectric rods in x direction and 16 columns in y direction. Here ε1 , ε2 denote the permittivity of dielectric rods and background materials, respectively, and lx , ly indicate the side length of the dielectric rods. The direction of incident wave parallels with the y axis. In what follows of date
Fig. 3. Transmission spectra with variation of side length ratio.
Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
2019
processing, normalized frequency (˛/) and normalized transmission coefficient were adopted in x-axis and y-axis, respectively, to possess the universality. In order to provide the optimal results, the H polarized wave which can present the wider PBG structure opposed to the E polarized wave was preferred. 3.2. Section-shaped selectivity of transmission properties Large numbers of references reported that the PBG width is lie on many different alterable factors such as dielectric rod shapes, falling ratio, etc. And then, the transmission properties of PC with four different section-shapes such as cylindrical, triangular, square, hexagonal-shaped air (ε1 = 1) hole are calculated, respectively. Where the filling ratio remains unchanged, namely f = 0.49, and ε1 , ε2 are equal to 1 and 10.5, respectively. The result is shown in Fig. 2. Among above PBG structures, the square-shaped PC width is the widest one while the triangular structure width is the smallest. So, the maximum PBG can be achieved by adjusting the parameters under the square structure.
Fig. 5. The PBG width versus ε2 for different ε1 .
3.3. Side length selectivity of transmission properties The filling ratio is fixed, changing the side length with lx and ly , and defining the side length ratio b = lx /ly . Considering f = 0.36, 0.41, 0.45, 0.49, the transmission characteristics were obtained. To each f, the value of b is focused over the range of 0.90–1.40, in which the results of b = 1.00, 1.21, 1.36 were selected to depict the difference obviously. The transmission spectra are plotted in Fig. 3. To validate the affection of the side length ratio to the PBG width exactly, the relation between PBG width and side length ratio was laid out in Fig. 4. The width of PBG does not change wide with the increasing of b all the while, but has a peak at b = 1.21 approximately. The maximum values of width are 0.125ωe , 0.152ωe , 0.173ωe and 0.196ωe , respectively, the width becomes wider with the increasing of the f. The changing tendency of corresponding central frequency is similar to that of PBG width. 3.4. Dielectric constant selectivity of transmission properties Fig. 6. Transmission spectra with (a) ε = 10 and (b) ε = 15.
The PBG width with the variation of ε1 = 1–4 and the corresponding ε2 (from 8 to 40) was calculated for filling ratio f = 0.49 and b = 1.21. It is shown in Fig. 5. We can see that the PBG width reaches a maximum of 0.152ωe for ε1 = 1 (air holes) and ε2 augmented to 13. For ε1 = 2–4, and the maximum width were obtained with 0.112ωe , 0.094ωe and 0.08ωe , and the corresponding ε2 reach to 22, 28 and 36, respectively. Also, the width of PC consisting of air holes is widest one among various dielectric rods PCs. The value of ω diminishes with the increasing of ε1 , and the maximal width for ε1 = 4 decreases to a half of that for ε1 = 1. However, the width ω decreases rapidly with the increasing of ε1 for the invitation of ε2 , exceptionally, ω (ε1 = 3) is less than ω (ε1 = 4) when ε2 is
Fig. 4. The PBG width versus b for different f.
above 33.5. The dispersion between ε2 and ε2 (ε = ε2 − ε1 ) alters bigger and bigger when the maximal BPG width appears. Thus it can be seen that the PBG width lies on the dispersion of dielectric constant, and has a peak value. To same ε, the transmission spectra were further calculated as in Fig. 6, where ε is equal to 10 and 15. Though ε is the same, the ω decreases rapidly with the increasing of ε1 and ε2 . It is revealed that the PBG width is determined not only the ε but also the ε1 and ε2 , respectively. 4. Conclusion By using the plane-wave expansion method, the PBG of 2D photonic crystal with triangular lattice was investigated. Transmission spectra were obtained by theoretical calculations. The PBG width of PC structure with quadrangular air holes provided with a peak value (0.196ωe ) at f = 0.49 and b = 1.21. In the aspect of dielectric constant, the PBG does not become wide monotonically with the increasing of ε2 , but possess of a maximum also. To augment the quadrangular rods ε1 , the ε2 increases dramatically when the maximal width appears, and the dispersion ε becomes larger accordantly. However, the maximal width diminishes promptly with the increasing of ε, and the bigger the quadrangular rods dielectric is, the narrower the PBG width is to fixed ε. Thus, people can adjust the side length ratio of dielectric rods section
2020
Y. Ming-Bao et al. / Optik 123 (2012) 2017–2020
and dielectric constant dispersion to achieve the wider PBG width, the larger dielectric constant will be not preferred any more. Although the theoretical results are obtained by taking example for a triangular lattice in this paper, it can be substantially applied to any kind of 2D or 3D PC with quadrangular rods structure. References [1] E. Yablonovich, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58 (20) (1987) 2059–2062. [2] S. John, Strong localization of photons in certain disordered dielectric superlattice, Phys. Rev. Lett. 58 (23) (1987) 2486–2489. [3] J.B. Pendry, Calculating photonic band structure, Phys. Condens. Matter 8 (1996) 1086–1108. [4] T. Baba, D. Mori, Slow light engineering in photonic crystals, Appl. Phys. 40 (2007) 2659–2665. [5] F.S.-S. Chien, J.B. Tu, W.-F. Hsien, Tight-binding theory for coupled photonic crystal waveguides, Phys. Rev. B 75 (2007), 125113/1–125113/8. [6] L. Dai, C. Jiang, Photonic crystal slow light waveguides with large delaybandwidth product, Appl. Phys. B 95 (2009) 101–105. [7] M. Qiu, He Sailing, Numerical method for computing defect modes in twodimensional photonic crystal with dielectric or metallic inclusions, Phys. Rev. B 61 (2000) 12871–12876.
[8] R.E.P. de Oliveira, C.J.S. de Matos, J.G. Hayashi, et al., Pressure sensing based on non-conventional air-guiding transmission windows in hollowcore photonic crystal fibers, J. Lightwave Technol. 27 (11) (2009) 1605– 1609. [9] S. Noda, M. Yokoyama, M. Imada, A. Chutinan, M. Mochizuki, Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design, Science 293 (2001) 1123–1125. [10] H. Kurt, The directional emission sensitivity of photonic crystal waveguides to air hole removal, Appl. Phys. B 95 (2009) 341–344. [11] S. Takayama, H. Kitagawa, Y. Tanaka, T. Asano, S. Noda, Experimental demonstration of complete photonic band gap in two-dimensional photonic crystal slabs, Appl. Phys. Lett. 87 (2005), 061107/1–061107/3. [12] Park J. Hwang, Y. Lee, J. Kim, Super-lattice photonic band-gap single-cell operating in the lowest-order whispering gallery mode, Appl. Phys. Lett. 80 (21) (2002) 3883–3885. [13] M. Maldovan, E.L. Thomas, C.W. Carter, Layer-by-layer diamond-like woodpile structure with a large photonic band gap, Appl. Phys. Lett. 84 (3) (2004) 362–364. [14] M. Che, Z.-Y. Li, Analysis of photonic crystal waveguide bends by a plane-wave transfer-matrix method, Phys. Rev. B 77 (2008), 125138/1–125138/8. [15] S. Xiao, L. Shen, S. He, A plane-wave expansion method based on the effective medium theory for calculating the band structure of a two-dimensional photonic crystal, Phys. Lett. A 312 (2) (2003) 132–138.