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Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs
G Model IJLEO-54703; No. of Pages 4
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs Xiao Chen ∗ , Xin-Yuan Liang, Yi-Quan Wang, Shuai Feng College of Science, Minzu University of China, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 5 August 2013 Accepted 5 February 2014 Available online xxx Keywords: Photonic crystal Octagonal quasicrystal Photonic bandgap Organic polystyrene Light location Microcavity.
a b s t r a c t Based on the polystyrene material of low refractive index, light localization in photonic bandgaps of two kinds of 2D octagonal quasiperiodic photonic crystal slabs are investigated in theory, including the air-rod polystyrene slab and polystyrene-rod slab. The properties of bandgaps and localized modes in both two defect-free patterns are analyzed in detail. When a single-point defect is introduced into two quasiperiodic structures, the position of emerging defect modes and the red-shifting of resonant modes in wavelength are observed quite differently when the defect microcavity is increased in size. This difference is caused by the competition of two physical mechanisms, which are the effect of defect energy levels caused by defects introduced into photonic crystals and the resonance of modes in the defect cavity. These results will provide theoretical support for experimental fabrication of organic lightemitting quasiperiodic photonic crystal devices. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Following the pioneer works of John and Yablonovitch on periodic photonic crystals (PCs), quasi-periodic photonic crystals (QPCs) which consist of two or more materials arranged in quasiperiodic patterns have received much attention because of their remarkable properties. The structures of QPCs are not periodic but have long-range orientational order, so that it gives rise to a more isotropic Brillouin zone with smaller modulations that depress the gap [1]. Due to the greater rotational symmetry in QPCs, photonic bandgaps (PBGs) are independent of the incident light direction, which have potential applications in photonics [2–5]. Since the quasi-periodic arrangement has many in-equivalent local environments, the properties of defects are more complex and interesting than those created in conventional periodic PCs and may offer more flexibility in tuning the defect state properties [6–8]. For example, the light localization modes may occur in defect-free QPCs, which is useful for designing novel PC fibers and microcavities [9–11]. Besides, the dielectric constant necessary for the complete PBGs in QPCs is small. In the case of octagonal quasi-periodic lattice of dielectric rods in air, the first PBG for the TM polarization stays open down to the dielectric constant as small as ε = 1.6 (n = 1.26) [12], while the threshold of the dielectric constant in dodecagonal QPCs is down to ε = 1.35 [13]. This property makes it possible that
many PBG-based photonic devices can be fabricated from resourceful SiO2 (n = 1.45) or organic polymer materials. As we know, most of the PCs are made from III–V semiconductors. In contrast to inorganic optoelectronic materials, organic polymer materials, especially conjugated polymers, have advantages over the light-emitting property in the visible spectrum, the ease of fabrication and high optical nonlinearity. They have, therefore, become promising candidates as PC nano-lasers or light-emitting diodes [14]. Although the dielectric constant of organic materials is small, the combination of the light-emitting polymer and quasi-periodic structures with higher-level symmetry would provide better active layers, and more efficient, uniform in-plane confinement in all directions. Hence, it is beneficial for optoelectronic devices to achieve ultralow lasing threshold and high slope efficiency. In this paper, we investigate the physical properties of PBGs and localized modes in 2D 8-fold symmetry QPC slabs at low-index contrast. By introducing a single-point microcavity into QPCs, we observe the dependence of localized resonant modes on the distribution of air-rods around a microcavity in polystyrene (PS) slab is quite different from that in PS-rods slab. The physical mechanism behind the difference and the fundamental parameters governing PBGs and resonant modes are discussed in detail. 2. Simulation and results
∗ Corresponding author. E-mail address: [email protected] (X. Chen).
We design two kinds of 2D 8-fold symmetry organic QPC patterns, which are the air-rod PS slab and PS-rod slab, respectively.
http://dx.doi.org/10.1016/j.ijleo.2014.02.041 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Fig. 1. Schematic patterns of 2D octagonal QPC slabs with air-rods in PS material (a) and PS-rods in air (b). The transmission spectra of Gaussian light in the corresponding structures (on the right).
Based on finite-difference time-domain (FDTD) method, we simulate the transmission of quasi-TE-polarized Gaussian light in the defect-free air-rod PS slab and quasi-TM light in the defect-free PSrod slab (see Fig. 1). The physical parameters of octagonal QPCs used in calculation are set as follows: the dielectric constant of PS material ε = 2.53 (n = 1.59), the lattice constant a = 260 nm, the rod radius r = 65 nm and the slab thickness h = 400 nm. Fig. 1 shows that even though the refractive index of PS is low, the bandgap still occurs in two kinds of slabs in the visible spectrum. Due to the high rotational symmetry of QPCs, several localized modes within the gap are observed in both defect-free structures. The origin of these peaks in QPC is attributed to the competition between two spatial structural properties: self-similarity and non-periodicity. If the effect of disorder resulting from the non-periodicity is dominant, the electromagnetic waves with some frequencies can be localized. The detailed explanation has been referred to our previous work in reference [9]. For the air-rod pattern, the PBG as shown in Fig. 1a is ranged from 532 to 619 nm and three peaks within the gap lie at 547, 556 and 589 nm, respectively. This spectral region is exactly corresponding to the fluorescence of conjugated PS at room temperature, which makes it possible for the fabrication of PC lasers and light-emitting diodes based on polymer as active layers. Different from that in the air-rod QPC slab, the bandgap of the PS-rod pattern covers the bluish-green region (442–495 nm) and only one mode is at 480 nm inside (see Fig. 1b). For all of the modes are at the band edge, the light localization is not obvious in the electric-field distribution. Furthermore, we introduce a defect into QPC structures by removing a central scatter to form a single-defect photonic microcavity. Fig. 2 shows the corresponding transmission spectra of the modified octagonal QPC slabs with the thickness of 400 nm. Without a central air-rod, the PBG and resonant modes are almost the same as those in defect-free structures, including the location, width and transmission (see Fig. 2). In comparison, for the PS-rod structure in Fig. 2b, the original peak at 480 nm and PBG also stay the same, while a new defect mode at 486 nm arises from the lowfrequency band edge. The difference in property of two patterns is caused by the different average refractive index, even though the symmetry is the same. Based on the single-defect microcavity, we then further shift the nearest-neighbor eight scatters around the cavity outward by
Fig. 2. Transmission spectra of 2D single-defect microcavity polymer octagonal QPC slabs by removing a central rod. Inset: schematic geometry of defect air-rods in PS material (a) and PS rods in air (b).
0.2a. It is clear to see in Fig. 3a that two resonant modes in the gap are obviously red-shifting from 547 and 556 to 554 and 562 nm, respectively. Notice that the mode at 562 nm is moving close to the gap center and the quality factor Q of a cavity is increased significantly. Fig. 3a shows the field pattern of electromagnetic waves at 562 nm which is well confined in the cavity. The localized state at 562 nm satisfies the constructive interference condition at the cavity boundary, so that even under the low refractive index, it still leads to the formation of the standing wave with in-plane confinement in all directions. According to Figs. 2a and 3a, it is therefore determined that the influence of the air-rods around a microcavity on the light localization is more effective than that of a central air-rod. For the PS-rod structure, however, the movement of nearest eight dielectric rods outward in Fig. 3b hardly affects the PBG and modes. Then, we further remove the nearest eight scatters to construct a larger microcavity. It shows in Fig. 4a that three modes keep red-shifting to 567, 569 and 599 nm, and a new mode emerges at 546 nm from the high-frequency band edge. However, the redshifting of resonant modes are not obvious in the PS-rod slab (see Fig. 4b). According to the simulated results above, we conclude two differences in property for two QPC slabs. One is that with the increase in the defect size, the red-shifting of resonant peaks in wavelength for the air-rod PS slab is more distinct than that for the PS-rod slab. The other is that a new defect mode emerges from the highfrequency band edge for the air-rod structure, while for the PS-rod structure, a mode arises from the low-frequency edge. The physical mechanism behind the differences can be explained by the band theory of solid physics. As we know, photonic crystals to manipulate beams are in the same way as semiconductors to control
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Fig. 3. Transmission spectra of 2D modified polymer octagonal QPC slabs by removing the central rod and by shifting eight nearest-neighbor scatters outward by 0.2a. Inset: schematic geometry of air-rods in PS material (a) and PS-rods in air (b). The right picture in (a) is the pattern of well-confined electromagnetic waves at 562 nm in a microcavity
electrons, which are both based on the modulation of bandgaps. The high-frequency band edge is called the air band and the lowfrequency edge is the dielectric band [15]. It is quite similar to the conduction band and valence band in semiconductors. In general, the electric energy of high-frequency modes (air bandgap) is mostly distributed in the low-dielectric region, so that the modes are quite sensitive to the variation of the low-dielectric domain. And for lowfrequency modes close to the dielectric bandgap, they are sensitive to the high-dielectric region. If defects or impurities are introduced into the lattices of semiconductors, the energy levels caused by defects or impurities will occur in the forbidden band. Photonic crystals are similar to semiconductors in effect. If a defect involves the addition of extra high dielectric material (a ‘dielectric defect’ as in the case of removal of air-rods in the dielectric material), the cavity mode drops from the air band and red-shifts in wavelength cross the gap to the dielectric band by increasing the amount of dielectric defect. It is like the property by doping donor atoms in semiconductors. Meanwhile, if the defect is caused by removing the high dielectric medium (like an “air defect” by removing dielectric rods in structure), a defect mode occurs from the dielectric band and blue-shifts in wavelength to the air band when the air-defect size is increased. It is exactly like the bandgap modulation by introducing acceptor atoms in semiconductors. Therefore, we can easily tune the defect modes within the gap by modifying the defect region in photonic crystals on purpose. The reasons why the red-shift in the air-rod pattern is more evident than that in the dielectric-rod pattern lie in the interaction of two physical mechanisms. For the air-rod structure, an increase in the amount of dielectric defect makes the modes redshifting in wavelength, similar to the property by doping donor atoms in semiconductors. On the other hand, the wavelengths of those standing waves that satisfy the resonance condition are proportional to the size of a microcavity. Therefore, these two reasons positively promote the red-shifting more evident when the defect size is increased. For the dielectric-rod structure, however, the two
Fig. 4. Transmission spectra of the single-defect microcavity QPC slabs by removing the central nine rods to form a larger cavity.
mechanisms interact in a destructive way, so that the location of modes consequently hardly moved.
3. Conclusions Based on the PS material with low refractive index, we investigate light propagation in PBGs of 2D organic octagonal QPC air-rod PS slab and PS-rod slab, respectively. The simulated results show that even in low-index contrast, both the bandgap and resonant modes are observed in the visible region for two kinds of defectfree structures, which is exactly corresponding to the fluorescence of conjugated polystyrene at room temperature. When a singlepoint defect is introduced into two quasiperiodic structures, the position of emerging defect resonant modes and the red-shifting in wavelength are both quite different when the defect microcavity is increased. The difference in property is the competition consequence of two physical mechanisms, which are the effect of energy levels caused by defects in photonic crystals and the resonance of modes in the defect cavity. For the air-rod PS structure, the two mechanisms interact positively, so that it makes the red-shifting more evident. However, for the PS-rod structure, the interaction of two mechanisms is destructive, leading to the position of modes almost unchanged. These results will provide the theoretical support to modulate the defect modes in PBG and help to better guide the fabrication of light-emitting QPC devices in experiments.
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Acknowledgments This work was supported by the National Science Foundation of China (11204387), the Key Project of Science and Technology from the Ministry of Education of China (212205), the “985 Project” (Grant Nos. 98507-010009 and 98504-012004), “211 Project” of Ministry of Education of China.
[6] [7]
[8] [9]
References [1] C. Mikael, Rechtsman, Hyeong-Chai Jeong, M. Paul, Chaikin, et al., Optimized structures for photonic quasicrystals, Phys. Rev. Lett. 101 (7) (2008) 073902. [2] C. Jin, B. Cheng, B. Man, et al., Band gap and wave guiding effect in a quasiperiodic photonic crystal, Appl. Phys. Lett. 75 (13) (1999) 1848–1850. [3] C. Rochstuhl, F. Lederer, The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals, New J. Phys. 206 (8) (2006) 206. [4] Jianling Yin, Xuguang Huang, Songhao Liu, et al., Photonic bandgap properties of 8-fold symmetric photonic quasicrystals, Opt. Commun. 269 (2007) 385–388. [5] Masashi Hase, Hiroshi Miyazaki, Mitsuru Egashira, et al., Isotropic photonic band gap and anisotropic structures in transmission spectra of
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[11] [12] [13] [14] [15]
two-dimensional fivefold and eightfold symmetric quasiperiodic photonic crystals, Phys. Rev. B 66 (21) (2002) 14205. Y.S. Chan, C.T. Chan, Z.Y. Liu, Photonic band gaps in two dimensional photonic quasicrystals, Phys. Rev. Lett. 80 (5) (1998) 956–959. C. Jin, X. Meng, B. Cheng, et al., Photonic gap in amorphous photonic materials, in: Summaries of papers presented at the Conference on Lasers and ElectroOptics, 2001, pp. 587–588. Y.Q. Wang, Z.F. Feng, X.S. Xu, et al., Uncoupled defect mode in a twodimensional quasiperiodic photonic crystal, Europhys. Lett. 64 (2003) 185–189. Y. Wang, X. Hu, X. Xu, et al., Localized modes in defect-free dodecagonal quasiperiodic photonic crystals, Phys. Rev. B 68 (16) (2003) 165106. M. Notomi, H. Suzuki, T. Tamamura, et al., Lasing action due to the twodimensional quasiperiodicity of photonic quasicrystals with a penrose lattice, Phys. Rev. Lett. 92 (12) (2004) 123906. K. Wang, Light wave states in two-dimensional quasiperiodic media, Phys. Rev. B 73 (23) (2003) 235122. J. Romero-Vivas, D. Chigrin, A. Lavrinenko, et al., Resonant add-drop filter based on a photonic quasicrystal, Opt. Express 13 (3) (2005) 826–835. P.N. Dyachenko, Y.V. Miklyaev, Band structure calculation of 2D photonic pseudoquasicrystals obtainable by holographic lithograph, SPIE 6182 (2006) 61822. Fabio, Biancalana, All-optical diode action with quasiperiodic photonic crystals, J. Appl. Phys. 104 (9) (2008) 3010299. J.D. Joannopoulos, P.R. Villeneuve, S.H. Fan, Photonic crystals: putting a new twist on light, Nature 386 (13) (1997) 143.
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs Xiao Chen ∗ , Xin-Yuan Liang, Yi-Quan Wang, Shuai Feng College of Science, Minzu University of China, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 5 August 2013 Accepted 5 February 2014 Available online xxx Keywords: Photonic crystal Octagonal quasicrystal Photonic bandgap Organic polystyrene Light location Microcavity.
a b s t r a c t Based on the polystyrene material of low refractive index, light localization in photonic bandgaps of two kinds of 2D octagonal quasiperiodic photonic crystal slabs are investigated in theory, including the air-rod polystyrene slab and polystyrene-rod slab. The properties of bandgaps and localized modes in both two defect-free patterns are analyzed in detail. When a single-point defect is introduced into two quasiperiodic structures, the position of emerging defect modes and the red-shifting of resonant modes in wavelength are observed quite differently when the defect microcavity is increased in size. This difference is caused by the competition of two physical mechanisms, which are the effect of defect energy levels caused by defects introduced into photonic crystals and the resonance of modes in the defect cavity. These results will provide theoretical support for experimental fabrication of organic lightemitting quasiperiodic photonic crystal devices. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Following the pioneer works of John and Yablonovitch on periodic photonic crystals (PCs), quasi-periodic photonic crystals (QPCs) which consist of two or more materials arranged in quasiperiodic patterns have received much attention because of their remarkable properties. The structures of QPCs are not periodic but have long-range orientational order, so that it gives rise to a more isotropic Brillouin zone with smaller modulations that depress the gap [1]. Due to the greater rotational symmetry in QPCs, photonic bandgaps (PBGs) are independent of the incident light direction, which have potential applications in photonics [2–5]. Since the quasi-periodic arrangement has many in-equivalent local environments, the properties of defects are more complex and interesting than those created in conventional periodic PCs and may offer more flexibility in tuning the defect state properties [6–8]. For example, the light localization modes may occur in defect-free QPCs, which is useful for designing novel PC fibers and microcavities [9–11]. Besides, the dielectric constant necessary for the complete PBGs in QPCs is small. In the case of octagonal quasi-periodic lattice of dielectric rods in air, the first PBG for the TM polarization stays open down to the dielectric constant as small as ε = 1.6 (n = 1.26) [12], while the threshold of the dielectric constant in dodecagonal QPCs is down to ε = 1.35 [13]. This property makes it possible that
many PBG-based photonic devices can be fabricated from resourceful SiO2 (n = 1.45) or organic polymer materials. As we know, most of the PCs are made from III–V semiconductors. In contrast to inorganic optoelectronic materials, organic polymer materials, especially conjugated polymers, have advantages over the light-emitting property in the visible spectrum, the ease of fabrication and high optical nonlinearity. They have, therefore, become promising candidates as PC nano-lasers or light-emitting diodes [14]. Although the dielectric constant of organic materials is small, the combination of the light-emitting polymer and quasi-periodic structures with higher-level symmetry would provide better active layers, and more efficient, uniform in-plane confinement in all directions. Hence, it is beneficial for optoelectronic devices to achieve ultralow lasing threshold and high slope efficiency. In this paper, we investigate the physical properties of PBGs and localized modes in 2D 8-fold symmetry QPC slabs at low-index contrast. By introducing a single-point microcavity into QPCs, we observe the dependence of localized resonant modes on the distribution of air-rods around a microcavity in polystyrene (PS) slab is quite different from that in PS-rods slab. The physical mechanism behind the difference and the fundamental parameters governing PBGs and resonant modes are discussed in detail. 2. Simulation and results
∗ Corresponding author. E-mail address: [email protected] (X. Chen).
We design two kinds of 2D 8-fold symmetry organic QPC patterns, which are the air-rod PS slab and PS-rod slab, respectively.
http://dx.doi.org/10.1016/j.ijleo.2014.02.041 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Fig. 1. Schematic patterns of 2D octagonal QPC slabs with air-rods in PS material (a) and PS-rods in air (b). The transmission spectra of Gaussian light in the corresponding structures (on the right).
Based on finite-difference time-domain (FDTD) method, we simulate the transmission of quasi-TE-polarized Gaussian light in the defect-free air-rod PS slab and quasi-TM light in the defect-free PSrod slab (see Fig. 1). The physical parameters of octagonal QPCs used in calculation are set as follows: the dielectric constant of PS material ε = 2.53 (n = 1.59), the lattice constant a = 260 nm, the rod radius r = 65 nm and the slab thickness h = 400 nm. Fig. 1 shows that even though the refractive index of PS is low, the bandgap still occurs in two kinds of slabs in the visible spectrum. Due to the high rotational symmetry of QPCs, several localized modes within the gap are observed in both defect-free structures. The origin of these peaks in QPC is attributed to the competition between two spatial structural properties: self-similarity and non-periodicity. If the effect of disorder resulting from the non-periodicity is dominant, the electromagnetic waves with some frequencies can be localized. The detailed explanation has been referred to our previous work in reference [9]. For the air-rod pattern, the PBG as shown in Fig. 1a is ranged from 532 to 619 nm and three peaks within the gap lie at 547, 556 and 589 nm, respectively. This spectral region is exactly corresponding to the fluorescence of conjugated PS at room temperature, which makes it possible for the fabrication of PC lasers and light-emitting diodes based on polymer as active layers. Different from that in the air-rod QPC slab, the bandgap of the PS-rod pattern covers the bluish-green region (442–495 nm) and only one mode is at 480 nm inside (see Fig. 1b). For all of the modes are at the band edge, the light localization is not obvious in the electric-field distribution. Furthermore, we introduce a defect into QPC structures by removing a central scatter to form a single-defect photonic microcavity. Fig. 2 shows the corresponding transmission spectra of the modified octagonal QPC slabs with the thickness of 400 nm. Without a central air-rod, the PBG and resonant modes are almost the same as those in defect-free structures, including the location, width and transmission (see Fig. 2). In comparison, for the PS-rod structure in Fig. 2b, the original peak at 480 nm and PBG also stay the same, while a new defect mode at 486 nm arises from the lowfrequency band edge. The difference in property of two patterns is caused by the different average refractive index, even though the symmetry is the same. Based on the single-defect microcavity, we then further shift the nearest-neighbor eight scatters around the cavity outward by
Fig. 2. Transmission spectra of 2D single-defect microcavity polymer octagonal QPC slabs by removing a central rod. Inset: schematic geometry of defect air-rods in PS material (a) and PS rods in air (b).
0.2a. It is clear to see in Fig. 3a that two resonant modes in the gap are obviously red-shifting from 547 and 556 to 554 and 562 nm, respectively. Notice that the mode at 562 nm is moving close to the gap center and the quality factor Q of a cavity is increased significantly. Fig. 3a shows the field pattern of electromagnetic waves at 562 nm which is well confined in the cavity. The localized state at 562 nm satisfies the constructive interference condition at the cavity boundary, so that even under the low refractive index, it still leads to the formation of the standing wave with in-plane confinement in all directions. According to Figs. 2a and 3a, it is therefore determined that the influence of the air-rods around a microcavity on the light localization is more effective than that of a central air-rod. For the PS-rod structure, however, the movement of nearest eight dielectric rods outward in Fig. 3b hardly affects the PBG and modes. Then, we further remove the nearest eight scatters to construct a larger microcavity. It shows in Fig. 4a that three modes keep red-shifting to 567, 569 and 599 nm, and a new mode emerges at 546 nm from the high-frequency band edge. However, the redshifting of resonant modes are not obvious in the PS-rod slab (see Fig. 4b). According to the simulated results above, we conclude two differences in property for two QPC slabs. One is that with the increase in the defect size, the red-shifting of resonant peaks in wavelength for the air-rod PS slab is more distinct than that for the PS-rod slab. The other is that a new defect mode emerges from the highfrequency band edge for the air-rod structure, while for the PS-rod structure, a mode arises from the low-frequency edge. The physical mechanism behind the differences can be explained by the band theory of solid physics. As we know, photonic crystals to manipulate beams are in the same way as semiconductors to control
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Fig. 3. Transmission spectra of 2D modified polymer octagonal QPC slabs by removing the central rod and by shifting eight nearest-neighbor scatters outward by 0.2a. Inset: schematic geometry of air-rods in PS material (a) and PS-rods in air (b). The right picture in (a) is the pattern of well-confined electromagnetic waves at 562 nm in a microcavity
electrons, which are both based on the modulation of bandgaps. The high-frequency band edge is called the air band and the lowfrequency edge is the dielectric band [15]. It is quite similar to the conduction band and valence band in semiconductors. In general, the electric energy of high-frequency modes (air bandgap) is mostly distributed in the low-dielectric region, so that the modes are quite sensitive to the variation of the low-dielectric domain. And for lowfrequency modes close to the dielectric bandgap, they are sensitive to the high-dielectric region. If defects or impurities are introduced into the lattices of semiconductors, the energy levels caused by defects or impurities will occur in the forbidden band. Photonic crystals are similar to semiconductors in effect. If a defect involves the addition of extra high dielectric material (a ‘dielectric defect’ as in the case of removal of air-rods in the dielectric material), the cavity mode drops from the air band and red-shifts in wavelength cross the gap to the dielectric band by increasing the amount of dielectric defect. It is like the property by doping donor atoms in semiconductors. Meanwhile, if the defect is caused by removing the high dielectric medium (like an “air defect” by removing dielectric rods in structure), a defect mode occurs from the dielectric band and blue-shifts in wavelength to the air band when the air-defect size is increased. It is exactly like the bandgap modulation by introducing acceptor atoms in semiconductors. Therefore, we can easily tune the defect modes within the gap by modifying the defect region in photonic crystals on purpose. The reasons why the red-shift in the air-rod pattern is more evident than that in the dielectric-rod pattern lie in the interaction of two physical mechanisms. For the air-rod structure, an increase in the amount of dielectric defect makes the modes redshifting in wavelength, similar to the property by doping donor atoms in semiconductors. On the other hand, the wavelengths of those standing waves that satisfy the resonance condition are proportional to the size of a microcavity. Therefore, these two reasons positively promote the red-shifting more evident when the defect size is increased. For the dielectric-rod structure, however, the two
Fig. 4. Transmission spectra of the single-defect microcavity QPC slabs by removing the central nine rods to form a larger cavity.
mechanisms interact in a destructive way, so that the location of modes consequently hardly moved.
3. Conclusions Based on the PS material with low refractive index, we investigate light propagation in PBGs of 2D organic octagonal QPC air-rod PS slab and PS-rod slab, respectively. The simulated results show that even in low-index contrast, both the bandgap and resonant modes are observed in the visible region for two kinds of defectfree structures, which is exactly corresponding to the fluorescence of conjugated polystyrene at room temperature. When a singlepoint defect is introduced into two quasiperiodic structures, the position of emerging defect resonant modes and the red-shifting in wavelength are both quite different when the defect microcavity is increased. The difference in property is the competition consequence of two physical mechanisms, which are the effect of energy levels caused by defects in photonic crystals and the resonance of modes in the defect cavity. For the air-rod PS structure, the two mechanisms interact positively, so that it makes the red-shifting more evident. However, for the PS-rod structure, the interaction of two mechanisms is destructive, leading to the position of modes almost unchanged. These results will provide the theoretical support to modulate the defect modes in PBG and help to better guide the fabrication of light-emitting QPC devices in experiments.
Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041
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Acknowledgments This work was supported by the National Science Foundation of China (11204387), the Key Project of Science and Technology from the Ministry of Education of China (212205), the “985 Project” (Grant Nos. 98507-010009 and 98504-012004), “211 Project” of Ministry of Education of China.
[6] [7]
[8] [9]
References [1] C. Mikael, Rechtsman, Hyeong-Chai Jeong, M. Paul, Chaikin, et al., Optimized structures for photonic quasicrystals, Phys. Rev. Lett. 101 (7) (2008) 073902. [2] C. Jin, B. Cheng, B. Man, et al., Band gap and wave guiding effect in a quasiperiodic photonic crystal, Appl. Phys. Lett. 75 (13) (1999) 1848–1850. [3] C. Rochstuhl, F. Lederer, The effect of disorder on the local density of states in two-dimensional quasi-periodic photonic crystals, New J. Phys. 206 (8) (2006) 206. [4] Jianling Yin, Xuguang Huang, Songhao Liu, et al., Photonic bandgap properties of 8-fold symmetric photonic quasicrystals, Opt. Commun. 269 (2007) 385–388. [5] Masashi Hase, Hiroshi Miyazaki, Mitsuru Egashira, et al., Isotropic photonic band gap and anisotropic structures in transmission spectra of
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Please cite this article in press as: X. Chen, et al., Theoretical study of light localization in photonic bandgaps of organic octagonal quasiperiodic photonic crystal slabs, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.02.041