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Multiple slow light bands in photonic crystal coupled resonator optical waveguides constructed with a portion of photonic quasicrystals
Physics Letters A 375 (2011) 712–715
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Multiple slow light bands in photonic crystal coupled resonator optical waveguides constructed with a portion of photonic quasicrystals Yun Shen a,∗ , Guoping Wang b , Guoping Yu b , Jiwu Fu a , Lin Chen c a b c
Department of Physics, Nanchang University, Nanchang 330031, China Key Laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education and Department of Physics, Wuhan University, Wuhan 430072, China Wuhan National Laboratory for Optoelectronics, Huazhong University of Science & Technology, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 23 November 2010 Accepted 5 December 2010 Available online 9 December 2010 Communicated by R. Wu Keywords: Multiple slow light bands Complex photonic crystals Photonic quasicrystal Coupled resonator optical waveguides
a b s t r a c t Coupled resonator optical waveguides (CROWs) in complex two-dimensional (2D) photonic crystals (PCs) constructed with a portion of 12-fold photonic quasicrystals (PQs) are proposed. We show that enhanced transmission and slow light can be simultaneously achieved in such waveguides as well as general CROWs. Moreover, due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs, multiple slow light bands can be flexibly obtained in CROWs constructed with complex 2D PCs. Our results may lead to the development of a variety of novel ultracompact devices for photonic integrated circuits. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Coupled resonator optical waveguides (CROWs) [1–4] constructed in photonic crystals (PCs) have attracted much attention since they were proposed in 1999 [1]. In which, the high efficient transmission [5–7], large optical field amplitude, and low group velocity can be achieved [8,9]. Due to those prominent properties, such waveguides can serve as delay line [10,11], wavelength filter [12], and demultiplexer [13–15], also can be used for pulse compression [11,15,16], enhancement of nonlinear effects [8], etc. Recently, the particular CROWs structures constructed with a chain of PC interlaced microcavity are proposed to realize multiple slow light bands [17], which may lead to the development of a variety of novel devices such as multichannel modulators, switches, tunable amplifiers and lasers, for photonic integrated circuits. In these particular CROWs structures, the interlaced microcavity is a combination of several PC microcavities, each of which has a particular defect radius. On the other hand, photonic quasicrystals (PQs) allow for a higher degree of flexibility and tunability for defect mode properties than conventional PCs. In PQs, the arrangement does not have translational symmetry and the cylinders have different local environments. Thus, the defect mode properties can be tuned by changing the nature and size of the defects, like in periodic
*
Corresponding author. Tel./fax: +86 791 3969238. E-mail address: [email protected] (Y. Shen).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.12.006
PCs, but also by removing cylinder from different locations. Which offers more flexibility and leads to a large variety of possibilities [18,19]. However, the exact theoretical prediction on the photonic band structure of PQs is currently a numerical challenge due to the lack of periodicity [20]. Additionally, entire infinite aperiodic structure is practically impossible, while periodic one is essential for optical implementation such as nanocavity array laser [21], PC array negative refraction [22], and CROWs. Recently, particular structures of complex 2D PCs constructed with a portion of 12fold PQs were reported in both numerical [20] and experimental [23] work. It is shown that complex PCs can indeed produce similar photonic properties of the original 12-fold PQs while maintain periodic scattering to light in long range order. In this Letter, CROWs in complex 2D PCs constructed with a portion of 12-fold PQs are proposed. Finite-difference time-domain (FDTD) simulations show that enhanced transmission and slow light can be simultaneously achieved. Moreover, multiple slow light bands can be flexibly obtained in CROWs constructed with complex 2D PCs due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs. 2. Optical properties of complex 2D PCs and structural defect cavities Fig. 1 shows the proposed complex 2D PCs constructed with a portion of 12-fold symmetric PQs (red solid lines). The whole structure is a perfect equal triangular lattice with 7 cylinders per
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
Fig. 1. Schematic geometry of the proposed complex 2D PCs constructed with a portion of 12-fold symmetric PQs (red solid lines). The whole structure is a perfect equal triangular lattice with 7 cylinders per unit cell (centered hexagons with black √ dotted lines). The period in the x-direction is (1 + 3 )a, where a is the lattice constant. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
unit cell (centered hexagons with black dotted lines), which can be defined as Wigner–Seitz primitive cell. All cylinders that compose the cell have five closet neighbors except the central one surrounded by six, leading √ to different local environment. Period in the x-direction is (1 + 3 )a (a is the lattice constant). Previous studies [20] have theoretically demonstrated that the perfect complex 2D PCs show significant advantages over the simple triangular structures in photonic band gap (PBG) properties and can indeed produce similar photonic properties of the original 12-fold PQs. Analyzing the geometry of the complex 2D PCs, we can see because of different local environment, the arrangement of cylinders in Wigner–Seitz primitive cell does not have translational symmetry. So in infinite complex 2D PCs, when we remove central cylinder A (see Fig. 1), the structural defect cavity is different from that by removing one of the six surrounding cylinders, for example, cylinder B (see Fig. 1). Similarly, removal of two cylinders, such as A and B, B and C, B and D, B and E, four different structural defect cavities can be obtained. Also, more complex and interesting defect cavities can be offered by removing more cylinders. Thus, complex 2D PCs can offer a great variety of possibilities for creating and controlling the number of defect cavity modes. In the following, we perform FDTD numerical simulation to demonstrate the optical properties of complex 2D PCs and its defect cavities, then the optical properties of CROWs constructed with defect cavities. In our simulation, cylinders are embedded in air of dielectric constant εb = 1 in TE polarization (electric field parallel to rods axis), with dielectric constant εa = 13 and radii R = 0.22a, where a = 709 nm is set according to experiment [23]. Besides, other conditions need to be met. Periodic boundary conditions are touched with the slab sample’s upper and lower edges, perpendicular to y-axis. Perfectly matched layer absorbing boundary conditions applied to terminate the computation space are far away from sample’s front and back edges, perpendicular to x-axis. And two lines as the sampling incident and transmitted windows, extending from the upper to lower edges are used for transmission coefficient measurement. One is close to the front edge and the other adjacent to the back surface of the slab sample. Then, the incident and transmitted energy fluxes of electromagnetic waves are calculated and summed up to yield the transmission coefficient. Samples of perfect complex 2D PCs, defect cavity I and defect cavity II are shown in Fig. 2(a). The distance between upper and
713
Fig. 2. (a) Samples of perfect complex 2D PCs, defect cavity I and defect cavity II. (b) Transmission spectra of perfect complex 2D PCs sample (black solid curve), cavity I (blue solid curve) and cavity II (red dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
lower, front and back edges defined as width, length is 6.71 um, 5.81 um, respectively. Here, as examples of the defect cavities, cavity I and cavity II are severally constructed by removal of cylinders B–C and cylinder B from a Wigner–Seitz primitive cell. Fig. 2(b) illustrates the transmission spectra of the perfect complex 2D PCs, cavity I and cavity II separately in black solid curve, blue curve and red dotted curve. For the complex 2D PCs [black solid curve in Fig. 2(b)], we can see two spectral gaps extending from 0.2348 to 0.3993, 0.5 to 0.561 in normalized frequency range, respectively. As to cavity I [blue solid curve in Fig. 2(b)], three peaks a, b and c appear at normalized frequency of 0.3377, 0.3912 and 0.5270 with transmission of 9.292 × 10−5 , 1.109 × 10−3 and 5.614 × 10−3 , respectively. As regards cavity II [red dotted curve in Fig. 2(b)], there are two peaks d and e, at normalized frequency of 0.3663 and 0.5126, with individual transmission of 3.496 × 10−4 and 4.64 × 10−3 . As peaks indicate the existence of localized modes [4], three kinds of localized modes exist in cavity I and two in cavity II. Same localized modes coupling, waveguide modes come into being in structure of CROWs [12]. 3. Optical properties of CROWs constructed in complex 2D PCs Sample of CROW constructed with five cavities I is shown in Fig. 3(a), in which, the width and length of the slab sample is 6.71 um and 9.69 um, respectively. The calculated transmission spectrum of this structural CROW for TE polarization is displayed in Fig. 3(b). From Fig. 3(b) we can see three minibands marked as a, b, c, displaying spectral oscillation ranging from normalized frequency of 0.3315 to 0.3428, 0.3863 to 0.3974, 0.5198 to 0.5333, with the maximum value of transmission 3.067 × 10−2 , 3.416 × 10−2 , 7.489 × 10−3 , respectively. Here, the spectral oscillation is attributed to the effect of finite length of CROW in the propagation direction of the light [4]. Compared to transmission spectrum of cavity I [blue curves in Fig. 2(b)], it’s illustrated that the minibands are located around the localized modes correspondingly, and higher transmission are realized. It’s well known that the properties of CROW generally can be simply deduced from those of individual defect cavities, and there are overall correspondences between CROW bands and localized modes of coupled defect cavi-
714
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
Fig. 4. Group velocities of light propagating in the CROW shown in Fig. 3(a). The maximum value of a, b, and c, corresponding to the minibands a, b, and c in Fig. 3(b), is 0.09699, 0.09527, and 0.11590, respectively.
Fig. 3. Sample of CROW constructed with five cavities I (a) and its transmission spectrum (b). The electric field distributions of waveguide modes of bands a–c in (b) are shown in (c).
ties [12]. These are confirmed in above comparison. Fig. 3(c) shows the electric field distributions of waveguide modes of bands a–c in Fig. 3(b). The three different spatial distributions in Fig. 3(c) mean that three waveguide modes can be offered by such CROW. The group velocity corresponding to the minibands can be calculated by adopting tight-binding approximation [4]. Keeping only the nearest-neighbor coupling, the dispersion relationship for a CROW waveguide mode is
ω( K ) = Ω 1 + κ cos( K Λ) ,
(1)
where ω( K ), K , and Ω are the angular frequency, wave vector along the direction of the waveguide, and angular eigenfrequency of solitary cavity, respectively. κ and Λ are the coupling strength and distance between two nearest-neighbor cavities. Owing to the periodicity structure of CROWs, the wave vector can be folded into the region [0, π /Λ]. Thus, the width of the miniband of the waveguide mode is
ω = 2|κ |Ω.
(2)
From Eqs. (1) and (2), we can obtain
υ g ( K ) = −ΩΛκ sin( K Λ) = −
ωκ Λ sin( K Λ). 2 |κ |
(3)
For CROW shown in Fig. 3(a) in our simulation, the distance Λ is 1937 nm. According to the minibands width shown in Fig. 3(b), the calculated group velocities of the waveguide modes are displayed in Fig. 4. The maximum value of a, b, and c, corresponding to the minibands a, b, and c, is 0.09699, 0.09527, and 0.11590, respectively. These indicate three slow light bands are realized in this CROW. Sample of CROW consisted of five repeated interlaced cavities is shown in Fig. 5(a), in which the interlaced cavities is a combination of cavity I and cavity II. The width, length of the slab sample is 6.71 um, 19.37 um, respectively. The calculated transmission spectrum of the structure for TE polarization is displayed in Fig. 5(b). It shows five high discrete peaks marked a, d, b, e, and c with spectral oscillation ranging from normalized frequency of 0.3361 to 0.3370, 0.3642 to 0.3662, 0.3924 to 0.3940, 0.5094 to 0.5113, and 0.5268 to 0.5299, corresponding to transmission up to 6.340 × 10−3 , 7.729 × 10−3 , 2.674 × 10−2 , 1.441 × 10−3 , and 4.320 × 10−3 , respectively. The transmission spectra at normalized frequencies around the five peaks are calculated with a
Fig. 5. Sample of CROW constructed with cavities I and cavities II alternately (a) and its transmission spectrum (b). For peaks a, d, b, e, and c, the enlarged spectrum is displayed in panels (c), (d), (e), (f), and (g), respectively; the group velocities and electric field distributions of waveguide modes are shown in (h) and (i), respectively.
high resolution and illustrated in Fig. 5(c)–(g), respectively. The results show that the five peaks are indeed minibands, indicating five waveguide modes in the special CROW. Compared to localized modes a, b, c of cavity I [blue curve in Fig. 2(b)] and d, e of cavity II [red dotted curve in Fig. 2(b)], the minibands a, b, c, and d, e in the CROW are located around the localized modes correspondingly with higher transmission. According to the minibands width in Fig. 5(b) and Eqs. (1)–(3), the calculated group velocities of the waveguide modes are displayed in Fig. 5(h). In the calculation, the distance Λ is 3874 nm as the interlace cavity is the combination of cavities I and cavities II. Fig. 5(h) illustrates that five slow light bands are realized in this CROW, the maximum value of curves a, d, b, e and c, in accordance with the minibands a, d, b, e and c in Fig. 5(b), is 0.01545, 0.03433, 0.02747, 0.03262 and 0.05321,
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
respectively. Fig. 5(i) shows the electric field distributions of the waveguide modes of bands a–e in Fig. 5(b), and demonstrates five waveguide modes are obtained in such CROW. The operation of interlaced-cavity CROW [Fig. 5(a)] is analogous to that of normal-cavity CROW [Fig. 3(a)], where photons hop from one defect to the next, except that there are now multiple cavities. Thus, light couples from one defect cavity to the next of the same type, skipping over the intervening cavities. This is demonstrated in Fig. 5(i), in which, modes of bands a, b, and c are localized to cavities I, d and e are localized to cavities II. Here, the skipping behavior means an increased distance Λ and a decreased κ in comparison with that in normal-cavity CROW. As variation of κ [and so ω in Eq. (2)] is greater than that of Λ, slower υ g [Eq. (3)] can be obtained. It can be demonstrated by comparing the group velocity of a, b, c in Fig. 4 with those in Fig. 5(h). Additionally, as overall correspondences exist between CROW bands and localized modes of coupled defect cavities [12,17], interlaced-cavity CROW can exhibit more possibilities of slow light bands than normalcavity CROW because the former contains multiple cavities, each of which has different localized modes. Moreover, different defect cavities can be flexibly presented in the complex 2D PCs in our Letter, therefore large variety of possibilities of slow light bands can be flexibly obtained in normal-cavity CROW by varying the type of defect cavities, and in interlaced-cavity CROW by varying the combination of different cavities. 4. Summary In conclusion, CROWs in complex 2D PCs constructed with a portion of 12-fold PQs are proposed and their optical properties are demonstrated by using FDTD simulation. The results show that enhanced transmission and slow light bands can be simultaneously achieved in such waveguides as well as general CROWs. Moreover, due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs, multiple slow light bands can be flexibly obtained in different kinds
715
of CROWs in complex 2D PCs. Our results may lead to the development of a variety of novel ultracompact devices such as multichannel modulators, switches, and multichannel tunable amplifiers for photonic integrated circuits. Acknowledgements We thank W.R. Hu, D.H. Fan and Y.J. Shen for their help. This work is supported in part by the National Natural Science Foundation of China (Grant No. 60967003), the Nanchang University Foundation, China (Grant No. Z03355), and the Key laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education, in Wuhan University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
A. Yariv, Y. Xu, R.K. Lee, A. Scherer, Opt. Lett. 24 (1999) 711. Y. Shen, G.P. Wang, Opt. Express 17 (2009) 12807. J.K. Poon, L. Zhu, G.A. DeRose, A. Yariv, Opt. Lett. 31 (2006) 456. Y. Wang, J. Liu, B. Zhang, S. Feng, Z.Y. Li, Phys. Rev. B 73 (2006) 155107. Z.H. Zhu, W.M. Ye, J.R. Ji, X.D. Yuan, C. Zen, Appl. Phys. B 86 (2007) 327. M. Bayindir, B. Temelkuran, E. Ozbay, Phys. Rev. B 61 (2000) R11855. A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, Phys. Rev. Lett. 77 (1996) 3787. M. Soljacic, J.D. Joannopoulos, Nature Mater. 3 (2004) 211. M. Bayindir, B. Temelkuran, E. Ozbay, Phys. Rev. Lett. 84 (2000) 2140. J.K.S. Poon, J. Scheuer, Y. Xu, A. Yariv, J. Opt. Soc. Am. B 21 (2004) 1665. A. Martinez, A. Garcia, P. Sanchis, J. Marti, J. Opt. Soc. Am. A 20 (2003) 147. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, U. Oesterle, Opt. Lett. 26 (2001) 1019. A. Maritinez, J. Marti, J. Bravo-abad, Fiber Integrated Opt. 22 (2003) 151. M. Bayindir, B. Temelkuran, E. Ozbay, Appl. Phys. Lett. 77 (2000) 3902. M. Bayindir, E. Ozbay, Opt. Express 10 (2002) 1279. T. Baba, Nature Photon. 2 (2008) 465. S.P. Anderson, A.R. Shroff, P.M. Fauchet, Adv. Opt. Technol. 2008 (2008) 293531. S.S.M. Cheng, L.M. Li, C.T. Chan, Z.Q. Zhang, Phys. Rev. B 59 (1999) 4091. Y.S. Chan, C.T. Chan, Z.Y. Liu, Phys. Rev. Lett. 80 (1998) 956. Y. Yang, G.P. Wang, Opt. Express 15 (2007) 5991. H. Altug, J. Vuckovic, Opt. Express 13 (2005) 8819. A. Locatelli, M. Conforti, D. Modotto, C. DeAngelis, Opt. Lett. 31 (2006) 1343. Y. Yang, Q. Li, G.P. Wang, Appl. Phys. Lett. 93 (2008) 061112.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Multiple slow light bands in photonic crystal coupled resonator optical waveguides constructed with a portion of photonic quasicrystals Yun Shen a,∗ , Guoping Wang b , Guoping Yu b , Jiwu Fu a , Lin Chen c a b c
Department of Physics, Nanchang University, Nanchang 330031, China Key Laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education and Department of Physics, Wuhan University, Wuhan 430072, China Wuhan National Laboratory for Optoelectronics, Huazhong University of Science & Technology, Wuhan 430074, China
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 23 November 2010 Accepted 5 December 2010 Available online 9 December 2010 Communicated by R. Wu Keywords: Multiple slow light bands Complex photonic crystals Photonic quasicrystal Coupled resonator optical waveguides
a b s t r a c t Coupled resonator optical waveguides (CROWs) in complex two-dimensional (2D) photonic crystals (PCs) constructed with a portion of 12-fold photonic quasicrystals (PQs) are proposed. We show that enhanced transmission and slow light can be simultaneously achieved in such waveguides as well as general CROWs. Moreover, due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs, multiple slow light bands can be flexibly obtained in CROWs constructed with complex 2D PCs. Our results may lead to the development of a variety of novel ultracompact devices for photonic integrated circuits. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Coupled resonator optical waveguides (CROWs) [1–4] constructed in photonic crystals (PCs) have attracted much attention since they were proposed in 1999 [1]. In which, the high efficient transmission [5–7], large optical field amplitude, and low group velocity can be achieved [8,9]. Due to those prominent properties, such waveguides can serve as delay line [10,11], wavelength filter [12], and demultiplexer [13–15], also can be used for pulse compression [11,15,16], enhancement of nonlinear effects [8], etc. Recently, the particular CROWs structures constructed with a chain of PC interlaced microcavity are proposed to realize multiple slow light bands [17], which may lead to the development of a variety of novel devices such as multichannel modulators, switches, tunable amplifiers and lasers, for photonic integrated circuits. In these particular CROWs structures, the interlaced microcavity is a combination of several PC microcavities, each of which has a particular defect radius. On the other hand, photonic quasicrystals (PQs) allow for a higher degree of flexibility and tunability for defect mode properties than conventional PCs. In PQs, the arrangement does not have translational symmetry and the cylinders have different local environments. Thus, the defect mode properties can be tuned by changing the nature and size of the defects, like in periodic
*
Corresponding author. Tel./fax: +86 791 3969238. E-mail address: [email protected] (Y. Shen).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.12.006
PCs, but also by removing cylinder from different locations. Which offers more flexibility and leads to a large variety of possibilities [18,19]. However, the exact theoretical prediction on the photonic band structure of PQs is currently a numerical challenge due to the lack of periodicity [20]. Additionally, entire infinite aperiodic structure is practically impossible, while periodic one is essential for optical implementation such as nanocavity array laser [21], PC array negative refraction [22], and CROWs. Recently, particular structures of complex 2D PCs constructed with a portion of 12fold PQs were reported in both numerical [20] and experimental [23] work. It is shown that complex PCs can indeed produce similar photonic properties of the original 12-fold PQs while maintain periodic scattering to light in long range order. In this Letter, CROWs in complex 2D PCs constructed with a portion of 12-fold PQs are proposed. Finite-difference time-domain (FDTD) simulations show that enhanced transmission and slow light can be simultaneously achieved. Moreover, multiple slow light bands can be flexibly obtained in CROWs constructed with complex 2D PCs due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs. 2. Optical properties of complex 2D PCs and structural defect cavities Fig. 1 shows the proposed complex 2D PCs constructed with a portion of 12-fold symmetric PQs (red solid lines). The whole structure is a perfect equal triangular lattice with 7 cylinders per
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
Fig. 1. Schematic geometry of the proposed complex 2D PCs constructed with a portion of 12-fold symmetric PQs (red solid lines). The whole structure is a perfect equal triangular lattice with 7 cylinders per unit cell (centered hexagons with black √ dotted lines). The period in the x-direction is (1 + 3 )a, where a is the lattice constant. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
unit cell (centered hexagons with black dotted lines), which can be defined as Wigner–Seitz primitive cell. All cylinders that compose the cell have five closet neighbors except the central one surrounded by six, leading √ to different local environment. Period in the x-direction is (1 + 3 )a (a is the lattice constant). Previous studies [20] have theoretically demonstrated that the perfect complex 2D PCs show significant advantages over the simple triangular structures in photonic band gap (PBG) properties and can indeed produce similar photonic properties of the original 12-fold PQs. Analyzing the geometry of the complex 2D PCs, we can see because of different local environment, the arrangement of cylinders in Wigner–Seitz primitive cell does not have translational symmetry. So in infinite complex 2D PCs, when we remove central cylinder A (see Fig. 1), the structural defect cavity is different from that by removing one of the six surrounding cylinders, for example, cylinder B (see Fig. 1). Similarly, removal of two cylinders, such as A and B, B and C, B and D, B and E, four different structural defect cavities can be obtained. Also, more complex and interesting defect cavities can be offered by removing more cylinders. Thus, complex 2D PCs can offer a great variety of possibilities for creating and controlling the number of defect cavity modes. In the following, we perform FDTD numerical simulation to demonstrate the optical properties of complex 2D PCs and its defect cavities, then the optical properties of CROWs constructed with defect cavities. In our simulation, cylinders are embedded in air of dielectric constant εb = 1 in TE polarization (electric field parallel to rods axis), with dielectric constant εa = 13 and radii R = 0.22a, where a = 709 nm is set according to experiment [23]. Besides, other conditions need to be met. Periodic boundary conditions are touched with the slab sample’s upper and lower edges, perpendicular to y-axis. Perfectly matched layer absorbing boundary conditions applied to terminate the computation space are far away from sample’s front and back edges, perpendicular to x-axis. And two lines as the sampling incident and transmitted windows, extending from the upper to lower edges are used for transmission coefficient measurement. One is close to the front edge and the other adjacent to the back surface of the slab sample. Then, the incident and transmitted energy fluxes of electromagnetic waves are calculated and summed up to yield the transmission coefficient. Samples of perfect complex 2D PCs, defect cavity I and defect cavity II are shown in Fig. 2(a). The distance between upper and
713
Fig. 2. (a) Samples of perfect complex 2D PCs, defect cavity I and defect cavity II. (b) Transmission spectra of perfect complex 2D PCs sample (black solid curve), cavity I (blue solid curve) and cavity II (red dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
lower, front and back edges defined as width, length is 6.71 um, 5.81 um, respectively. Here, as examples of the defect cavities, cavity I and cavity II are severally constructed by removal of cylinders B–C and cylinder B from a Wigner–Seitz primitive cell. Fig. 2(b) illustrates the transmission spectra of the perfect complex 2D PCs, cavity I and cavity II separately in black solid curve, blue curve and red dotted curve. For the complex 2D PCs [black solid curve in Fig. 2(b)], we can see two spectral gaps extending from 0.2348 to 0.3993, 0.5 to 0.561 in normalized frequency range, respectively. As to cavity I [blue solid curve in Fig. 2(b)], three peaks a, b and c appear at normalized frequency of 0.3377, 0.3912 and 0.5270 with transmission of 9.292 × 10−5 , 1.109 × 10−3 and 5.614 × 10−3 , respectively. As regards cavity II [red dotted curve in Fig. 2(b)], there are two peaks d and e, at normalized frequency of 0.3663 and 0.5126, with individual transmission of 3.496 × 10−4 and 4.64 × 10−3 . As peaks indicate the existence of localized modes [4], three kinds of localized modes exist in cavity I and two in cavity II. Same localized modes coupling, waveguide modes come into being in structure of CROWs [12]. 3. Optical properties of CROWs constructed in complex 2D PCs Sample of CROW constructed with five cavities I is shown in Fig. 3(a), in which, the width and length of the slab sample is 6.71 um and 9.69 um, respectively. The calculated transmission spectrum of this structural CROW for TE polarization is displayed in Fig. 3(b). From Fig. 3(b) we can see three minibands marked as a, b, c, displaying spectral oscillation ranging from normalized frequency of 0.3315 to 0.3428, 0.3863 to 0.3974, 0.5198 to 0.5333, with the maximum value of transmission 3.067 × 10−2 , 3.416 × 10−2 , 7.489 × 10−3 , respectively. Here, the spectral oscillation is attributed to the effect of finite length of CROW in the propagation direction of the light [4]. Compared to transmission spectrum of cavity I [blue curves in Fig. 2(b)], it’s illustrated that the minibands are located around the localized modes correspondingly, and higher transmission are realized. It’s well known that the properties of CROW generally can be simply deduced from those of individual defect cavities, and there are overall correspondences between CROW bands and localized modes of coupled defect cavi-
714
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
Fig. 4. Group velocities of light propagating in the CROW shown in Fig. 3(a). The maximum value of a, b, and c, corresponding to the minibands a, b, and c in Fig. 3(b), is 0.09699, 0.09527, and 0.11590, respectively.
Fig. 3. Sample of CROW constructed with five cavities I (a) and its transmission spectrum (b). The electric field distributions of waveguide modes of bands a–c in (b) are shown in (c).
ties [12]. These are confirmed in above comparison. Fig. 3(c) shows the electric field distributions of waveguide modes of bands a–c in Fig. 3(b). The three different spatial distributions in Fig. 3(c) mean that three waveguide modes can be offered by such CROW. The group velocity corresponding to the minibands can be calculated by adopting tight-binding approximation [4]. Keeping only the nearest-neighbor coupling, the dispersion relationship for a CROW waveguide mode is
ω( K ) = Ω 1 + κ cos( K Λ) ,
(1)
where ω( K ), K , and Ω are the angular frequency, wave vector along the direction of the waveguide, and angular eigenfrequency of solitary cavity, respectively. κ and Λ are the coupling strength and distance between two nearest-neighbor cavities. Owing to the periodicity structure of CROWs, the wave vector can be folded into the region [0, π /Λ]. Thus, the width of the miniband of the waveguide mode is
ω = 2|κ |Ω.
(2)
From Eqs. (1) and (2), we can obtain
υ g ( K ) = −ΩΛκ sin( K Λ) = −
ωκ Λ sin( K Λ). 2 |κ |
(3)
For CROW shown in Fig. 3(a) in our simulation, the distance Λ is 1937 nm. According to the minibands width shown in Fig. 3(b), the calculated group velocities of the waveguide modes are displayed in Fig. 4. The maximum value of a, b, and c, corresponding to the minibands a, b, and c, is 0.09699, 0.09527, and 0.11590, respectively. These indicate three slow light bands are realized in this CROW. Sample of CROW consisted of five repeated interlaced cavities is shown in Fig. 5(a), in which the interlaced cavities is a combination of cavity I and cavity II. The width, length of the slab sample is 6.71 um, 19.37 um, respectively. The calculated transmission spectrum of the structure for TE polarization is displayed in Fig. 5(b). It shows five high discrete peaks marked a, d, b, e, and c with spectral oscillation ranging from normalized frequency of 0.3361 to 0.3370, 0.3642 to 0.3662, 0.3924 to 0.3940, 0.5094 to 0.5113, and 0.5268 to 0.5299, corresponding to transmission up to 6.340 × 10−3 , 7.729 × 10−3 , 2.674 × 10−2 , 1.441 × 10−3 , and 4.320 × 10−3 , respectively. The transmission spectra at normalized frequencies around the five peaks are calculated with a
Fig. 5. Sample of CROW constructed with cavities I and cavities II alternately (a) and its transmission spectrum (b). For peaks a, d, b, e, and c, the enlarged spectrum is displayed in panels (c), (d), (e), (f), and (g), respectively; the group velocities and electric field distributions of waveguide modes are shown in (h) and (i), respectively.
high resolution and illustrated in Fig. 5(c)–(g), respectively. The results show that the five peaks are indeed minibands, indicating five waveguide modes in the special CROW. Compared to localized modes a, b, c of cavity I [blue curve in Fig. 2(b)] and d, e of cavity II [red dotted curve in Fig. 2(b)], the minibands a, b, c, and d, e in the CROW are located around the localized modes correspondingly with higher transmission. According to the minibands width in Fig. 5(b) and Eqs. (1)–(3), the calculated group velocities of the waveguide modes are displayed in Fig. 5(h). In the calculation, the distance Λ is 3874 nm as the interlace cavity is the combination of cavities I and cavities II. Fig. 5(h) illustrates that five slow light bands are realized in this CROW, the maximum value of curves a, d, b, e and c, in accordance with the minibands a, d, b, e and c in Fig. 5(b), is 0.01545, 0.03433, 0.02747, 0.03262 and 0.05321,
Y. Shen et al. / Physics Letters A 375 (2011) 712–715
respectively. Fig. 5(i) shows the electric field distributions of the waveguide modes of bands a–e in Fig. 5(b), and demonstrates five waveguide modes are obtained in such CROW. The operation of interlaced-cavity CROW [Fig. 5(a)] is analogous to that of normal-cavity CROW [Fig. 3(a)], where photons hop from one defect to the next, except that there are now multiple cavities. Thus, light couples from one defect cavity to the next of the same type, skipping over the intervening cavities. This is demonstrated in Fig. 5(i), in which, modes of bands a, b, and c are localized to cavities I, d and e are localized to cavities II. Here, the skipping behavior means an increased distance Λ and a decreased κ in comparison with that in normal-cavity CROW. As variation of κ [and so ω in Eq. (2)] is greater than that of Λ, slower υ g [Eq. (3)] can be obtained. It can be demonstrated by comparing the group velocity of a, b, c in Fig. 4 with those in Fig. 5(h). Additionally, as overall correspondences exist between CROW bands and localized modes of coupled defect cavities [12,17], interlaced-cavity CROW can exhibit more possibilities of slow light bands than normalcavity CROW because the former contains multiple cavities, each of which has different localized modes. Moreover, different defect cavities can be flexibly presented in the complex 2D PCs in our Letter, therefore large variety of possibilities of slow light bands can be flexibly obtained in normal-cavity CROW by varying the type of defect cavities, and in interlaced-cavity CROW by varying the combination of different cavities. 4. Summary In conclusion, CROWs in complex 2D PCs constructed with a portion of 12-fold PQs are proposed and their optical properties are demonstrated by using FDTD simulation. The results show that enhanced transmission and slow light bands can be simultaneously achieved in such waveguides as well as general CROWs. Moreover, due to higher degree of flexibility and tunability of PQs for defect mode properties compared to conventional periodic PCs, multiple slow light bands can be flexibly obtained in different kinds
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of CROWs in complex 2D PCs. Our results may lead to the development of a variety of novel ultracompact devices such as multichannel modulators, switches, and multichannel tunable amplifiers for photonic integrated circuits. Acknowledgements We thank W.R. Hu, D.H. Fan and Y.J. Shen for their help. This work is supported in part by the National Natural Science Foundation of China (Grant No. 60967003), the Nanchang University Foundation, China (Grant No. Z03355), and the Key laboratory of Acoustic and Photonic Materials and Devices, Ministry of Education, in Wuhan University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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