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Dispersion optimization of slow light in slotted photonic crystal waveguide by selective air holes infiltration
Optik 125 (2014) 1967–1970
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Dispersion optimization of slow light in slotted photonic crystal waveguide by selective air holes infiltration Sheng-xi Jiao a , Yong Zhao b,∗ , Ya-nan Zhang b , Qi Wang b a b
Northeast Dianli University, School of Automation Engineering, Jilin 132012, China Northeastern University, College of Information Science and Engineering, Shenyang 110819, China
a r t i c l e
a b s t r a c t
i n f o
Article history: Received 5 May 2013 Accepted 28 September 2013
Keywords: Slow-light Slotted photonic crystal waveguide Dispersion Liquid infiltration
This work proposed a methodology based on the liquid infiltration of slotted photonic crystal waveguide (SPCW). By choosing the refractive index that infiltrated in the first and second rows of air holes adjacent to the slot, respectively, SPCW was optimized to possess wideband slow light with large group index and low dispersion. The properties of SPCW were numerically simulated by plane wave expansion (PWE) method and finite-difference time-domain (FDTD) method. Simulation results showed that the designed SPCW could control the group index for the same SPCW with the nearly constant group index of 50, 68, 81, 150, and 200 over 7.5 nm, 5.5 nm, 3.1 nm, 1.65 nm, and 1.15 nm. In addition, we demonstrated that this post-fabrication liquid infiltrated technology has the potential for realizing reconfigurable and tunable SPCW, in which the flexible wavelength range of SPCW can also be controlled by different liquid infiltration. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Slow light with a remarkably low group velocity has recently attracted wide attention because it is regarded as a promising approach for time-domain processing of optical signal and spatial compression of optical energy [1,2]. It also offers the possibility for miniaturization sensors with high sensitivity [3,4]. Photonic crystal waveguide (PCW) is especially attractive for generating slow light, which has many advantages such as room-temperature operation, great potential for wide-bandwidth, and realizing slow light at the scale of the optical wavelength [5,6]. Particularly, the slotted photonic crystal waveguide (SPCW) could combine the ability to confine light in the nano-scale slot with the slow light enhancement available from PCW, which is very good for the practical applications [7–9]. However, in practical applications, the high group index slow light in SPCW is usually accompany with large group velocity dispersion (GVD) [10–12]. As the useful bandwidth lies below the silica light-line, it is a challenge to tailor the dispersion band for high group index transmission with a wide bandwidth and it is also difficult to linearize the dispersion band close to the band edge where the GVD diverges. Very recently, a number of structural-based technologies have been proposed to optimize the slow light properties of SPCW, such as modifying the sizes of air holes [13] or adjusting the positions of air holes [14]. With appropriate optimization, the realization of
∗ Corresponding author. E-mail address: [email protected] (Y. Zhao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.067
wideband slow light with enhanced group index and low GVD, has been demonstrated. However, all of the previous structural-based optimized methods require very high fabrication tolerances and are irreversible once performed, which would limit the development of photonic crystal devices in practical applications. At the same time, the post-fabrication technology, which can be used to optimize the dispersion property of PCW after the fabrication, has attracted considerable attention [15–18]. Post-fabrication technology can not only relaxes the constraint on the fabrication precision as compared to previous structural-based schemes for dispersion optimization, but also offers a flexible technology to reconfigure photonic crystal devices. Many advances have been made on postfabrication of the optical properties of photonic crystal devices, such as by infiltration of optofluidic [15], liquid crystal [16], and polymer [17]. Particularly, the infiltration of liquid into the air holes of photonic crystal has been greatly investigated and demonstrated [18]. For photonic crystal, it is naturally suitable for liquid infiltrating due to its sub-wavelength scale air holes that can house liquids. The degree of freedom in the liquid selection combined with the reversibility of the infiltration operation makes liquid infiltrating a flexible, efficient and versatile technology for post-fabrication photonic crystal. The theoretical design has been presented that either high group index slow light with wide bandwidth [19], or tunable slow light [20] has been achieved by simply changing the refractive index of liquids that infiltrated in the air holes of photonic crystal waveguide. In this paper, we have demonstrated, for the first time, dispersion optimization of slow light in slotted photonic crystal waveguide by simply choosing the refractive index that infiltrated
1968
S.-x. Jiao et al. / Optik 125 (2014) 1967–1970
Fig. 1. The basic structure of SPCW.
in the first and second rows of air holes adjacent to the slot with different liquids, respectively. By careful optimizing, we obtained a structure that presents a serial of tunable slow light with wide bandwidth, high group index and low dispersion. 2. Physical model The intact structure of our proposed SPCW is shown in Fig. 1. It is a triangular lattice air-bridge photonic crystal slab consisting of circular air holes in a silicon-on-insulator (SOI) substrate (n = 3.48). The waveguide is formed by removing the central row of air holes along the X direction with an air slot of width ω0 = 0.32a, and the radius of air hole is r = 0.30a, where a = 447 nm is the lattice constant. The triangular lattice is employed because of its large transverse electric band gap and it is expected to serve as a good platform for photonic integrated circuits and miniaturized optical devices. Hole-array-based SPCW proposed in this paper will be easier to fabrication, and it is able to selectively remove the material underneath the waveguide to form a similar air-bridge membrane, which will be better to reduce the vertical leakage into the substrate than the pillar-array-based SPCW in practical applications. Specially, the first and second rows of air holes adjacent to the slot are marked with blue and red color, and the parameters n1 and n2 denote the refractive index of liquid that infiltrated in each row.
The SPCW is optimized through simulations performed by 2D plane wave expansion (PWE) using the MIT’s freely available software MPB [21], which is precise enough to estimate the slow light property of SPCW [22]. In numerical simulation, the effective refractive index of 2.87 is used for the 220 nm thick slab [23]. As shown in Fig. 2(a), the supercell is selected with a lateral length of 1 row of air holes in the X direction and 10 rows of air holes in the Y direction for eigenmodes calculation. Fig. 2(b) presents the dispersion diagram of the proposed SPCW when the polarization is considered to be transverse-electric (TE)-like polarized modes. In this case, the effective index in the slot is lower than that in the surrounding slab. Therefore, the even mode (represented by the red line) has a positive slope compared with the standard W1type PCW, and it concentrates most of its energy in the slot due to symmetric constraint. However, the odd mode (represented by the blue line) is pushed toward higher frequencies, and the energy will be leaked in the slab mode. Considering the practical applications, only the flat band edge of the even mode will be considered as the guided mode and be discussed in this paper. Besides, what should be mentioned is that only the dispersion region below the light line could confine light well in the vertical direction of the slab. The electric field distribution of the even mode is shown in Fig. 2(c). The simulation is based on the 2D finite-difference time-domain (FDTD) method [24] by using the MIT’s freely available software MEEP, while the operating frequency of input light is below the light line. It certifies that most of the energy could be confined in the slot as the light propagating forward, on the contrary, just a small part of light energy penetrates toward transverse to the propagation direction. The group velocity g = dω/dk = c/ng of the light waves can be calculated from the slope of the dispersion curve, where ω is the light frequency, k is the wave vector, ng is the group index, and c is the light velocity in vacuum. Fig. 2(d) shows the group index curve of the basic SPCW. It is clearly that the group index varies rapidly in the slow light region. The problems of such a slow light SPCW are the narrow bandwidth and high GVD, which makes the slow light of SPCW be highly frequency dependent and easily interfered by surrounding environment As we know, the index guided and gap guided modes coexist in one guided mode of SPCW. An anticrossing between these two types of modes determines the local shape of the mode dispersion diagram [25]. When the wave vector
Fig. 2. (a) The periodic supercell for simulation analysis; (b) the dispersion diagram of basic SPCW; (c) the electric field distribution of basic SPCW; and (d) the group index curve of basic SPCW.
S.-x. Jiao et al. / Optik 125 (2014) 1967–1970 (b) 0.290 Normalized Frequency (ω a/2 π c)
Normalized Frequency (ω a/2π c)
(a) 0.290 0.285 0.280 0.275 0.270 0.265 0.30
1969
n1=1.33 n1=1.38 n1=1.43 n1=1.48 n1=1.53 n1=1.58 0.35 0.40 0.45 Wave Vector (ka/2π)
0.50
0.285
n2=1.33 n2=1.43 n2=1.53 n2=1.63 n2=1.73 n2=1.83
0.280
0.275
0.270
0.35
0.40 0.45 Wave Vector (ka/2π)
0.50
Fig. 3. Relationship of the dispersion curve with (a) the refractive index of n1 from 1.33 to 1.58 when n2 = 0.05 and (b) the refractive index of n2 from 1.33 to 1.83 when n1 = 0.1.
As our optimization goal in this paper is to optimize the dispersion property to obtain tunable slow light with high group index and wide bandwidth and low GVD, the method of adjusting the refractive index of the first and second rows of air holes adjacent to the slot is adopted by liquid infiltration. The values of ng and GVD can be obtained from the first and second derivative of the dispersion curve. So we firstly survey the relationship of the dispersion curve on the refractive index of the first two rows. Fig. 3(a) illustrates the dispersion curve shift for n1 ranging from 1.33 to 1.58 with a step of 0.05, while the structure parameters are identical with the basic SPCW. As seen, the dispersion curve shifts to lower frequency with n1 increased, and the shift becomes more rapidly near the tail of the curve. The shift of dispersion curve can be understood by the basic electromagnetic variation equation c/n = f for light, from which it can be found that the refractive index and frequency are inversely related for a given wavelength of light, thus stimulates the concept of liquid infiltration on the dispersion optimization. By enlarging the refractive index of the infiltrated liquid, the dispersion mode shifts to lower frequency and the energy of the mode will be minimized. Fig. 3(b) shows the shifting tendency of the dispersion curve with n2 gradually changed from 1.33 to 1.83 with a step of 0.1, while other structure parameters are the same as the basic SPCW. In this case, the dispersion curve undergoes the same trend with those of increasing n1, namely, the dispersion curve shifts also to lower frequency with n2 increased. However, it is the middle of the curve that varies distinctly, while the tail of the diagram shows only a slight variation. So, it is possible to linearize regions of the dispersion curve over a large range, which is corresponding to regimes of wideband slow light with high group index and low dispersion, by properly increasing n1 to decrease the slope of the tail of the curve and adjusting appropriate n2 to decrease the slope of the middle of the curve. Besides, the infiltrated liquid can be freely infiltrated into any air holes of SPCW after fabrication to reconfigure the infiltrated structure, and the refractive index of
the infiltrated liquid spans most of the refractive index window, thus providing the possibility for the tuning of slow light operating wavelength in a large range. Fig. 4(a) illustrates the dispersion curve for different infiltrated refractive index of n1 and n2. It can be found that there exists a flat region. Besides, the frequency range and the slope of flat dispersion curve for different liquid infiltration are different. Fig. 4(b) shows the corresponding group index curve. Unlike the basic SPCW case, the slow light of the infiltrated SPCW possesses a “wideband slow light” window where the group index is nearly constant. As n1 is increasing and n2 is decreasing, the optimized group index curve shifts to higher working wavelength, and the corresponding group index is increased. Following previous literature, the group (a) Normalized Frequency (ωa/2πc)
3. Numerical simulation and dispersion optimization
Fig. 4. Dispersion curves (a) and the corresponding wideband group index curves (b) for different sets of n1 and n2; (c) the group index curve and the corresponding group index dispersion curve for ng = 150.
0.286
0.284
0.282
n1=1.406, n2=1.634 n1=1.416, n2=1.645 n1=1.426, n2=1.660 n1=1.435, n2=1.672
0.28 0.278 0.35
0.4 0.45 Wave Vector (ka/2π)
0.5
(b) 300 F1: n1=1.406, n2=1.634 F2: n1=1.416, n2=1.645 F3: n1=1.426, n2=1.660 F4: n1=1.435, n2=1.672
250
Group Index
is small, the guided mode is well bound to the waveguide core, and the slope of the dispersion curve is large, which indicates a small ng . Thus, this region is regarded to be index-guided region and the mode in this region is called index-guided mode. However, when the wave vector is large, the guided mode will penetrate into the air holes of photonic crystal lattice, and the slope of the dispersion curve will decreased rapidly, which indicates a large ng . Hence, this region is regarded to be gap-guided region, and this mode is gapguided mode. The shape, size, position or the refractive index of air holes adjacent to the air slot will determine the physics conjunction of index guided and gap guided modes. Adjusting these parameters can change the intrinsic interaction of the index guided and gap guided modes, and it is possible to linearize the dispersion curve of the slow mode in SPCW, and extend the bandwidth of slow light accordingly.
200 150 100 50 0
1564
1566
1568
1570 1572 Wavelength (nm)
1574
1576
1578
Fig. 5. Dispersion curves (a) and the corresponding tunable group index curves (b) for different sets of n1 and n2.
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S.-x. Jiao et al. / Optik 125 (2014) 1967–1970
index is considered as a constant in ±10% range. For nearly constant group indexes of 50, 68, 81, 150, and 200, their corresponding bandwidths can reach 7.5 nm, 5.5 nm, 3.1 nm, 1.65 nm, and 1.15 nm, respectively. To investigate the group velocity dispersion property, we also calculate the GVD value. Take ng = 150 for example, Fig. 4(c) shows the group index curve and the corresponding group velocity dispersion curve. As expected, the GVD value of flat region is on the order of 106 ps2 /km, which is one order of magnitude lower than the basic SPCW in Fig. 2(d), and it can be suitable for the practical application requirement. As mentioned above, the working wavelength of wideband slow light can also be tuned by the liquid infiltration technology. To verify this issue, the corresponding group index characteristics of the SPCW as a function of wavelength for different refractive index of the infiltrated liquid are also investigated. As shown in Fig. 5(a), we demonstrate the shift of the flat region of the dispersion curve as the suitable increase of the refractive index of the infiltrated liquid in the first and the second air holes. Naturally, wideband and high group index slow light can be realized, and the working wavelength can be simultaneously tuned in the same SPCW by suitable liquid infiltration in the slow light region. For the working wavelength of 1565 nm, 1567 nm, 1569 nm, and 1571 nm, wideband slow light are realized with the group index reach up to ng = 150. 4. Conclusion In this paper, a novel post-fabrication scheme based on the selective liquid infiltration of SPCW to produce nearly tunable slow light with wide band-width, high group index and low dispersion has been discussed. We numerically demonstrated that this approach allows one to control the group index ranging from 50 to 200 at the same SPCW project, and the working wavelength can be tuned from 1565 nm to 1571 nm with the group index stable at high group index of 150, simply by selecting two liquids with suitable refractive index to infiltrate into the first and second rows of air holes adjacent to the slot, respectively. The degree of freedom in the liquid choice combined with the reversibility and reconfiguration of the infiltration technology provides a post-fabrication scheme, which opens a flexible, efficient and versatile method to optimize the properties of slow light in SPCW, and relaxes the constraint on the fabrication accuracy as compared to previous structuralbased methods for slow light dispersion optimization. Moreover, we could also engineering the slow properties of SPCW by using controllable magnetic fluid or liquid crystal to opens up a new potential application for reconfigurable SPCW, in which the flexible wavelength range of SPCW can also be controlled after the liquid infiltration. Anyway, the preliminary results of this study have provided important theoretical basis for the potential application offered by the SPCW in future optical technologies. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 61203206 and 61273059,
the Fundamental Research Funds for the Central Universities under Grant 110204002 and 110804003, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20120042120038. References [1] T.F. Krauss, Why do we need slow light, Nat. Photonics 2 (8) (2008) 448–450. [2] Y. Zhao, H.W. Zhao, X.Y. Zhang, et al., New mechanisms of slow light and their applications, Opt. Laser Technol. 41 (5) (2009) 517–525. [3] H. Wen, M. Terrel, S.H. Fan, et al., Sensing with slow light in fiber Bragg gratings, IEEE Sens. J. 12 (1) (2012) 156–163. [4] M.J.F. Digonnet, H. Wen, M.A. Terrel, et al., Slow light in fiber sensors, Proc. SPIE 8273 (2012) 82730W. [5] T.F. Krauss, Slow light in photonic crystal waveguides, J. Phys. D: Appl. Phys. 40 (9) (2007) 2666–2670. [6] Y.A. Vlasov, M. O’Boyle, H.F. Hamann, et al., Active control of slow light on a chip with photonic crystal waveguides, Nature 438 (2005) 65–69. [7] V.R. Almeida, Q.F. Xu, C.A. Barrios, et al., Guiding and confining light in void nanostructure, Opt. Lett. 29 (11) (2004) 1209–1211. [8] C. Caer, X. Le Roux, E. Cassan, Enhanced localization of light in slow wave slot photonic crystal waveguides, Opt. Lett. 37 (17) (2012) 3660–3662. [9] A. Di Falco, L. O’Faolain, T.F. Krauss, Photonic crystal slotted slab waveguides, Photonics Nanostruct. Fundam. Appl. 6 (1) (2008) 38–41. [10] S.A. Schulz, L. O’Faolain, D.M. Beggs, et al., Dispersion engineered slow light in photonic crystals: a comparison, J. Opt. 12 (10) (2010) 104004. [11] J. Pedersen, S.S. Xiao, N. Mortensen, Limits of slow light in photonic crystals, Phys. Rev. B 78 (15) (2008) 153101. [12] W.W. Song, R.A. Integlia, W. Jiang, Slow light loss due to roughness in photonic crystal waveguides: an analytic approach, Phys. Rev. B 82 (23) (2010) 235306. [13] H. Kurt, K. Üstün, L. Ayas, Study of different spectral regions and delay bandwidth relation in slow light photonic crystal waveguides, Opt. Express 18 (26) (2010) 26965–26977. [14] Y. Zhao, Y.N. Zhang, D. Wu, et al., Wideband slow light with large group index and low dispersion in slotted photonic crystal waveguide, J. Lightwave Technol. 30 (17) (2012) 2812–2817. [15] S.H. Kim, J.H. Choi, S.K. Lee, et al., Optofluidic integration of a photonic crystal nanolaser, Opt. Express 16 (9) (2008) 6515–6527. [16] A.C. Bedoya, S. Mahmoodian, C. Monat, et al., Liquid crystal dynamics in a photonic crystal cavity created by selective microfluidic infiltration, Opt. Express 18 (26) (2010) 27280–27290. [17] C.Y. Lin, X.L. Wang, S. Chakravarty, et al., Electro-optic polymer infiltrated silicon photonic crystal slot waveguide modulator with 23 dB slow light enhancement, Appl. Phys. Lett. 97 (2010) 093304. [18] M.G. Scullion, A. Di Falco, T.F. Krauss, Slotted photonic crystal cavities with integrated microfluidics for biosensing applications, Biosens. Bioelectron. 27 (2011) 101–105. [19] S.Y. Lü, J.L. Zhao, D. Zhang, Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration, Appl. Opt. 49 (20) (2010) 3930–3934. [20] J. Wu, Y.P. Li, C. Peng, et al., Numerical demonstration of slow light tuning in slotted photonic crystal waveguide using microfluidic infiltration, Opt. Commun. 284 (2011) 2149–2152. [21] S.G. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods for maxwell’s equations in a planewave basic, Opt. Express 8 (2000) 173–190. [22] S.P. Guo, S. Albin, Simple plane wave implementation for photonic crystal calculations, Opt. Express 11 (2) (2003) 167–175. [23] J. Li, T.P. White, L. O’Faolain, et al., Systematic design of flat band slow light in photonic crystal waveguides, Opt. Express 16 (9) (2008) 6227–6232. [24] A.F. Oskooi, D. Roundy, M. Ibanescu, et al., Meep: a flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181 (3) (2010) 687–702. [25] A.Y. Petrov, M. Eich, Zero dispersion at small group velocities in photonic crystal waveguides, Appl. Phys. Lett. 85 (21) (2004) 4866–4868.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Dispersion optimization of slow light in slotted photonic crystal waveguide by selective air holes infiltration Sheng-xi Jiao a , Yong Zhao b,∗ , Ya-nan Zhang b , Qi Wang b a b
Northeast Dianli University, School of Automation Engineering, Jilin 132012, China Northeastern University, College of Information Science and Engineering, Shenyang 110819, China
a r t i c l e
a b s t r a c t
i n f o
Article history: Received 5 May 2013 Accepted 28 September 2013
Keywords: Slow-light Slotted photonic crystal waveguide Dispersion Liquid infiltration
This work proposed a methodology based on the liquid infiltration of slotted photonic crystal waveguide (SPCW). By choosing the refractive index that infiltrated in the first and second rows of air holes adjacent to the slot, respectively, SPCW was optimized to possess wideband slow light with large group index and low dispersion. The properties of SPCW were numerically simulated by plane wave expansion (PWE) method and finite-difference time-domain (FDTD) method. Simulation results showed that the designed SPCW could control the group index for the same SPCW with the nearly constant group index of 50, 68, 81, 150, and 200 over 7.5 nm, 5.5 nm, 3.1 nm, 1.65 nm, and 1.15 nm. In addition, we demonstrated that this post-fabrication liquid infiltrated technology has the potential for realizing reconfigurable and tunable SPCW, in which the flexible wavelength range of SPCW can also be controlled by different liquid infiltration. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Slow light with a remarkably low group velocity has recently attracted wide attention because it is regarded as a promising approach for time-domain processing of optical signal and spatial compression of optical energy [1,2]. It also offers the possibility for miniaturization sensors with high sensitivity [3,4]. Photonic crystal waveguide (PCW) is especially attractive for generating slow light, which has many advantages such as room-temperature operation, great potential for wide-bandwidth, and realizing slow light at the scale of the optical wavelength [5,6]. Particularly, the slotted photonic crystal waveguide (SPCW) could combine the ability to confine light in the nano-scale slot with the slow light enhancement available from PCW, which is very good for the practical applications [7–9]. However, in practical applications, the high group index slow light in SPCW is usually accompany with large group velocity dispersion (GVD) [10–12]. As the useful bandwidth lies below the silica light-line, it is a challenge to tailor the dispersion band for high group index transmission with a wide bandwidth and it is also difficult to linearize the dispersion band close to the band edge where the GVD diverges. Very recently, a number of structural-based technologies have been proposed to optimize the slow light properties of SPCW, such as modifying the sizes of air holes [13] or adjusting the positions of air holes [14]. With appropriate optimization, the realization of
∗ Corresponding author. E-mail address: [email protected] (Y. Zhao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.067
wideband slow light with enhanced group index and low GVD, has been demonstrated. However, all of the previous structural-based optimized methods require very high fabrication tolerances and are irreversible once performed, which would limit the development of photonic crystal devices in practical applications. At the same time, the post-fabrication technology, which can be used to optimize the dispersion property of PCW after the fabrication, has attracted considerable attention [15–18]. Post-fabrication technology can not only relaxes the constraint on the fabrication precision as compared to previous structural-based schemes for dispersion optimization, but also offers a flexible technology to reconfigure photonic crystal devices. Many advances have been made on postfabrication of the optical properties of photonic crystal devices, such as by infiltration of optofluidic [15], liquid crystal [16], and polymer [17]. Particularly, the infiltration of liquid into the air holes of photonic crystal has been greatly investigated and demonstrated [18]. For photonic crystal, it is naturally suitable for liquid infiltrating due to its sub-wavelength scale air holes that can house liquids. The degree of freedom in the liquid selection combined with the reversibility of the infiltration operation makes liquid infiltrating a flexible, efficient and versatile technology for post-fabrication photonic crystal. The theoretical design has been presented that either high group index slow light with wide bandwidth [19], or tunable slow light [20] has been achieved by simply changing the refractive index of liquids that infiltrated in the air holes of photonic crystal waveguide. In this paper, we have demonstrated, for the first time, dispersion optimization of slow light in slotted photonic crystal waveguide by simply choosing the refractive index that infiltrated
1968
S.-x. Jiao et al. / Optik 125 (2014) 1967–1970
Fig. 1. The basic structure of SPCW.
in the first and second rows of air holes adjacent to the slot with different liquids, respectively. By careful optimizing, we obtained a structure that presents a serial of tunable slow light with wide bandwidth, high group index and low dispersion. 2. Physical model The intact structure of our proposed SPCW is shown in Fig. 1. It is a triangular lattice air-bridge photonic crystal slab consisting of circular air holes in a silicon-on-insulator (SOI) substrate (n = 3.48). The waveguide is formed by removing the central row of air holes along the X direction with an air slot of width ω0 = 0.32a, and the radius of air hole is r = 0.30a, where a = 447 nm is the lattice constant. The triangular lattice is employed because of its large transverse electric band gap and it is expected to serve as a good platform for photonic integrated circuits and miniaturized optical devices. Hole-array-based SPCW proposed in this paper will be easier to fabrication, and it is able to selectively remove the material underneath the waveguide to form a similar air-bridge membrane, which will be better to reduce the vertical leakage into the substrate than the pillar-array-based SPCW in practical applications. Specially, the first and second rows of air holes adjacent to the slot are marked with blue and red color, and the parameters n1 and n2 denote the refractive index of liquid that infiltrated in each row.
The SPCW is optimized through simulations performed by 2D plane wave expansion (PWE) using the MIT’s freely available software MPB [21], which is precise enough to estimate the slow light property of SPCW [22]. In numerical simulation, the effective refractive index of 2.87 is used for the 220 nm thick slab [23]. As shown in Fig. 2(a), the supercell is selected with a lateral length of 1 row of air holes in the X direction and 10 rows of air holes in the Y direction for eigenmodes calculation. Fig. 2(b) presents the dispersion diagram of the proposed SPCW when the polarization is considered to be transverse-electric (TE)-like polarized modes. In this case, the effective index in the slot is lower than that in the surrounding slab. Therefore, the even mode (represented by the red line) has a positive slope compared with the standard W1type PCW, and it concentrates most of its energy in the slot due to symmetric constraint. However, the odd mode (represented by the blue line) is pushed toward higher frequencies, and the energy will be leaked in the slab mode. Considering the practical applications, only the flat band edge of the even mode will be considered as the guided mode and be discussed in this paper. Besides, what should be mentioned is that only the dispersion region below the light line could confine light well in the vertical direction of the slab. The electric field distribution of the even mode is shown in Fig. 2(c). The simulation is based on the 2D finite-difference time-domain (FDTD) method [24] by using the MIT’s freely available software MEEP, while the operating frequency of input light is below the light line. It certifies that most of the energy could be confined in the slot as the light propagating forward, on the contrary, just a small part of light energy penetrates toward transverse to the propagation direction. The group velocity g = dω/dk = c/ng of the light waves can be calculated from the slope of the dispersion curve, where ω is the light frequency, k is the wave vector, ng is the group index, and c is the light velocity in vacuum. Fig. 2(d) shows the group index curve of the basic SPCW. It is clearly that the group index varies rapidly in the slow light region. The problems of such a slow light SPCW are the narrow bandwidth and high GVD, which makes the slow light of SPCW be highly frequency dependent and easily interfered by surrounding environment As we know, the index guided and gap guided modes coexist in one guided mode of SPCW. An anticrossing between these two types of modes determines the local shape of the mode dispersion diagram [25]. When the wave vector
Fig. 2. (a) The periodic supercell for simulation analysis; (b) the dispersion diagram of basic SPCW; (c) the electric field distribution of basic SPCW; and (d) the group index curve of basic SPCW.
S.-x. Jiao et al. / Optik 125 (2014) 1967–1970 (b) 0.290 Normalized Frequency (ω a/2 π c)
Normalized Frequency (ω a/2π c)
(a) 0.290 0.285 0.280 0.275 0.270 0.265 0.30
1969
n1=1.33 n1=1.38 n1=1.43 n1=1.48 n1=1.53 n1=1.58 0.35 0.40 0.45 Wave Vector (ka/2π)
0.50
0.285
n2=1.33 n2=1.43 n2=1.53 n2=1.63 n2=1.73 n2=1.83
0.280
0.275
0.270
0.35
0.40 0.45 Wave Vector (ka/2π)
0.50
Fig. 3. Relationship of the dispersion curve with (a) the refractive index of n1 from 1.33 to 1.58 when n2 = 0.05 and (b) the refractive index of n2 from 1.33 to 1.83 when n1 = 0.1.
As our optimization goal in this paper is to optimize the dispersion property to obtain tunable slow light with high group index and wide bandwidth and low GVD, the method of adjusting the refractive index of the first and second rows of air holes adjacent to the slot is adopted by liquid infiltration. The values of ng and GVD can be obtained from the first and second derivative of the dispersion curve. So we firstly survey the relationship of the dispersion curve on the refractive index of the first two rows. Fig. 3(a) illustrates the dispersion curve shift for n1 ranging from 1.33 to 1.58 with a step of 0.05, while the structure parameters are identical with the basic SPCW. As seen, the dispersion curve shifts to lower frequency with n1 increased, and the shift becomes more rapidly near the tail of the curve. The shift of dispersion curve can be understood by the basic electromagnetic variation equation c/n = f for light, from which it can be found that the refractive index and frequency are inversely related for a given wavelength of light, thus stimulates the concept of liquid infiltration on the dispersion optimization. By enlarging the refractive index of the infiltrated liquid, the dispersion mode shifts to lower frequency and the energy of the mode will be minimized. Fig. 3(b) shows the shifting tendency of the dispersion curve with n2 gradually changed from 1.33 to 1.83 with a step of 0.1, while other structure parameters are the same as the basic SPCW. In this case, the dispersion curve undergoes the same trend with those of increasing n1, namely, the dispersion curve shifts also to lower frequency with n2 increased. However, it is the middle of the curve that varies distinctly, while the tail of the diagram shows only a slight variation. So, it is possible to linearize regions of the dispersion curve over a large range, which is corresponding to regimes of wideband slow light with high group index and low dispersion, by properly increasing n1 to decrease the slope of the tail of the curve and adjusting appropriate n2 to decrease the slope of the middle of the curve. Besides, the infiltrated liquid can be freely infiltrated into any air holes of SPCW after fabrication to reconfigure the infiltrated structure, and the refractive index of
the infiltrated liquid spans most of the refractive index window, thus providing the possibility for the tuning of slow light operating wavelength in a large range. Fig. 4(a) illustrates the dispersion curve for different infiltrated refractive index of n1 and n2. It can be found that there exists a flat region. Besides, the frequency range and the slope of flat dispersion curve for different liquid infiltration are different. Fig. 4(b) shows the corresponding group index curve. Unlike the basic SPCW case, the slow light of the infiltrated SPCW possesses a “wideband slow light” window where the group index is nearly constant. As n1 is increasing and n2 is decreasing, the optimized group index curve shifts to higher working wavelength, and the corresponding group index is increased. Following previous literature, the group (a) Normalized Frequency (ωa/2πc)
3. Numerical simulation and dispersion optimization
Fig. 4. Dispersion curves (a) and the corresponding wideband group index curves (b) for different sets of n1 and n2; (c) the group index curve and the corresponding group index dispersion curve for ng = 150.
0.286
0.284
0.282
n1=1.406, n2=1.634 n1=1.416, n2=1.645 n1=1.426, n2=1.660 n1=1.435, n2=1.672
0.28 0.278 0.35
0.4 0.45 Wave Vector (ka/2π)
0.5
(b) 300 F1: n1=1.406, n2=1.634 F2: n1=1.416, n2=1.645 F3: n1=1.426, n2=1.660 F4: n1=1.435, n2=1.672
250
Group Index
is small, the guided mode is well bound to the waveguide core, and the slope of the dispersion curve is large, which indicates a small ng . Thus, this region is regarded to be index-guided region and the mode in this region is called index-guided mode. However, when the wave vector is large, the guided mode will penetrate into the air holes of photonic crystal lattice, and the slope of the dispersion curve will decreased rapidly, which indicates a large ng . Hence, this region is regarded to be gap-guided region, and this mode is gapguided mode. The shape, size, position or the refractive index of air holes adjacent to the air slot will determine the physics conjunction of index guided and gap guided modes. Adjusting these parameters can change the intrinsic interaction of the index guided and gap guided modes, and it is possible to linearize the dispersion curve of the slow mode in SPCW, and extend the bandwidth of slow light accordingly.
200 150 100 50 0
1564
1566
1568
1570 1572 Wavelength (nm)
1574
1576
1578
Fig. 5. Dispersion curves (a) and the corresponding tunable group index curves (b) for different sets of n1 and n2.
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index is considered as a constant in ±10% range. For nearly constant group indexes of 50, 68, 81, 150, and 200, their corresponding bandwidths can reach 7.5 nm, 5.5 nm, 3.1 nm, 1.65 nm, and 1.15 nm, respectively. To investigate the group velocity dispersion property, we also calculate the GVD value. Take ng = 150 for example, Fig. 4(c) shows the group index curve and the corresponding group velocity dispersion curve. As expected, the GVD value of flat region is on the order of 106 ps2 /km, which is one order of magnitude lower than the basic SPCW in Fig. 2(d), and it can be suitable for the practical application requirement. As mentioned above, the working wavelength of wideband slow light can also be tuned by the liquid infiltration technology. To verify this issue, the corresponding group index characteristics of the SPCW as a function of wavelength for different refractive index of the infiltrated liquid are also investigated. As shown in Fig. 5(a), we demonstrate the shift of the flat region of the dispersion curve as the suitable increase of the refractive index of the infiltrated liquid in the first and the second air holes. Naturally, wideband and high group index slow light can be realized, and the working wavelength can be simultaneously tuned in the same SPCW by suitable liquid infiltration in the slow light region. For the working wavelength of 1565 nm, 1567 nm, 1569 nm, and 1571 nm, wideband slow light are realized with the group index reach up to ng = 150. 4. Conclusion In this paper, a novel post-fabrication scheme based on the selective liquid infiltration of SPCW to produce nearly tunable slow light with wide band-width, high group index and low dispersion has been discussed. We numerically demonstrated that this approach allows one to control the group index ranging from 50 to 200 at the same SPCW project, and the working wavelength can be tuned from 1565 nm to 1571 nm with the group index stable at high group index of 150, simply by selecting two liquids with suitable refractive index to infiltrate into the first and second rows of air holes adjacent to the slot, respectively. The degree of freedom in the liquid choice combined with the reversibility and reconfiguration of the infiltration technology provides a post-fabrication scheme, which opens a flexible, efficient and versatile method to optimize the properties of slow light in SPCW, and relaxes the constraint on the fabrication accuracy as compared to previous structuralbased methods for slow light dispersion optimization. Moreover, we could also engineering the slow properties of SPCW by using controllable magnetic fluid or liquid crystal to opens up a new potential application for reconfigurable SPCW, in which the flexible wavelength range of SPCW can also be controlled after the liquid infiltration. Anyway, the preliminary results of this study have provided important theoretical basis for the potential application offered by the SPCW in future optical technologies. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 61203206 and 61273059,
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