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Design of a novel wideband single-mode waveguide in a photonic crystal slab structure
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Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147 www.elsevier.com/locate/photonics
Design of a novel wideband single-mode waveguide in a photonic crystal slab structure M. Khatibi Moghadam a,b,*, Mir Mojtaba Mirsalehi a, Amir Reza Attari a a
b
Electrical Engineering Department, Ferdowsi University of Mashhad, Iran Communications and Computer Research Center, Ferdowsi University of Mashhad, Iran
Received 26 December 2007; received in revised form 14 May 2008; accepted 21 May 2008 Available online 25 May 2008
Abstract We propose a novel photonic crystal slab waveguide that provides a single-mode band with large bandwidth. The proposed waveguide is obtained by introducing a line defect in a triangular lattice of air holes in a dielectric slab. This line defect consists of holes which are not located in the original lattice points. The plane wave expansion (PWE) method is used to extract the band diagram of guiding modes and based on the results, photonic crystal holes in the defect row and its adjacent rows are modified to maximize the waveguide bandwidth. We show that using the proposed structure, a single-mode bandwidth of 17% can be achieved. # 2008 Elsevier B.V. All rights reserved. PACS : 42.70.Qs; 42.25.Bs Keywords: Photonic crystal; Single-mode waveguide; PWE method; Slab structure
1. Introduction Recently, there has been a great interest in the properties of photonic crystals (PCs), which prohibit the propagation of light for frequencies within a photonic band gap (PBG) [1,2]. Using PCs, new methods are found to control the propagation of light and to construct integrated optical devices [1]. Due to the difficulties associated with the fabrication of threedimensional (3D) photonic crystals, two-dimensional (2D) photonic crystal slabs are more attractive [3]. The commonly used PC slab is a dielectric slab perforated with a 2D triangular lattice of air holes. This type of PC provides a large PBG for TE modes [4]. The
* Corresponding author. E-mail address: [email protected] (M. Khatibi Moghadam). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.05.001
field distribution of TE modes in a PC waveguide (PCW) is similar to that of a conventional optical waveguide. Therefore, the hole type PC can be efficiently coupled to conventional waveguides [1]. Waveguides are essential parts of integrated optical circuits. Because of their unique advantages, PCWs are suitable for designing optical circuits. An interesting property of a PCW is its capability of having sharp bends with high transmission efficiencies [5]. Typically, a PCW is built by introducing a line defect in the periodic structure of a PC [6]. This line defect is usually formed by eliminating one row of holes in a hole type structure. An ideal PC slab waveguide should have several properties. In many applications, it is desired that the waveguide supports a wide single-mode bandwidth. Most PCWs are not single mode over a wide bandwidth. Therefore, various methods have been proposed in the literature to increase the single-mode bandwidth of PCWs [7,8].
M. Khatibi Moghadam et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147
To extract the dispersion relations of guided modes in PC slab structures, we need to perform a 3D analysis. In modal analysis of PC slab waveguides, it is found that, if the thickness of the slab causes the single index guiding mode, the 2D analysis with the effective index, neff, provides approximately the same results as the 3D analysis [8–10]. For example, for a PC slab with a lattice constant of a, a slab height of 0.6a, an air hole radius of 0.3a and a refractive index of 3.4, the 3D analysis can be replaced with a 2D analysis of PC, provided that an effective index of 2.76 is used [9]. To design a wideband single-mode PCW, research has focused on properties of guided modes in different structures. Single-line-defect waveguides in PC slabs were fabricated into an SOI wafer by Baba et al. [11]. The results and calculation of scattering loss suggest that the propagation loss can be 1–2 orders of magnitude lower than in the index waveguide. Consequently, the use of a PC waveguide is advantageous not only at bends, branches, etc., but also at straight parts in a photonic chip. Jamois et al. gave a review on the properties of silicon-based 2D PCs and their application for wave guiding [12]. They showed that the modes in PC waveguides are confined by the total internal reflection similar to the ridge waveguide. In addition, there is a second guiding mechanism based on the existence of PBG, which confines modes in the defect region. Yamada et al. have established a simple interpretation for dispersion curves of guided modes and have proposed a novel waveguide with a high group velocity [8]. They considered a silicon PC slab with SiO2 cladding. Considering light line limitation, the normalized bandwidth of their proposed structure became as large as 6% [8]. They also experimentally investigated the waveguide characteristics and groupvelocity dispersion of line defects in the PC slabs as a function of defect width [13]. Using a dielectric waveguide placed between two PC slabs, Lau and Fan have designed a single-mode line defect waveguide with a bandwidth approaching 13% [14]. In 2001, Adibi et al. showed that the main guiding mechanisms in dielectric-core PC waveguides are total internal reflection and distributed Bragg reflection. Considering this point, efficient coupling between a PC waveguide and a dielectric slab waveguide with similar slab properties can be achieved [15]. To obtain a wideband single-mode PC waveguide, they increased the radius of air holes next to the guiding region from r = 0.3a to r = 0.4a, where a represents the lattice constant of the PC [16]. In another paper, they demonstrated that by modifying the periodicity of only two rows of air holes adjacent to the guiding region, wider single-mode
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bandwidth can be achieved [17]. Moreover, Søndegaard and Lavrinenko presented a design principle for making large bandwidth PC waveguides. As a specific example, a single-mode guidance, which covers 60% of PBG, was designed [18]. In 2005, Yang et al. investigated the effects of modulation in width of line defect and size of holes in a PC slab with hexagonal elements and proved that they can be used to obtain single-mode waveguides [19]. It has been shown that dispersion diagram of a PCW mode can be controlled by the geometry of air holes next to the waveguide region [16–20]. In this paper, we propose a novel designing method for a PC slab waveguide that provides a large single-mode bandwidth and high group velocity. 2. Guided modes analysis of a photonic crystal slab waveguide Consider a 2D triangular lattice of air holes in a dielectric slab. The lattice constant and the radius of air holes are represented by a and r, respectively. The dielectric material is assumed to be Si with a refractive index of 3.4 at the wavelength of 1.55 mm. We consider only TE modes. The band diagram calculation shows that the largest complete PBG occurs for a hole radius of 0.47a [1]. To have the largest band gap just for TE modes, the best choice is r = 0.43a [1]. However, considering manufacturing problems and vertical loss in PC slabs, the radius of air holes should be smaller. The hole radius of r = 0.3a is widely used in the literature and will be used throughout this paper. The best choice for the slab height is h = 0.6a that results in the largest band gap for this structure [4]. Also, air claddings are assumed on both sides of the slab to obtain strong confinement of light inside the slab [4]. As pointed out before, 2D analysis of the structure using an effective index produces approximately the same results as the 3D analysis. This approximation reduces the computational resources (CPU time and memory) significantly. The TE-like band structure of the above PC slab is compared with the corresponding 2D band structure in Fig. 1. As shown in this figure, the 2D PC with neff = 2.76 has the same PBG as the PC slab. To extract the photonic band structures and waveguide modes, we first consider a unit cell of PC (or a supercell of PC waveguide) and then use 2D plane wave expansion (2D PWE) method with Bloch boundary conditions [1]. Also, to calculate the distribution of electromagnetic fields for a given mode, we use the 2D finite difference time domain (2D FDTD) method [21,22].
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Fig. 1. TE-like band structure of the PC slab shown by solid lines and the corresponding 2D band structure with neff = 2.76, shown with circles. The dotted lines show the air light lines for air cladding.
Fig. 2(a) shows a simple waveguide which is created by introducing a line defect along GK direction. As depicted in Fig. 2(b), the investigated PC structure has a PBG in normalized frequency range of 0.259–0.326 (c/ a), where c is the light velocity in free space. Fig. 2(b) also illustrates the guided modes in the waveguide
structure. The solid line shows the symmetric (even) mode and the dashed line is related to the asymmetric (odd) mode. The gray area represents the continuum of extended modes above the air light line and the two horizontal dash-dotted lines indicate the lower and upper frequencies of the band gap. In Fig. 2(b), the symmetric waveguide mode spans from frequency 0.265 to 0.286 (c/a), where there is no other waveguide mode. Therefore, this waveguide provides single-mode bandwidth of about 8% normalized to the center frequency. Light propagation along a PC waveguide can be caused by two mechanisms: index guiding and gap guiding. The low group velocity is a result of gap guiding when interacts with index guiding. Since there is a periodic index structure around the waveguide, it creates strong distributed reflections that cause large dispersion and low bandwidth. Due to the confinement mechanism, a gap-guided mode expands in the crystal region, and the expanded region is generally larger than that of an index-guiding mode [8]. Therefore, the gapguided modes see higher longitudinal perturbation and so they have a lower group velocity. Yamada et al. investigated this property of PC waveguides and demonstrated that adding holes, which are not located in original lattice points, to a PC waveguide will cancel the longitudinal perturbation and also reduce the averaged index in the waveguide region [8]. Fig. 3(a) illustrates a PC waveguide realized by introducing air holes with a radius of rs = 0.24a at shifted points equal to half of the lattice constant. As shown in Fig. 3(b), three waveguide modes exist in the PC band gap and the single-mode bandwidth of even guided modes is about 10.6%. In comparison with the previous structure, the second mode in this structure has higher group velocity and so larger single-mode bandwidth. It can be seen that the single-mode bandwidth of second mode is limited by the odd mode (third mode). In the next section, we propose some modification in order to change the position of guided modes and obtain higher single-mode bandwidth. 3. Increasing the waveguide bandwidth
Fig. 2. (a) Waveguide structure created in PC slab along the GK direction and a typical super cell (dashed box) which is used for PWE analysis and (b) waveguide modes. The odd mode is represented by dashed line and the even mode is shown by solid line. The gray region represents the continuum of extended modes above the air light line and two horizontal dash-dotted lines indicate the lower and upper frequencies of PC band gap.
To increase the waveguide bandwidth, we first perform an optimization on the structure shown in Fig. 3(a) to obtain the optimum value of rs. Fig. 4 indicates that a maximum normalized bandwidth of 14.3% can be achieved at rs = 0.27a. As shown in Fig. 3(b), due to the air cladding in the proposed structure, the bandwidth of the second even mode is limited by the odd mode and not by the light
M. Khatibi Moghadam et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147
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Fig. 4. Normalized bandwidth versus rs for the waveguide structure shown in Fig. 3(a).
Fig. 3. (a) Hole shifted structure to increase the group velocity. (b) Even guided modes with 10.6% single-mode bandwidth are represented with solid lines (first and second modes) and the odd mode is represented with dashed line (third mode). The gray region is the continuum of extended modes above the air light line. Two horizontal dash-dotted lines indicate the lower and upper frequencies of the band gap.
line. Therefore, if we increase the frequency of asymmetric mode without any changes in the position of symmetric modes, we can increase the single-mode bandwidth of the waveguide. For this purpose, we analyze the electromagnetic field distribution of the guided modes. The odd and both even modes are distributed in the crystal region. However, the FDTD analysis indicates that the odd mode is distributed in a larger region than the even modes. Therefore, we predict that if we change the radius of holes adjacent to the line defect it affects mainly the odd mode. We propose a new waveguide structure in which air holes of the line defect are shifted (as shown in Fig. 3(a)) and the radii of holes next to the line defect (r0 ) are different
from the radii of other holes in the PC structure (r). This waveguide is shown in Fig. 5(a). Also, dispersion curves of the waveguide modes are illustrated for different values of r0. As expected, increasing the radius of adjacent holes causes reduction of the average refractive index, and so the waveguide modes shift upward. Note that the odd mode is shifted more than the even mode and, therefore, the single-mode bandwidth of waveguide increases. To obtain the optimum value of r0, we set rs = 0.27a and calculate the band structure and waveguide bandwidth for different values of r0. Fig. 5(b) indicates that the normalized bandwidth can get as large as 20% at r0 = 0.36a. So far, we have optimized the geometrical parameters of the structure (r0 and rs) separately. In fact, to find the optimum values of r0 and rs corresponding to the largest bandwidth, we should perform a simultaneous optimization on both parameters. The results of this optimization are depicted in Fig. 6. The calculation results show that a maximum bandwidth of 21.5% can be achieved at r0 = 0.36a and rs = 0.26a. 4. 3D analysis of the proposed waveguide In order to design a wideband waveguide, we have analyzed 2D structures as an approximation for a PC slab and we have proposed a novel structure with 21.5% bandwidth. To find the exact bandwidth of this structure, it is required to perform a 3D analysis. The numerical resolution of the lattice is an important parameter in 3D analysis. Using a finer numerical grid increases the accuracy of the calculation at the expense of longer computational time. In fact, in the PWE method, the
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Fig. 7. The effect of grid size in the 3D analysis of the bandwidth of proposed waveguide structure.
Fig. 5. (a) Dispersion curves for different values of r0 (rs = 0.27a). Solid lines, dashed lines, and dotted lines correspond to r0 = 0.3a, r0 = 0.32a, and r0 = 0.34a, respectively. (b) Variation of normalized bandwidth versus r0. Fig. 8. Waveguide modes of the proposed waveguide structure computed by 2D PWE using neff = 2.76 shown by dashed dotted lines and results obtained by 3D PWE shown by solid lines.
calculation time increases rapidly with the number of points. Therefore, it is logical to begin with a rough resolution and reduce it as necessary to obtain consistent results. As shown in Fig. 7, the value of the bandwidth converges to 17% when we choose the 32 points. So the exact bandwidth of the proposed PC waveguide obtained by 3D analysis is 4.5% less than the value predicted by the 2D analysis. Fig. 8 shows the guided modes for both 2D and 3D analyses. 5. Conclusion
0
Fig. 6. Simultaneous optimization of r and rs, for obtaining the maximum bandwidth, considering 25 points for each parameter. The label of each contour indicates the value of the normalized bandwidth.
In this paper, we introduced a new method of designing wideband single-mode waveguide in PC slab. We showed that the properties of the guided modes in a PC waveguide can be modified by changing the radius of air holes that are adjacent to the line defect. We noticed
M. Khatibi Moghadam et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147
that in a shifted hole waveguide, this change affects the odd mode more than the even modes and it can be used to increase the single-mode frequency region. Using 2D PWE method as an approximation, we optimized the structure to obtain the widest bandwidth. The 2D FDTD method was applied to calculate the field intensity distribution. Finally, we used the 3D PWE method to obtain a more accurate value of the bandwidth. The results showed that the proposed PC slab waveguide has a normalized single-mode bandwidth of 17%. Acknowledgement The authors thank the reviewers for their helpful suggestions. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 1995. [2] S. Noda, T. Baba, Roadmap on Photonic Crystal, Kluwer Academic Publisher, 2003. [3] J.D. Joannopoulos, S.G. Johnson, Photonic Crystals: The Road from Theory to Practice, Kluwer Academic Publisher, 2002. [4] S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, L.A. Kolodziejski, Phys. Rev. B 60 (1999) 5751. [5] J.C. Mekis, I. Chen, S. Kurland, P.R. Fan, J.D. Villeneuve, Joannopoulos, Phys. Rev. Lett. 77 (1996) 3787.
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[6] S.G. Johnson, P.R. Villeneuve, S. Fan, J.D. Joannopoulos, Phys. Rev. B 62 (2000) 8212. [7] M. Loncar, J. Vuckovic, A. Scherer, J. Opt. Soc. Am. B 18 (2001) 1362. [8] K. Yamada, H. Morita, A. Shinya, M. Notomi, Opt. Commun. 198 (2001) 395. [9] A. Chutinan, S. Noda, Phys. Rev. B 62 (2000) 4488. [10] A. Chutinan, M. Okano, S. Noda, Appl. Phys. Lett. 80 (2002) 1698. [11] T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, A. Sakai, IEEE J. Quantum Electron. 38 (2002) 341. [12] C. Jamois, R.B. Wehrspohn, L.C. Andreani, C. Hermannd, O. Hess, U. Go¨sele, Photonics Nanostruct. Fundam. Appl. 1 (2003) 13. [13] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, I. Yokohama, Phys. Rev. Lett. 87 (2001) 253902–253911. [14] W.T. Lau, S. Fan, Appl. Phys. Lett. 81 (2002) 3915. [15] A. Adibi, Y. Xu, R.K. Lee, A. Yariv, A. Scherer, Phys. Rev. B. 64 (2001) 033308. [16] A. Adibi, R.K. Lee, Y. Xu, A. Yariv, A. Scherer, Electron. Lett. 36 (2000) 1376. [17] A. Jafarpour, A. Adibi, Y. Xu, R.K. Lee, Phys. Rev. B 68 (2003) 233102. [18] T. Søndegaard, A.V. Lavrinenko, Opt. Commun. 203 (2002) 263. [19] L. Yang, J. Motohisa, T. Fukui, Jpn. J. Appl. Phys. 44 (2005) 2531. [20] M. Khatibi Moghadam, M.M. Mirsalehi, A.R. Attari, ICEE2007, Tehran, Iran, (2007), p. 146. [21] C.P. Yu, H.C. Chang, Opt. Express 12 (2002) 6165. [22] A. Lavrinenko, P.I. Borel, L.H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, H.M.H. Chong, Opt. Express 12 (2004) 234.
Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147 www.elsevier.com/locate/photonics
Design of a novel wideband single-mode waveguide in a photonic crystal slab structure M. Khatibi Moghadam a,b,*, Mir Mojtaba Mirsalehi a, Amir Reza Attari a a
b
Electrical Engineering Department, Ferdowsi University of Mashhad, Iran Communications and Computer Research Center, Ferdowsi University of Mashhad, Iran
Received 26 December 2007; received in revised form 14 May 2008; accepted 21 May 2008 Available online 25 May 2008
Abstract We propose a novel photonic crystal slab waveguide that provides a single-mode band with large bandwidth. The proposed waveguide is obtained by introducing a line defect in a triangular lattice of air holes in a dielectric slab. This line defect consists of holes which are not located in the original lattice points. The plane wave expansion (PWE) method is used to extract the band diagram of guiding modes and based on the results, photonic crystal holes in the defect row and its adjacent rows are modified to maximize the waveguide bandwidth. We show that using the proposed structure, a single-mode bandwidth of 17% can be achieved. # 2008 Elsevier B.V. All rights reserved. PACS : 42.70.Qs; 42.25.Bs Keywords: Photonic crystal; Single-mode waveguide; PWE method; Slab structure
1. Introduction Recently, there has been a great interest in the properties of photonic crystals (PCs), which prohibit the propagation of light for frequencies within a photonic band gap (PBG) [1,2]. Using PCs, new methods are found to control the propagation of light and to construct integrated optical devices [1]. Due to the difficulties associated with the fabrication of threedimensional (3D) photonic crystals, two-dimensional (2D) photonic crystal slabs are more attractive [3]. The commonly used PC slab is a dielectric slab perforated with a 2D triangular lattice of air holes. This type of PC provides a large PBG for TE modes [4]. The
* Corresponding author. E-mail address: [email protected] (M. Khatibi Moghadam). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.05.001
field distribution of TE modes in a PC waveguide (PCW) is similar to that of a conventional optical waveguide. Therefore, the hole type PC can be efficiently coupled to conventional waveguides [1]. Waveguides are essential parts of integrated optical circuits. Because of their unique advantages, PCWs are suitable for designing optical circuits. An interesting property of a PCW is its capability of having sharp bends with high transmission efficiencies [5]. Typically, a PCW is built by introducing a line defect in the periodic structure of a PC [6]. This line defect is usually formed by eliminating one row of holes in a hole type structure. An ideal PC slab waveguide should have several properties. In many applications, it is desired that the waveguide supports a wide single-mode bandwidth. Most PCWs are not single mode over a wide bandwidth. Therefore, various methods have been proposed in the literature to increase the single-mode bandwidth of PCWs [7,8].
M. Khatibi Moghadam et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 142–147
To extract the dispersion relations of guided modes in PC slab structures, we need to perform a 3D analysis. In modal analysis of PC slab waveguides, it is found that, if the thickness of the slab causes the single index guiding mode, the 2D analysis with the effective index, neff, provides approximately the same results as the 3D analysis [8–10]. For example, for a PC slab with a lattice constant of a, a slab height of 0.6a, an air hole radius of 0.3a and a refractive index of 3.4, the 3D analysis can be replaced with a 2D analysis of PC, provided that an effective index of 2.76 is used [9]. To design a wideband single-mode PCW, research has focused on properties of guided modes in different structures. Single-line-defect waveguides in PC slabs were fabricated into an SOI wafer by Baba et al. [11]. The results and calculation of scattering loss suggest that the propagation loss can be 1–2 orders of magnitude lower than in the index waveguide. Consequently, the use of a PC waveguide is advantageous not only at bends, branches, etc., but also at straight parts in a photonic chip. Jamois et al. gave a review on the properties of silicon-based 2D PCs and their application for wave guiding [12]. They showed that the modes in PC waveguides are confined by the total internal reflection similar to the ridge waveguide. In addition, there is a second guiding mechanism based on the existence of PBG, which confines modes in the defect region. Yamada et al. have established a simple interpretation for dispersion curves of guided modes and have proposed a novel waveguide with a high group velocity [8]. They considered a silicon PC slab with SiO2 cladding. Considering light line limitation, the normalized bandwidth of their proposed structure became as large as 6% [8]. They also experimentally investigated the waveguide characteristics and groupvelocity dispersion of line defects in the PC slabs as a function of defect width [13]. Using a dielectric waveguide placed between two PC slabs, Lau and Fan have designed a single-mode line defect waveguide with a bandwidth approaching 13% [14]. In 2001, Adibi et al. showed that the main guiding mechanisms in dielectric-core PC waveguides are total internal reflection and distributed Bragg reflection. Considering this point, efficient coupling between a PC waveguide and a dielectric slab waveguide with similar slab properties can be achieved [15]. To obtain a wideband single-mode PC waveguide, they increased the radius of air holes next to the guiding region from r = 0.3a to r = 0.4a, where a represents the lattice constant of the PC [16]. In another paper, they demonstrated that by modifying the periodicity of only two rows of air holes adjacent to the guiding region, wider single-mode
143
bandwidth can be achieved [17]. Moreover, Søndegaard and Lavrinenko presented a design principle for making large bandwidth PC waveguides. As a specific example, a single-mode guidance, which covers 60% of PBG, was designed [18]. In 2005, Yang et al. investigated the effects of modulation in width of line defect and size of holes in a PC slab with hexagonal elements and proved that they can be used to obtain single-mode waveguides [19]. It has been shown that dispersion diagram of a PCW mode can be controlled by the geometry of air holes next to the waveguide region [16–20]. In this paper, we propose a novel designing method for a PC slab waveguide that provides a large single-mode bandwidth and high group velocity. 2. Guided modes analysis of a photonic crystal slab waveguide Consider a 2D triangular lattice of air holes in a dielectric slab. The lattice constant and the radius of air holes are represented by a and r, respectively. The dielectric material is assumed to be Si with a refractive index of 3.4 at the wavelength of 1.55 mm. We consider only TE modes. The band diagram calculation shows that the largest complete PBG occurs for a hole radius of 0.47a [1]. To have the largest band gap just for TE modes, the best choice is r = 0.43a [1]. However, considering manufacturing problems and vertical loss in PC slabs, the radius of air holes should be smaller. The hole radius of r = 0.3a is widely used in the literature and will be used throughout this paper. The best choice for the slab height is h = 0.6a that results in the largest band gap for this structure [4]. Also, air claddings are assumed on both sides of the slab to obtain strong confinement of light inside the slab [4]. As pointed out before, 2D analysis of the structure using an effective index produces approximately the same results as the 3D analysis. This approximation reduces the computational resources (CPU time and memory) significantly. The TE-like band structure of the above PC slab is compared with the corresponding 2D band structure in Fig. 1. As shown in this figure, the 2D PC with neff = 2.76 has the same PBG as the PC slab. To extract the photonic band structures and waveguide modes, we first consider a unit cell of PC (or a supercell of PC waveguide) and then use 2D plane wave expansion (2D PWE) method with Bloch boundary conditions [1]. Also, to calculate the distribution of electromagnetic fields for a given mode, we use the 2D finite difference time domain (2D FDTD) method [21,22].
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Fig. 1. TE-like band structure of the PC slab shown by solid lines and the corresponding 2D band structure with neff = 2.76, shown with circles. The dotted lines show the air light lines for air cladding.
Fig. 2(a) shows a simple waveguide which is created by introducing a line defect along GK direction. As depicted in Fig. 2(b), the investigated PC structure has a PBG in normalized frequency range of 0.259–0.326 (c/ a), where c is the light velocity in free space. Fig. 2(b) also illustrates the guided modes in the waveguide
structure. The solid line shows the symmetric (even) mode and the dashed line is related to the asymmetric (odd) mode. The gray area represents the continuum of extended modes above the air light line and the two horizontal dash-dotted lines indicate the lower and upper frequencies of the band gap. In Fig. 2(b), the symmetric waveguide mode spans from frequency 0.265 to 0.286 (c/a), where there is no other waveguide mode. Therefore, this waveguide provides single-mode bandwidth of about 8% normalized to the center frequency. Light propagation along a PC waveguide can be caused by two mechanisms: index guiding and gap guiding. The low group velocity is a result of gap guiding when interacts with index guiding. Since there is a periodic index structure around the waveguide, it creates strong distributed reflections that cause large dispersion and low bandwidth. Due to the confinement mechanism, a gap-guided mode expands in the crystal region, and the expanded region is generally larger than that of an index-guiding mode [8]. Therefore, the gapguided modes see higher longitudinal perturbation and so they have a lower group velocity. Yamada et al. investigated this property of PC waveguides and demonstrated that adding holes, which are not located in original lattice points, to a PC waveguide will cancel the longitudinal perturbation and also reduce the averaged index in the waveguide region [8]. Fig. 3(a) illustrates a PC waveguide realized by introducing air holes with a radius of rs = 0.24a at shifted points equal to half of the lattice constant. As shown in Fig. 3(b), three waveguide modes exist in the PC band gap and the single-mode bandwidth of even guided modes is about 10.6%. In comparison with the previous structure, the second mode in this structure has higher group velocity and so larger single-mode bandwidth. It can be seen that the single-mode bandwidth of second mode is limited by the odd mode (third mode). In the next section, we propose some modification in order to change the position of guided modes and obtain higher single-mode bandwidth. 3. Increasing the waveguide bandwidth
Fig. 2. (a) Waveguide structure created in PC slab along the GK direction and a typical super cell (dashed box) which is used for PWE analysis and (b) waveguide modes. The odd mode is represented by dashed line and the even mode is shown by solid line. The gray region represents the continuum of extended modes above the air light line and two horizontal dash-dotted lines indicate the lower and upper frequencies of PC band gap.
To increase the waveguide bandwidth, we first perform an optimization on the structure shown in Fig. 3(a) to obtain the optimum value of rs. Fig. 4 indicates that a maximum normalized bandwidth of 14.3% can be achieved at rs = 0.27a. As shown in Fig. 3(b), due to the air cladding in the proposed structure, the bandwidth of the second even mode is limited by the odd mode and not by the light
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Fig. 4. Normalized bandwidth versus rs for the waveguide structure shown in Fig. 3(a).
Fig. 3. (a) Hole shifted structure to increase the group velocity. (b) Even guided modes with 10.6% single-mode bandwidth are represented with solid lines (first and second modes) and the odd mode is represented with dashed line (third mode). The gray region is the continuum of extended modes above the air light line. Two horizontal dash-dotted lines indicate the lower and upper frequencies of the band gap.
line. Therefore, if we increase the frequency of asymmetric mode without any changes in the position of symmetric modes, we can increase the single-mode bandwidth of the waveguide. For this purpose, we analyze the electromagnetic field distribution of the guided modes. The odd and both even modes are distributed in the crystal region. However, the FDTD analysis indicates that the odd mode is distributed in a larger region than the even modes. Therefore, we predict that if we change the radius of holes adjacent to the line defect it affects mainly the odd mode. We propose a new waveguide structure in which air holes of the line defect are shifted (as shown in Fig. 3(a)) and the radii of holes next to the line defect (r0 ) are different
from the radii of other holes in the PC structure (r). This waveguide is shown in Fig. 5(a). Also, dispersion curves of the waveguide modes are illustrated for different values of r0. As expected, increasing the radius of adjacent holes causes reduction of the average refractive index, and so the waveguide modes shift upward. Note that the odd mode is shifted more than the even mode and, therefore, the single-mode bandwidth of waveguide increases. To obtain the optimum value of r0, we set rs = 0.27a and calculate the band structure and waveguide bandwidth for different values of r0. Fig. 5(b) indicates that the normalized bandwidth can get as large as 20% at r0 = 0.36a. So far, we have optimized the geometrical parameters of the structure (r0 and rs) separately. In fact, to find the optimum values of r0 and rs corresponding to the largest bandwidth, we should perform a simultaneous optimization on both parameters. The results of this optimization are depicted in Fig. 6. The calculation results show that a maximum bandwidth of 21.5% can be achieved at r0 = 0.36a and rs = 0.26a. 4. 3D analysis of the proposed waveguide In order to design a wideband waveguide, we have analyzed 2D structures as an approximation for a PC slab and we have proposed a novel structure with 21.5% bandwidth. To find the exact bandwidth of this structure, it is required to perform a 3D analysis. The numerical resolution of the lattice is an important parameter in 3D analysis. Using a finer numerical grid increases the accuracy of the calculation at the expense of longer computational time. In fact, in the PWE method, the
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Fig. 7. The effect of grid size in the 3D analysis of the bandwidth of proposed waveguide structure.
Fig. 5. (a) Dispersion curves for different values of r0 (rs = 0.27a). Solid lines, dashed lines, and dotted lines correspond to r0 = 0.3a, r0 = 0.32a, and r0 = 0.34a, respectively. (b) Variation of normalized bandwidth versus r0. Fig. 8. Waveguide modes of the proposed waveguide structure computed by 2D PWE using neff = 2.76 shown by dashed dotted lines and results obtained by 3D PWE shown by solid lines.
calculation time increases rapidly with the number of points. Therefore, it is logical to begin with a rough resolution and reduce it as necessary to obtain consistent results. As shown in Fig. 7, the value of the bandwidth converges to 17% when we choose the 32 points. So the exact bandwidth of the proposed PC waveguide obtained by 3D analysis is 4.5% less than the value predicted by the 2D analysis. Fig. 8 shows the guided modes for both 2D and 3D analyses. 5. Conclusion
0
Fig. 6. Simultaneous optimization of r and rs, for obtaining the maximum bandwidth, considering 25 points for each parameter. The label of each contour indicates the value of the normalized bandwidth.
In this paper, we introduced a new method of designing wideband single-mode waveguide in PC slab. We showed that the properties of the guided modes in a PC waveguide can be modified by changing the radius of air holes that are adjacent to the line defect. We noticed
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that in a shifted hole waveguide, this change affects the odd mode more than the even modes and it can be used to increase the single-mode frequency region. Using 2D PWE method as an approximation, we optimized the structure to obtain the widest bandwidth. The 2D FDTD method was applied to calculate the field intensity distribution. Finally, we used the 3D PWE method to obtain a more accurate value of the bandwidth. The results showed that the proposed PC slab waveguide has a normalized single-mode bandwidth of 17%. Acknowledgement The authors thank the reviewers for their helpful suggestions. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 1995. [2] S. Noda, T. Baba, Roadmap on Photonic Crystal, Kluwer Academic Publisher, 2003. [3] J.D. Joannopoulos, S.G. Johnson, Photonic Crystals: The Road from Theory to Practice, Kluwer Academic Publisher, 2002. [4] S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, L.A. Kolodziejski, Phys. Rev. B 60 (1999) 5751. [5] J.C. Mekis, I. Chen, S. Kurland, P.R. Fan, J.D. Villeneuve, Joannopoulos, Phys. Rev. Lett. 77 (1996) 3787.
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