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Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits
Optics Express, Vol. 10, Issue 24, pp. 1391-1397 (2002)
http://dx.doi.org/10.1364/OE.10.001391
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– Abstract
1. Introduction
2. Coupling technique and structures
3. Simulation results and discussion
4. Conclusion
– References and links
– Abstract
1. Introduction
2. Coupling technique and structures
3. Simulation results and discussion
4. Conclusion
– References and links
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Abstract
In this paper, a mode matching technique for highly efficient coupling between dielectric silica waveguides (SWG) and planar photonic crystal (PPC) waveguides based on setting localized defects in a PPC tapered waveguide is reported. The introduction of multiple defects is designed properly depending on mode mismatching arising from the different widths of the SWG and the PPC waveguide. The procedure to obtain the optimum defects configuration is described. Transmission efficiencies above 80% at a wavelength of 1.55μm are reported improving significantly the transmission efficiencies achieved with conventional PPC tapered structures without defects. Furthermore, the feasibility of the coupling technique for both input/output coupling over a large frequency band is shown.
© 2002 Optical Society of America
[Optical Society of America ]
1. Introduction
The control of the flow of light due to the photonic band gap (PBG) effect in photonic crystals makes these materials one of the most promising approaches to achieve highly integrated photonic circuits. Although to achieve full control of light propagation a three-dimensional (3D) photonic crystal is needed, planar photonic crystals (PPC) have been proposed to reduce manufacturing complexity [1]. By creating line defects in PPC structures, well-confined guided modes appear inside the PBG because light propagation is not allowed out of the defect. One of the major challenges to develop reliable PPC circuits is the minimization of the coupling losses between conventional silica waveguides (SWG) and PPC waveguides. Hence, several coupling structures and techniques have been proposed in the last years such as grating-coupler-based structures [2–4], tapered waveguides [5,6], a J-coupler structure [7] and PPC tapers [8–10]. Among all the proposed solutions, one of the most promising approaches are PPC tapers mainly due to its small coupling length and high coupling efficiencies achieved over a large frequency range.
T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996). [CrossRef]
D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-pale grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE Journal of Quantum Electronics 38, 949–955 (2002). [CrossRef]
Y. Xu, R. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757 (2000). [CrossRef]
A. Mekis and J.D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” IEEE J. Lightwave Technol. 19, 861–865 (2001). [CrossRef]
D.W. Prather, J. Murakowski, S. Shi, S. Venkataram, A. Sharkawy, C. Chen, and D. Pustai, “High-effciency coupling structure for a single-line-defect photonic-crystal waveguide,” Opt. Lett. 27, 1601–1603 (2002). [CrossRef]
Ph. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguides,” Opt. Express 10, 354 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-354 [CrossRef] [PubMed]
PPC coupling structures were designed and experimentally verified in [8–9] by varying gradually the rod sizes for efficient coupling between PPC waveguides of different widths. Furthermore, to couple a wide SWG to a typically narrower PPC waveguide several tapered structures realized by broadening the PPC waveguide for efficient profile matching were investigated in [10]. A novel coupling technique based on setting a single localized defect within a 0.5μm-long PPC tapered waveguide structure was outlined in [11], which improves significantly the transmission results using only a conventional PPC taper [10]. However, wider PPC tapers are required for efficient mode profile matching to wider dielectric waveguides. In this paper, it is shown that the proposed coupling technique can also be employed with wider PPC tapers but, in this case, a new defects configuration must be designed for the required PPC taper to maximize the transmission efficiency. The introduction of multiple localized defects is investigated as well as their effect on the frequency transmission spectra when both coupling sides (input and output) of a PPC waveguide are considered. By setting properly the defects mode matching at the interface between the SWG and the PPC waveguide is attained, which improves significantly the transmission efficiency compared to the PPC taper without defects.
Ph. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguides,” Opt. Express 10, 354 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-354 [CrossRef] [PubMed]
T.D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. 26, 1102–1104 (2001). [CrossRef]
P. Sanchis, J. Martí, A. García, A. Martínez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. 38, 961–962 (2002). [CrossRef]
T.D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. 26, 1102–1104 (2001). [CrossRef]
2. Coupling technique and structures
The PPC structure considered here is a two-dimensional (2D) triangular array of dielectric rods of lattice constant a surrounded by a homogeneous dielectric medium. Rods have a refractive index value of 3.45, which corresponds to Silicon (Si) at 1.55μm, and a radius of R= 0.2a. The surrounding medium in the PPC has an index value of 1.45, which corresponds to Silica (SiO2) at 1.55μm. This PPC has a PBG between the normalized frequencies ω1
=026(a/λ) and ω2
=036(a/λ) for TM polarized waves, calculated by employing a 2D plane wave expansion (PWE) method [12]. The PPC waveguide is created by removing all the rods of a row along the ΓK direction. A single-mode guided by a truly PBG guiding effect appears by forming the line defect in the PPC since the core of the PPC waveguide has a refractive index smaller than that of the surrounding medium. A lattice constant value of 465nm is chosen in order to fix the transmitted band around 1.55μm. For such lattice constant the PBG ranges from 1.29μm to 1.79μm for TM polarized waves. The SWG has a dielectric index of 1.45, a width of w, and the surrounding dielectric medium is air.
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]
The investigated coupling structure consists of a PPC taper (a discrete taper realized by removing some of the rods of the original PPC waveguide) where a set of different dielectric defect rods are introduced. Mode matching employing different PPC taper structures is achieved by choosing the optimum defect parameters within the PPC taper. In this study two
different PPC taper structures are considered: (a) a 2μm-wide/0.5μm-long PPC taper, and (b) a 4μm-wide/1μm-long PPC taper. Both structures, employed to couple light both into and out of a finite length PPC waveguide, are shown in Fig. 1(a) and Fig. 1(b), respectively. In the former case, only one defect was set for coupling to a 1.5μm-wide SWG [11]. However, in the case of coupling to a 3μm-wide SWG, the latter case achieves an efficiently mode profile matching but, in this case, better coupling performance is obtained by setting two defects with different radii (see Fig. 1(b)), as will be shown later.
P. Sanchis, J. Martí, A. García, A. Martínez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. 38, 961–962 (2002). [CrossRef]
Fig. 1. Schematic view of the structures considered. (a) a 2μm-wide/ 0.5μm-long and (b) a 4μm-wide/1μm-long planar photonic crystal (PPC) taper each one with a different defects configuration are used to couple light both into and out of a finite PPC waveguide from a silica waveguide (SWG). The lattice constant of the PPC is a and the radius of the rods is R.
3. Simulation results and discussion
Coupling losses between conventional SWG and PPC waveguides are mainly caused due to the mode mismatch derived from the different widths and propagation mechanisms in SWG and PPC waveguides. Although mode transformation is achieved by employing a PPC taper [10], the condition of adiabaticity is not satisfied resulting in a mode mismatch that decreases the coupling efficiency and increases reflection losses. The introduction of localized defects within the PPC taper alters the modal properties of the guided mode so that mode matching can be achieved by determining the optimum number of defects as well as their optimum parameters (radii and relative position within the PPC taper) improving, therefore, the transmission efficiency. Multiple localized defects were also employed in PPC waveguides to maximize the transmission efficiency in sharp bends [13]. But while the effect of introducing defects inside a PPC waveguide was modeled by means of the transmission line theory [13], a different approach is necessary for setting localized defects into PPC tapers due to the variation of the modal properties along the taper. A straightforward procedure based on numerical analysis is followed here. We first optimize the number and the relative position of the defects that should be placed into the PPC taper, and further the transmission is improved by optimizing the radius of each defect. The computational method used is based on a 2D finite difference time domain (FDTD) algorithm [14]. Perfectly matched layer (PML) conditions have been considered in the calculations to ensure no back reflection in the limit of the analyzed region [15].
T.D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. 26, 1102–1104 (2001). [CrossRef]
S. Boscolo, C. Conti, M. Midirio, and C.G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” IEEE J. Lightwave Technol. 20, 304–310 (2002). [CrossRef]
S. Boscolo, C. Conti, M. Midirio, and C.G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” IEEE J. Lightwave Technol. 20, 304–310 (2002). [CrossRef]
J. P. Berenger, “A perfectly matched layer for the absorption for electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
For the PPC taper shown in Fig. 1(a) a single defect with an optimized radius ropt
=0.5R and an optimum relative position of zopt
=0.6a was set to achieve a transmission efficiency above 80% for the input coupling from a SWG with w=1.5μm [11]. However, it was seen that the transmission efficiency decreased as the width of the SWG increased, although it was still more than 15% higher than the employed PPC taper without the defect. Figure 2 shows the normalized frequency response of a 16-row PPC waveguide coupled to an input and output SWG of two different widths, w=1.5μm and w=3μm, with and without the optimized defect placed within the PPC taper.
P. Sanchis, J. Martí, A. García, A. Martínez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. 38, 961–962 (2002). [CrossRef]
Fig. 2. Transmission spectra as a function of the normalized frequency for the structure shown in Fig.1(a) taking into account different SWG widths, w=1.5μm and w= 3μm, and with and without the proposed coupling technique.
The fundamental mode of the SWG is excited by a pulsed wave that propagates along the z-direction (see Fig. 1) and the transmission spectra is calculated with the overlap integral between the launched and the measured field at the output SWG. The resonances that appear in the transmission spectrum of the PPC taper without defect are due to the Fabry-Perot-like cavity created by the mode mismatching at the interfaces between the SWG and the PPC waveguide and thus the number of resonances depends on the PPC length. When the proposed coupling technique is employed those peaks in the response sharply diminish because a better mode matching at the interfaces of the PPC waveguide is achieved. It can be obtained that an average transmission level of 68.95% is achieved over the normalized frequency band ranging from 0.29(a/λ) to 0.31(a/λ) (corresponding to a transmission band from 1.5 to 1.6μm for the
lattice constant value considered) when the optimized defect is set and w=1.5μm. However, when w=3μm the average transmission level decreased up to 45.45%, although clearly improves still the 21.43% average transmission level achieved when the defect is not considered.
The optimum relative position of the defect placed within the PPC taper shown in Fig. 1(a) was obtained by varying the position of the defect along the z-axis and measuring the obtained transmission from a SWG to a PPC waveguide of a monochromatic continuous-wave with normalized power [11]. The same procedure was used for the 4μm-wide/1μm-long PPC taper, so that a single defect with radius R was initially employed for scanning the positions that result in relative maximum transmissions, at which the defects were set.
P. Sanchis, J. Martí, A. García, A. Martínez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. 38, 961–962 (2002). [CrossRef]
Fig. 3. (a) Normalized transmitted power as a function of the relative position of a localized defect in the z-axis normalized to the lattice constant and (b) as a function of the radius of the defects, rext
and rint
, normalized to the radius of the rods, R, and located the former at zext
=0.59a and the latter at zint
=1.52a, both for the PPC taper shown in Fig.1(b).
The normalized transmitted power obtained as a function of z/a for the case of coupling only one side of the PPC waveguide with the 4μm-wide/1μm-long taper to the SWG and for a fixed normalized frequency of 0.3(a/λ) is shown in Fig. 3(a). It can be seen that there are two z/a positions that provide relative transmission maximums: zext
=0.59a and zint
=1.52a. By setting two defects at both maximum positions depicted in Fig. 3 and then optimizing their radii the resulting transmission is further improved.
Fig. 4. Movies (both 1.6 MB) of the electric field intensity input coupling from the SWG to the PPC waveguide employing the PPC taper shown in Fig. 1(b), (a) without and (b) with the optimized two-defects configuration. [Media 2]
Figure 3(b) depicts the normalized transmitted power as a function of the defect radius (rdef
) normalized to the radius of the rods (R) for both defects the inner (rint
) and outer one (rext
) regarding the PPC tapered waveguide. The optimization procedure employed has been as follows. After fixing the two defects with radii R at the optimum z/a positions shown in Fig.3(a), we first varied rint
/R, maintaining the outer defect with a radius R. In Fig. 3(b) it can be observed that a maximum transmission of 82% is achieved for a rint
=0.37R (see lines with circles in Fig. 3(b)) improving the 74% achieved if only one defect is considered, as depicted in Fig. 3.
The transmission efficiency is further improved by optimizing the radius of the defect located at zext
=0.59a. Fig. 3(b) also shows the transmission as a function of the outer defect radius normalized to the radius of the rods in the PPC (rext
/R), by keeping fixed the inner defect with the optimum radius calculated previously. In Fig. 3(b) (see line with diamonds) it can be observed that the peak transmission improves up to 84% for a defect radius equal to rext
=0.86R. Therefore, by setting the two defects within the PPC taper at the optimum positions and with the optimum radii the peak transmission at 0.3(a/λ), which corresponds to a wavelength of 1.55μm for the lattice constant value of 465 nm, is enhanced up to 84% transmission from a nearly 40% transmission when a taper structure with no defects is considered. Figure 4 shows the electric field for the input coupling employing the 4μm-wide/1μm-long PPC taper with and without the optimized two-defects configuration obtained previously. It can be seen that a better coupling to the PPC waveguide is achieved when the proposed coupling technique is employed. It should also be noticed that the standing-wave pattern that appears in the input SWG even for the optimum coupling achieved with the two-defects configuration shown in Fig. 4(b) arises from the fact that there is still reflection since full mode matching is not achieved
Fig. 5. Transmission spectra as a function of the normalized frequency for the structure shown in Fig.1(b) for a SWG width of 3 μm with and without the proposed coupling technique. In the former case, two spectra are depicted showing the influence of the normalized frequency employed in the optimization procedure.
The transmission spectra against the normalized frequency for a 14-row-long PPC waveguide coupled to both an input and output SWG with a 1μm-long PPC taper using the two-defects configuration (see Fig. 1(b)) and without defects is depicted in Fig. 5. An average transmission level of 71.31% is achieved for a normalized frequency range from 0.29(a/λ) to 0.31(a/λ), which enhances the 17.98% average transmission level achieved with the PPC taper without defects and also the 45.45% average transmission level achieved with the PPC taper shown in Fig.1 (a) when the SWG width was 3μm. However, in Fig. 5 it can be observed that the bandwidth is reduced when compared to the results provided in Fig. 2 although it still satisfies bandwidth requirements for optical communications. This bandwidth reduction is derived from the larger number of defects needed to achieve high coupling efficiency for a wider SWG. Consequently the high coupling efficiency is achieved at the expense of bandwidth which becomes more sensitive to the normalized frequency employed to optimize the parameters of the defects. To demonstrate this fact the same optimization procedure was employed but for a normalized frequency of 0.32(a/λ). In this case, the optimum two-defects configuration was obtained for zint
=1.56a, rint
=0.43R and zext
=0.74a, rext
=0.86R. By using this defects configuration the transmission spectra is shifted towards the normalized frequency employed in the optimization procedure as it can be seen in Fig. 5.
4. Conclusion
In this paper, we report a coupling technique based on setting multiple localized defects within a PPC tapered waveguide. The coupling technique achieves mode matching at the interface between SWG and PPC waveguides reducing reflection losses and improving significantly the transmission efficiency over a large frequency band. An optimization procedure to chose the optimum number of defects and their radii, which will depend on the width and length of the PPC taper, has been proposed. The simulation results show that by setting properly two defects within a 4μm-wide/1μm-long PPC taper a transmission of 84% at a wavelength of 1.55μm can be achieved, which sharply enhances the 40% transmission obtained when no defects are considered. The proposed coupling technique can be easily generalized and applied to different PPC tapers required for efficient mode profile matching to SWG of arbitrarily width.
Acknowledgments
This work has been partially funded by the Spanish Ministry of Science and Technology under grant TIC2002-01553. P. Sanchis acknowledges the Spanish Ministry of Education, Culture and Sport for funding his grant.
References and Links
T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996). [CrossRef] | |
D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-pale grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE Journal of Quantum Electronics 38, 949–955 (2002). [CrossRef] | |
W. Kuang, C. Kim, A. Stapleton, and J.D. O’Brien, “Grating-assisted coupling of optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1604–1606 (2002). [CrossRef] | |
M.E. Potter and R.W. Ziolkowski, “Two compact structures for perpendicular coupling of optical signals between dielectric and photonic crystal waveguides,” Opt. Express , 10 691, (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-691 [CrossRef] [PubMed] | |
Y. Xu, R. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757 (2000). [CrossRef] | |
A. Mekis and J.D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” IEEE J. Lightwave Technol. 19, 861–865 (2001). [CrossRef] | |
D.W. Prather, J. Murakowski, S. Shi, S. Venkataram, A. Sharkawy, C. Chen, and D. Pustai, “High-effciency coupling structure for a single-line-defect photonic-crystal waveguide,” Opt. Lett. 27, 1601–1603 (2002). [CrossRef] | |
Ph. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguides,” Opt. Express 10, 354 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-354 [CrossRef] [PubMed] | |
A. Talneau, Ph. Lalanne, M. Agio, and C.M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. 27, 1522–1524 (2002) [CrossRef] | |
T.D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. 26, 1102–1104 (2001). [CrossRef] | |
P. Sanchis, J. Martí, A. García, A. Martínez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. 38, 961–962 (2002). [CrossRef] | |
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed] | |
S. Boscolo, C. Conti, M. Midirio, and C.G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” IEEE J. Lightwave Technol. 20, 304–310 (2002). [CrossRef] | |
A. Taflove, Computational Electrodynamics (Artech, Norwood, MA, 1995). | |
J. P. Berenger, “A perfectly matched layer for the absorption for electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef] |
OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(250.5300) Optoelectronics : Photonic integrated circuits
ToC Category:
Research Papers
History
Original Manuscript: October 29, 2002
Revised Manuscript: November 19, 2002
Published: December 2, 2002
Citation
Pablo Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, "Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits," Opt. Express 10, 1391-1397 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-24-1391
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References
- T. F. Krauss, R. M. De La Rue and S. Brand, "Two-dimensional photonic bandgap structures operating at nearinfrared wavelengths," Nature 383, 699-702 (1996). [CrossRef]
- D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel and R. Baets, "An out-of-pale grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers," IEEE J. Quantum Electron. 38, 949-955 (2002). [CrossRef]
- W. Kuang, C. Kim, A. Stapleton and J.D. O'Brien, "Grating-assisted coupling of optical fibers and photonic crystal waveguides," Opt. Lett. 27, 1604-1606 (2002). [CrossRef]
- M.E. Potter and R.W. Ziolkowski, "Two compact structures for perpendicular coupling of optical signals between dielectric and photonic crystal waveguides," Opt. Express 10, 691, (2002) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-691">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-691</a> [CrossRef] [PubMed]
- Y. Xu, R. Lee, and A. Yariv, "Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry," Opt. Lett. 25, 755-757 (2000). [CrossRef]
- A. Mekis and J. D. Joannopoulos, "Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides," IEEE J. Lightwave Technol. 19, 861-865 (2001). [CrossRef]
- D.W. Prather, J. Murakowski, S. Shi, S. Venkataram, A. Sharkawy, C. Chen and D. Pustai, "High-effciency coupling structure for a single-line-defect photonic-crystal waveguide," Opt. Lett. 27, 1601-1603 (2002). [CrossRef]
- Ph. Lalanne and A. Talneau, "Modal conversion with artificial materials for photonic-crystal waveguides," Opt. Express 10, 354 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-354">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-354</a> [CrossRef] [PubMed]
- A. Talneau, Ph. Lalanne, M. Agio and C.M. Soukoulis, "Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths," Opt. Lett. 27, 1522-1524 (2002) [CrossRef]
- T.D. Happ, M. Kamp and A. Forchel, "Photonic crystal tapers for ultracompact mode conversion," Opt. Lett. 26, 1102-1104 (2001). [CrossRef]
- P. Sanchis, J. Martí, A. García, A. Martínez and J. Blasco, "High efficiency coupling technique for planar photonic crystal waveguides," Electron. Lett. 38, 961-962 (2002). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
- S. Boscolo, C. Conti, M. Midirio and C.G. Someda, "Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length," IEEE J. Lightwave Technol. 20, 304-310 (2002). [CrossRef]
- A. Taflove, Computational Electrodynamics (Artech, Norwood, MA, 1995).
- J. P. Berenger, "A perfectly matched layer for the absorption for electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
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