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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 11 — May. 31, 2004
  • pp: 2356–2364
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New ring resonator configuration using hybrid photonic crystal and conventional waveguide structures

Seunghyun Kim, Jingbo Cai, Jianhua Jiang, and Gregory P. Nordin  »View Author Affiliations


Optics Express, Vol. 12, Issue 11, pp. 2356-2364 (2004)
http://dx.doi.org/10.1364/OPEX.12.002356


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Abstract

We propose a new method of realizing ring resonators based on hybrid photonic crystal and conventional waveguide structures. The proposed ring resonator configuration is advantageous compared with general ring resonator structures for its controllability of the quality (Q) factor, free spectral range (FSR), and full width at half maximum (FWHM) over a wide range. We show ring resonator structures based on a single mode waveguide with core and clad refractive indices of 1.5 and 1.465, respectively. A 35µm×50µm ring resonator has a free spectral range (FSR) of 14.1nm and a quality (Q) factor of 595 with high optical efficiency (92.7%). By decreasing the size of the ring resonator to 35µm×35µm, the FSR is increased to 19.8nm. Modifying the splitting ratio of the beam splitters permits the Q factor to be increased to 1600.

© 2004 Optical Society of America

1. Introduction

Ring resonators have been investigated for a variety of applications including add/drop and band pass filters, wavelength division multiplexer/demultiplexers, and all optical switches [1

1. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-Compact Si-SiO2 Microring Resonator Optical Channel Dropping Filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998). [CrossRef]

6

6. B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring Resonator Arrays for VLSI Photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000). [CrossRef]

]. Ring resonators reported in the literature are generally configured with circular or racetrack waveguides placed between two straight waveguides. Light of particular wavelengths are coupled from one straight waveguide to the other. These are called drop wavelengths. Two waveguide types are mainly used as platforms to build ring resonators. One is a low refractive index waveguide with low index contrast between the core and clad [3

3. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002). [CrossRef]

6

6. B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring Resonator Arrays for VLSI Photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000). [CrossRef]

]. This type of waveguide has advantages, such as low propagation loss, low coupling loss from the fiber, and low dispersion. However, since a very large bend radius is needed for a low index contrast waveguide to achieve a high efficiency bend, ring resonators based on such waveguides require a very large radius for the circular or racetrack waveguide, which in turn increases the overall size of the ring resonator and limits the maximum free spectral range (FSR) [3

3. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002). [CrossRef]

,4

4. G. Bourdon, G. Alibert, A. Beguin, B. Bellman, and E. Guiot, “Ultralow Loss Ring Resonators Using 3.5% Index-Contrast Ge-Doped Silica Waveguides,” IEEE Photon. Technol. Lett. 15, 709–711 (2003). [CrossRef]

]. Another waveguide type is a high core refractive index waveguide with high index contrast to the cladding [1

1. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-Compact Si-SiO2 Microring Resonator Optical Channel Dropping Filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998). [CrossRef]

,2

2. A. Vorckel, M. Monster, W. Henschel, P. H. Bolivar, and H. Kurz, “Asymmetrically Coupled Silicon-On-Insulator Microring Resonators for Compact Add-Drop Multiplexers,” IEEE Photon. Technol. Lett. 15, 921–923 (2003). [CrossRef]

]. In this case, the waveguide mode is strongly confined in the core region. Therefore, a high efficiency bend can be achieved with very small bend radius (often <10µm), which results in a small ring resonator structure. However, high index contrast waveguides generally suffer high fiber to waveguide coupling loss because the waveguide mode size (<2µm2) is so much smaller than that of a fiber (~50µm2) [1

1. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-Compact Si-SiO2 Microring Resonator Optical Channel Dropping Filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998). [CrossRef]

].

We have recently proposed high efficiency ultracompact waveguide bends, splitters, and polarizing beam splitters using hybrid photonic crystal (PhC) and conventional waveguide (CWG) structures [7

7. G. P. Nordin, S. Kim, J. Cai, and J. Jiang, “Hybrid integration of conventional waveguide and photonic crystal structures,” Opt. Express 10, 1334–1341 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1334. [CrossRef] [PubMed]

11

11. J. Cai, G. P. Nordin, S. Kim, and J. Jiang, “3-D Analysis of hybrid conventional waveguide/photonic crystal 90 degree bend” Appl. Opt. (to be published).

]. Compared with conventional PhC structures for bend and splitters [12

12. M. Tokushima and H. Yamada, “Light propagation in a photonic-crystal-slab line-defect waveguide,” IEEE J. Quantum Electron. 38, 753–759 (2002). [CrossRef]

14

14. D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. W. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1823 [CrossRef] [PubMed]

], such hybrid structures take advantage of low loss CWGs for propagation and use limited PhC regions to implement small-area bend and splitter functions. In this paper, we present a new ring resonator configuration based on these hybrid structures. Finite difference time domain (FDTD) simulations of a particular ring resonator design demonstrate a 2.5nm full width at half maximum (FWHM), drop port spectral width, a 14.1nm free spectral range (FSR), and a quality (Q) factor (λdrop/ΔλFWHM) of 595. This structure is used as a base ring resonator for further modification to show how increased FSR and Q factor can be achieved. A FSR of 19.8nm is realized by decreasing the light propagation distance in the ring resonator while the Q factor is increased to 1600 by changing power splitting ratio of the splitter structures.

2. Design of a ring resonator

We assume a two-dimensional (2-D) single mode waveguide (at λ=1.55µm) with a 2µm core width and core and clad refractive indices of 1.5 and 1.465. As shown in Fig. 1, a PhC region composed of a square Si (n=3.481) post array (100nm radius and 380nm lattice constant) is used to achieve a high efficiency 90 degree waveguide bend while a single row of Si posts tilted at 45 degree (100nm radius and 424nm period) functions as a high efficiency splitter. Both are designed to operate with TM polarized light (i.e., electric field out of the plane). Alternative structures could be designed to operate with TE polarized light. To numerically calculate the optical efficiencies of both the bend and splitter, a two-dimensional (2-D) finite difference time domain (FDTD) method [15

15. A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method” (Artech House, Boston, 1995).

] with Berenger perfectly matched layer boundary conditions [16

16. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]

] is used. The optical efficiency used throughout this paper is the ratio of the power at the output detector to the incident mode power. The waveguide mode source along the input waveguide for both the bend and splitter has an 8µm width as illustrated in Fig. 1. To monitor the output power, 8µm wide detectors are placed at the output waveguides.

Fig. 1. Hybrid PhC and CWG structures for (a) a high efficiency 90 degree bend and (b) a high efficiency beam splitter. Square inset in Fig. 1(a) indicates the first Brilluin zone of the PhC.

Fig. 2. Efficiency as a function of wavelength for TM polarization. Solid line (black) indicates the bend efficiency of the hybrid structure while other two lines (blue) correspond to the efficiencies at output channels of the splitter.
Fig. 3. A ring resonator structure constructed with hybrid PhC and CWG structures. To maximize the efficiency, the throughput port is shifted 120nm to the left of the nominal waveguide center line and the waveguide between the two splitters is similarly shifted up 120nm.

Figure 3 shows a ring resonator formed by the combination of hybrid 90 degree bend and splitter structures. We will refer to this as the base ring resonator structure. The area occupied by this ring resonator is 35.0µm×50.0µm. Due to the large computational requirements for the numerical simulation of ring resonators, a 2-D FDTD code is parallelized in order that the computation can be done by multiple CPUs. As shown in Fig. 3, an 8µm wide waveguide mode source is launched at the input waveguide and the efficiencies at both drop and throughput ports are calculated along the 8µm wide detector lines.

The spectral response at both the drop and throughput ports of the base structure are shown in Fig. 4. The drop wavelength efficiency is 92.7% with a 2.5nm FWHM, a 14.1nm FSR, and a Q factor of 595. The extinction ratio calculated (i.e., ratio of the throughput port power to the drop port power for the drop wavelength) is 13.6dB as shown in Fig. 4. A time snapshot of the electric field for the drop wavelength as calculated by our parallelized 2-D FDTD is shown in Fig. 5.

We define the fraction of the optical power incident at the input part of the ring resonator that is transmitted through the drop port as the drop port efficiency, ηdrop. An expression for ηdrop can be easily obtained with the method generally used to calculate the transmission of a Fabry-Perot resonator [17

17. B. E. A. Saleh and M. C. Teich, “Fundamentals of Photonics” (A Wiley-Interscience publication, New York, 1991), Chap. 9.

].

ηdrop=Ts2(1RsRb)21+[4RsRb(1RsRb)2]sin2ϕ2,
(1)
ϕ=θ+2πndλ0,
(2)

where the Ts and Rs are the efficiencies of the splitter at the output channels 1 and 2, Rb is the bend efficiency, θ corresponds to the total phase delay at the splitters and bends, and d is the light propagation distance in the ring resonator. The free space wavelength is λ 0 and n is the effective index of the waveguide.

As seen from Eq. (1), the efficiency at the drop port is maximum when sin(ϕ/2) goes to zero. Therefore, the drop wavelengths, which have maximum efficiency at the drop port, satisfy ϕ=2 (where m=0,1,2,3…). Since ϕ is a function of the light propagation distance d for the given waveguide and hybrid bend and splitter structures, the FSR of the ring resonator can be increased or decreased by changing d. Note that the Q factor of the base ring resonator can be changed by varying Rs or Rb. For fixed d, since the phase delay accompanied by the variation of Rs or Rb is negligible, the spectral response of the ring resonator at the drop port becomes shaper or wider depending upon Rs or Rb, which in turn changes the FWHM of the peak. As the Q factor is a function of the FWHM, the Q factor can therefore be changed by Rs or Rb. In order to increase the Q factor of the base ring resonator, we can only increase Rs since Rb is already nearly one.

Fig. 4. Spectral responses of the base ring resonator at the drop (solid line) and throughput (dashed line) ports. The computation is done by 8 PCs in a linux beowolf cluster.
Fig. 5. (1.55 MB) Movie of the electric field at the drop wavelength. The computation is done by 4 PCs with the parallelized 2-D FDTD. Yee cell size is 15nm and computation is done for 500,000 time steps.

3. Ring resonators with larger FSR and Q factor than the base structure.

As shown in Fig. 6, the FSR can be increased by decreasing the size of the ring resonator. In this case, the bends have been moved 15µm closer to the splitters such that the ring resonator occupies an area of 35µm×35µm.

The spectral response for the modified and base ring resonators is shown in Fig. 7 to facilitate comparison. The FSR is increased to 19.8nm and the drop wavelength efficiency is 91.0%. This wide FSR with high drop wavelength efficiency is an advantage of the proposed ring resonator configuration because this cannot otherwise be achieved with the low index contrast CWG ring resonator configurations. Note that the FSR can also be made arbitrarily small by simply increasing the size of the ring. The FSR is thus widely adjustable.

Fig. 6. Modified ring resonator geometry in order to increase the FSR.
Fig. 7. Spectral responses for the ring resonator which occupies 35µm×35µm (solid black curve) compared with the base ring resonator structure (dotted red curve).

The Q factor of the ring resonator can be also improved quite easily by increasing the power directed to channel 2 of the splitter (Rs in Eq. (1)) while keeping the overall splitter efficiency high. As shown in Fig. 8(a), this can be achieved by adding another row of Si posts to the splitter (300nm separation in the x direction between the two rows of posts). The output waveguide positions are adjusted to maximize the efficiency at each channel. The efficiencies as a function of wavelength for both channels are shown in Fig. 8(b) for several different designs and compared with those of a single row of Si posts.

Fig. 8. (a) Double Si post splitter structure with 100nm Si post radius and (b) spectral responses of single and double Si posts array splitters as a function of wavelength. The dotted red line corresponds to the single Si posts array splitter and the dashed (green) and solid (black) lines are for the double Si posts array splitters with 95nm and 100 nm Si post radii, respectively.

Efficiencies calculated at the drop port for ring resonators with double Si posts array splitters are shown in Fig. 9 together with those of the base ring resonator structure for comparison. The ring resonator with double layer splitters comprised of 100nm radius Si posts has a Q factor of 1600 and a FSR of 14.1nm. Since the efficiency along channel 1 (Ts in Eq. (1)) is decreased and the efficiencies of hybrid structures for bends and splitters are less than 1, the drop wavelength efficiency is decreased to the 74.8%. An approach to design a ring resonator with high Q factor and drop wavelength efficiency is increasing the overall ring resonator size which is not considered in this paper. Even though the FSR of the ring resonator will be decreased with this approach, the Q factor will be increased while keeping the high drop wavelength efficiency.

Fig. 9. The efficiencies at the drop port of the ring resonators using single Si posts array splitter (dotted red line), double Si posts array splitter with 95nm radius (dashed green line) and 100 nm radius (solid black line).

4. Summary

We have proposed a new ring resonator configuration using low refractive index contrast waveguides which has a relatively wide design freedom to achieve large FSR and Q factor. As examples, we first presented a 14.1nm FSR ring resonator with 92.7% drop wavelength efficiency. Then, we showed modified ring resonators with a 19.8nm FSR ring resonator and 91.0% drop wavelength efficiency, and a 1600 Q factor ring resonator with 14.1nm FSR. Our new ring resonator structure can be applied to a wide variety of waveguide configurations with proper design of the PhC bends and splitters. Such single ring resonators can be used as building blocks for many different functional devices, such as compact add/drop filters, dense wavelength division demultiplexers, all pass filters, and all optical switches.

Acknowledgments

This work was supported in part by DARPA Grant N66001-01-1-8938 and National Science Foundation Grant EPS-0091853.

References and links

1.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-Compact Si-SiO2 Microring Resonator Optical Channel Dropping Filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998). [CrossRef]

2.

A. Vorckel, M. Monster, W. Henschel, P. H. Bolivar, and H. Kurz, “Asymmetrically Coupled Silicon-On-Insulator Microring Resonators for Compact Add-Drop Multiplexers,” IEEE Photon. Technol. Lett. 15, 921–923 (2003). [CrossRef]

3.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002). [CrossRef]

4.

G. Bourdon, G. Alibert, A. Beguin, B. Bellman, and E. Guiot, “Ultralow Loss Ring Resonators Using 3.5% Index-Contrast Ge-Doped Silica Waveguides,” IEEE Photon. Technol. Lett. 15, 709–711 (2003). [CrossRef]

5.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, “An Eight-Channel Add-Drop Filter Using Vertically Coupled Microring Resonators over a Cross Grid,” IEEE Photon. Technol. Lett. 11, 691–693 (1999). [CrossRef]

6.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring Resonator Arrays for VLSI Photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000). [CrossRef]

7.

G. P. Nordin, S. Kim, J. Cai, and J. Jiang, “Hybrid integration of conventional waveguide and photonic crystal structures,” Opt. Express 10, 1334–1341 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1334. [CrossRef] [PubMed]

8.

S. Kim, G. P. Nordin, J. Jiang, and J. Cai, “Micro-genetic algorithm design of hybrid conventional waveguide and photonic crystal structures,” Optical Engineering (to be published).

9.

S. Kim, G. P. Nordin, J. Cai, and J. Jiang, “Ultracompact high-efficiency polarizing beam splitter with a hybrid photonic crystal and conventional waveguide structure,” Opt. Lett. 28, 2384–2386 (2003). [CrossRef] [PubMed]

10.

S. Kim, G. P. Nordin, J. Jiang, and J. Cai, “High Efficiency 90 Degree Silica Waveguide Bend Using an Air Hole Photonic Crystal Region,” IEEE Photon. Technol. Lett. (to be published).

11.

J. Cai, G. P. Nordin, S. Kim, and J. Jiang, “3-D Analysis of hybrid conventional waveguide/photonic crystal 90 degree bend” Appl. Opt. (to be published).

12.

M. Tokushima and H. Yamada, “Light propagation in a photonic-crystal-slab line-defect waveguide,” IEEE J. Quantum Electron. 38, 753–759 (2002). [CrossRef]

13.

S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, R. Houdre, and U. Oesterle., “Improved 60° bend transmission of submicron-width waveguides defined in two-dimensional photonic crystals,” J. Lightwave Technol. 20, 1198–1203 (2002). [CrossRef]

14.

D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. W. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1823 [CrossRef] [PubMed]

15.

A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method” (Artech House, Boston, 1995).

16.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]

17.

B. E. A. Saleh and M. C. Teich, “Fundamentals of Photonics” (A Wiley-Interscience publication, New York, 1991), Chap. 9.

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(130.1750) Integrated optics : Components
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Research Papers

History
Original Manuscript: April 7, 2004
Revised Manuscript: May 10, 2004
Published: May 30, 2004

Citation
Seunghyun Kim, Jingbo Cai, Jianhua Jiang, and Gregory Nordin, "New ring resonator configuration using hybrid photonic crystal and conventional waveguide structures," Opt. Express 12, 2356-2364 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2356


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References

  1. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, �??Ultra-Compact Si-SiO2 Microring Resonator Optical Channel Dropping Filters,�?? IEEE Photon. Technol. Lett. 10, 549-551 (1998). [CrossRef]
  2. A. Vorckel, M. Monster, W. Henschel, P. H. Bolivar, and H. Kurz, �??Asymmetrically Coupled Silicon-On-Insulator Microring Resonators for Compact Add-Drop Multiplexers,�?? IEEE Photon. Technol. Lett. 15, 921-923 (2003). [CrossRef]
  3. P. Rabiei, W. H. Steier, C. Zhang, L. R. Dalton, �??Polymer Micro-Ring Filters and Modulators,�?? J. Lightwave Technol. 20, 1968-1975 (2002). [CrossRef]
  4. G. Bourdon, G. Alibert, A. Beguin, B. Bellman, and E. Guiot, �??Ultralow Loss Ring Resonators Using 3.5% Index-Contrast Ge-Doped Silica Waveguides,�?? IEEE Photon. Technol. Lett. 15, 709-711 (2003). [CrossRef]
  5. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y. Kokubun, �??An Eight-Channel Add-Drop Filter Using Vertically Coupled Microring Resonators over a Cross Grid,�?? IEEE Photon. Technol. Lett. 11, 691- 693 (1999). [CrossRef]
  6. B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, �??Microring Resonator Arrays for VLSI Photonics,�?? IEEE Photon. Technol. Lett. 12, 323-325 (2000). [CrossRef]
  7. G. P. Nordin, S. Kim, J. Cai, and J. Jiang, �??Hybrid integration of conventional waveguide and photonic crystal structures,�?? Opt. Express 10, 1334-1341 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1334.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1334</a> [CrossRef] [PubMed]
  8. S. Kim, G. P. Nordin, J. Jiang, and J. Cai, �??Micro-genetic algorithm design of hybrid conventional waveguide and photonic crystal structures,�?? Optical Engineering (to be published).
  9. S. Kim, G. P. Nordin, J. Cai, and J. Jiang, �??Ultracompact high-efficiency polarizing beam splitter with a hybrid photonic crystal and conventional waveguide structure,�?? Opt. Lett. 28, 2384-2386 (2003). [CrossRef] [PubMed]
  10. S. Kim, G. P. Nordin, J. Jiang, and J. Cai, �??High Efficiency 90 Degree Silica Waveguide Bend Using an Air Hole Photonic Crystal Region,�?? IEEE Photon. Technol. Lett. (to be published).
  11. J. Cai, G. P. Nordin, S. Kim, and J. Jiang, �??3-D Analysis of hybrid conventional waveguide/photonic crystal 90 degree bend�?? Appl. Opt. (to be published).
  12. M. Tokushima and H. Yamada, �??Light propagation in a photonic-crystal-slab line-defect waveguide,�?? IEEE J. Quantum Electron. 38, 753-759 (2002). [CrossRef]
  13. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, R. Houdre, and U. Oesterle., �??Improved 60° bend transmission of submicron-width waveguides defined in two-dimensional photonic crystals,�?? J. Lightwave Technol. 20, 1198-1203 (2002). [CrossRef]
  14. D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. W. Prather, "Analysis of splitters for self-collimated beams in planar photonic crystals," Opt. Express 12, 1823-1831 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1823">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1823</a> [CrossRef] [PubMed]
  15. A. Taflove, �??Computational Electrodynamics: The Finite-Difference Time-Domain Method�?? (Artech House, Boston, 1995).
  16. J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
  17. B. E. A. Saleh and M. C. Teich, �??Fundamentals of Photonics�?? (A Wiley-Interscience publication, New York, 1991), Chap. 9.

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