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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 17 — Aug. 23, 2004
  • pp: 3934–3939
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Propagation loss reduction of photonic crystal slab waveguides by microspheres

Chii-Chang Chen, Ya-Lun Tsai, Che-Lung Hsu, and Jenq-Yang Chang  »View Author Affiliations


Optics Express, Vol. 12, Issue 17, pp. 3934-3939 (2004)
http://dx.doi.org/10.1364/OPEX.12.003934


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Abstract

Dielectric microspheres are theoretically studied to reduce the propagation loss of Si-based photonic crystal slab waveguides. Two-dimensional photonic crystal formed by etched air hole can act as a template for microsphere sedimentation. The analytical results show that the transmission of the photonic crystal slab waveguides with microspheres can be enhanced to be around twice that without microspheres.

© 2004 Optical Society of America

1. Introduction

Waveguides consisting of a linear row (rows) of defect in a photonic crystal have been intensively studied for realizing ultracompact photonic circuits. Slab waveguides are typical structures used to obtain the lateral and vertical light confinements by photonic crystal (PC) structures and by index guiding, respectively. The transmission of the PC slab waveguides (PCSWG) can be enhanced by modifying the lateral two-dimensional (2-D) PC structures. A. Chutinan et al. proposed that additional air-holes in PCSWGs can enlarge the high-transmission bandwidth by making the waveguides single moded at the bends. [1

1. A Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends,in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002). [CrossRef]

] S. J. McNab et al. reported a novel structure to embed the slab in polymer to reduce the fiber/waveguide coupling loss and a novel structure for decreasing the slab/PC waveguide coupling loss. [2

2. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit withmembrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927 [CrossRef] [PubMed]

] Moreover, A. Talneau et al. indicated that the transmission of bent waveguides can be more efficient for the single-missing-row defect supporting a lower number of modes. However, the center frequency of the mode in the single-missing-row defect is too close to the lower photonic band gap. Any slight process variation may shift the guiding mode into the allowed band. [3

3. A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 µm,” Appl. Phys. Lett. 80, 547–549 (2002). [CrossRef]

] Several missing rows may push the guided-mode frequency away from the lower band edge. [4

4. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos,” Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

] This behavior might form a multimode waveguide that produces larger in-plane loss. However, M. Augustin et al. demonstrate experimentally that the several-missing-row defect of PCSWGs in Nb2O5 can operate in the single mode. An extremely low propagation loss of 1.7dB/mm has been reported. [5

5. M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, A. Tunnermann, R. Iliew, C. Etrich, U. Peschel, and F. Lederer, “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett. 84, 663–665 (2004). [CrossRef]

] This work alleviates not only the problem of the in-plane loss of PCSWG, but the highly visible transparency of Nb2O5 shows the potential for use in the visible range.

For the vertical confinement, the light propagation characteristics of PCSWGs with different cladding layers have also been studied in the literature. [6

6. B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am B 15, 1155–1159 (1998). [CrossRef]

7

7. T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, “Light Propagation Characteristics of Straight Single-Line-Defect Waveguides in Photonic Crystal Slabs Fabricated Into a Silicon-on-Insulator Substrate,” IEEE J. Quantum Electron. 38, 743–752 (2002). [CrossRef]

] Air-bridge type (air cladding) PCSWGs [8

8. N. Kawai, K. Inoue, N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, and T. Takemori, “Confined Band Gap in an Air-Bridge Type of Two-Dimensional AlGaAs Photonic Crystal,” Phys. Rev. Lett. 86, 2289–2292 (2001). [CrossRef] [PubMed]

] provide very strong vertical mode confinement and can reduce many of the losses associated with leaky waveguides and scattering from the bottom of the etched holes. [6

6. B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am B 15, 1155–1159 (1998). [CrossRef]

] The study of W. Kuang et al. also indicates that air-bridge type PCSWGs are capable of lower-loss transmission over a limited range of frequencies. [9

9. W. Kuang, C. Kim, A. Stapleton, W. J. Kim, and J. D. O’Brien, “Calculated out-of-plane transmission loss for photonic-crystal slab waveguides,” Opt. Lett. 28, 1781–1783 (2003). [CrossRef] [PubMed]

] Experimentally, the propagation characteristics can be directly observed from the top of the PCSWGs [10

10. S. I. Bozhevolnyi, V. S. Volkov, J. Arentoft, A. Boltasseva, T. Sondergaard, and M. Kristensen, “Direct mapping of light propagation in photonic crystal waveguides,” Opt. Commun. 212, 51–55 (2002). [CrossRef]

13

13. M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000). [CrossRef]

]. Although, in Ref. 13, the out-of-plane loss of the air-bridge type PCSWG can almost disappear when the fiber-to-waveguide coupling is well manipulated, the simulation results in Ref. 6 reveal that the slight in-plane and out-of-plane losses will still take place at the edge of the missing row defect.

PCSWGs have also been developed for polarization filters [14

14. E. Chow, S.Y. Lin, S.G. Johnson, P.R. Villeneuve, J.D. Joannopoulos, J.R. Wendt, G.A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000). [CrossRef] [PubMed]

] or beamsplitter [15

15. 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]

] using the self-collimation phenomenon. The horizontal confinement of these waveguides is achieved by index guiding and self-guiding, respectively. The vertical confinement is obtained by index guiding for both cases. This study proposes a novel method for reducing the out-of-plane loss of the PCSWGs by stacking microspheres on the PCSWGs. A PCSWG which is similar to the polarization filter of Ref. 14 is studied, as shown in Fig. 1(a). In the photonic bandgap, the light in the TE mode can be blocked by the PC structure formed by the etched holes and the light in the TM mode can pass through the PC structure. However, the etched holes also cause the scattering loss of the light. To date, the sedimentation of microspheres is the easiest artificial method for forming the three-dimensional (3-D) PC structure which is generally face-center cubic (FCC). The templating of the initial surface on which the opal is formed has been a promising method for obtaining a well stacked structure. [16

16. F. Bresson, C. C. Chen, G. C. Chi, and Y. W. Chen, “Inclusion of defects in opal-like photonic crystals layers with a stop-band in the visible range,” Appl. Surf. Science 17, 281–288 (2003). [CrossRef]

17

17. V. Mizeikis, S. Juodkazis, A. Marcinkevicius, S. Matsuo, and H. Misawa, “Tailoring and characterization of photonic crystals,” J. of Photochemistry and Photobiology C 2, 35–69 (2001) [CrossRef]

] The etched air holes of the PCSWGs can form a template on which colloidal particles were sedimented to reduce the scattering loss of the light propagating in the PC structure.

Fig. 1. (a) Schematic drawing of PCSWGs (b) Effective index of the slab waveguide for different thickeness of the slab. (c) Fundamental mode of the slab waveguide.

2. Simulation procedure

For the simulation of the propagation loss, a PCSWG is assumed to be formed on silicon-on-insulator. The refractive indexes of Si and SiO2 are 3.46 and 1.46, respectively. The effective index of the slab waveguide is calculated by effective index method [17

17. V. Mizeikis, S. Juodkazis, A. Marcinkevicius, S. Matsuo, and H. Misawa, “Tailoring and characterization of photonic crystals,” J. of Photochemistry and Photobiology C 2, 35–69 (2001) [CrossRef]

] as shown in Fig. 1 (b). The waveguide is monomode for the TE-mode as the thickness of Si is up to 0.25µm. Therefore, the thicknesses of the silicon and the SiO2 bottom cladding are designed to be 0.2 and 0.5µm, respectively as shown in Fig. 1(a). The corresponding effective index of the slab waveguide is 2.74. Furthermore, the width of the waveguide is 5µm. The optical field of the fundamental mode of the slab waveguide without PC air holes is first calculated by the beam propagation method (BPM) [18

18. K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” (John Wiley & Sons Inc, New York, 2001), for the effective index method: pp. 20–35; for BPM: pp. 165–232; for FDTD: pp. 233–250.

] as shown in Fig. 1(c). 2-D hexagonally arranged air holes then are formed on the silicon layer as shown in Fig. 1(a). The period and the diameter of the air-holes are 400 and 240 nm, respectively. Moreover, the numbers of layers for the Γ-K and Γ-M directions are 11 and 9, respectively. The 2-D plane wave expansion method is used to calculate the PC band structure using the effective index. The simulation result shows that the normalized frequency of the bandgap for the TE mode is between 0.26 and 0.32. The TM mode has no bandgap. Therefore, in the bandgap of the TE mode, the device can block the light of the TE mode, and the light in the TM mode can pass through the PC structure. Since the period of the air-holes is 400nm, the corresponding wavelength of the bandgap ranges from 1.25 to 1.53µm. The microspheres with diameter of 400nm are stacked on the top of the PCSWG. Three layers of microspheres are assumed to be stacked in FCC structure on the PCSWG. Figure 2 shows the cross section of the PCSWG with the stacked microspheres at the X-Y plane. The air-holes serve as the template for the stacked microspheres, i.e. the Γ-K direction of the hexagonally arranged air-holes is aligned along the Γ-K direction of the microspheres in FCC structure. The 3-D finite-difference time-domain (FDTD) method [18

18. K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” (John Wiley & Sons Inc, New York, 2001), for the effective index method: pp. 20–35; for BPM: pp. 165–232; for FDTD: pp. 233–250.

] is performed to simulate the transmission spectra for the microspheres with the refractive indexes ranging from 1.3 to 1.59. With the optical field of the fundamental mode of the slab calculated by BPM, pulsed light sources in transverse electric (TE) and transverse magnetic (TM) mode are launched into the PCSWG, respectively. The waveguide lies on the X-Z plane, and the light propagates along the Z direction. The computational domain for X, Y and Z axes is as large as 100×225×220 cells for the PCSWGs with and without microspheres, corresponding to 8, 20 and 8 points per lattice constant a(=400nm). After 213 time steps, the transmission spectra can be obtained by Fourier transform of the pulse. The refractive indexes of the microspheres used in the simulation are 1.3, 1.4(PMMA), 1.46(silica), 1.5 and 1.59(polystyrene).

Fig. 2. Cross section of PCSWG with microspheres at X-Y plane. The etched holes act as a template for sedimentation of microspheres.

3. Results and discussion

Figures 3(a) and (b) show the transmission spectra of the PCSWGs with and without microspheres for the TE and TM modes, respectively. The bandgap of the PCSWG for the TE mode can be observed from 1.2 to 1.55µm. This range corresponds to the bandgap calculated by the effective index method and the plane wave expansion method. After stacking the microspheres with different refractive indexes on the PCSWG, the bandgaps in the transmission spectra do not shift significantly. This behavior shows that the microspheres do not affect the filtering properties of the photonic crystal structure. For the TM mode, the transmission spectra of the PCSWG with the microspheres are higher than that without the microspheres for wavelength from 1.3 to 1.7µm. The enhancement of the transmission improves with increasing the refractive index of the microspheres. The transmission ratio between the case without microsphere and with the microspheres of refractive index of 1.59 is shown in Fig. 2(b). The maximum transmission for the polystyrene microspheres is 2.25 times higher than that without microspheres at the wavelength of 1.45µm.

Fig. 3. Transmission spectra of PCSWG with microspheres of different refractive indexes (a) for the TE mode and (b) for the TM mode.

The simulation above studies the transmission spectra of the waveguides. To realize the reason of the loss reduction using the microsphere deposition, a continuous wave in the TM mode at the wavelength of 1.45µm is launched into the PCSWG without and with a single layer of polystyrene microspheres. Figure 4 shows the distribution of the electromagnetic field at the Y-Z plane. In zone I, the field is more intense for the case without microsphere. This behavior reveals that the reflection by the PC zone in the PCSWG without microsphere is severer and decreases the light transmission. In zone II, the light can be observed to be scattered into the silica bottom cladding. As the microspheres are stacked on the PCSWG, the vertical mode profile in the PC zone appears to become more vertically symmetrical and the scattering loss into the silica layer reduces significantly. In zone III, the transmission can be observed to be higher for the case with microspheres. Therefore, the enhancement of the transmission by stacking microspheres on the PCSWG originates from the reduction of the reflection by the PC zone and vertically symmetrizing the mode profile in the PC zone. The mode symmetrizing, one of the two reasons to enhance the transmission, may also be achieved by adding a silica top cladding layer on the PC zone. However, to avoid to filling the etched holed in the PC zone, this silica layer can not be deposited by the conventional chemical vapor deposition methods or electron gun deposition method. The wafer bonding technique to add a silica layer on the PC zone may be an alternative method. However, since the local bonding of the silica layer just on the PC zone is hard to realize and the wafer bonding should be performed for the whole photonic circuit, the silica layer added by wafer bonding may change the optical properties of the whole photonic circuit. Moreover, the microspheres can be deposited on the specific area by micropipette. Therefore, the microsphere deposition may be a better approach to reduce locally the loss in the photonic circuit without changing the properties of the whole photonic circuit.

Fig. 4. Propagation of light in the TM mode. The loss in the SiO2 layer is reduced by deposition of microspheres.

4. Conclusion

This work presents a novel method for reducing the propagation loss of the PCSWG by stacking the microspheres. The etched holes of the PCSWG can provide a template for assembling the microspheres in an FCC structure. The simulation results show that the bandgap of the PC structure in the TE mode is not significantly influenced by the microspheres and the transmission of the light in the TM mode can be largely enhanced. The enhancement increases with the refractive index of the microspheres. The reduction of the scattering loss by stacking microspheres on the PCSWG results from vertically symmetrizing the mode profile. Recently, the PCSWGs using self-collimating phenomenon have attracted increased attention owing to its potential to realize the nanophotonic circuit. The scattering loss of the waveguide can also be reduced by using the deposition of the microspheres.

References and links

1.

A Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends,in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002). [CrossRef]

2.

S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit withmembrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927 [CrossRef] [PubMed]

3.

A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 µm,” Appl. Phys. Lett. 80, 547–549 (2002). [CrossRef]

4.

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos,” Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

5.

M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, A. Tunnermann, R. Iliew, C. Etrich, U. Peschel, and F. Lederer, “High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,” Appl. Phys. Lett. 84, 663–665 (2004). [CrossRef]

6.

B. D’Urso, O. Painter, J. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am B 15, 1155–1159 (1998). [CrossRef]

7.

T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, “Light Propagation Characteristics of Straight Single-Line-Defect Waveguides in Photonic Crystal Slabs Fabricated Into a Silicon-on-Insulator Substrate,” IEEE J. Quantum Electron. 38, 743–752 (2002). [CrossRef]

8.

N. Kawai, K. Inoue, N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, and T. Takemori, “Confined Band Gap in an Air-Bridge Type of Two-Dimensional AlGaAs Photonic Crystal,” Phys. Rev. Lett. 86, 2289–2292 (2001). [CrossRef] [PubMed]

9.

W. Kuang, C. Kim, A. Stapleton, W. J. Kim, and J. D. O’Brien, “Calculated out-of-plane transmission loss for photonic-crystal slab waveguides,” Opt. Lett. 28, 1781–1783 (2003). [CrossRef] [PubMed]

10.

S. I. Bozhevolnyi, V. S. Volkov, J. Arentoft, A. Boltasseva, T. Sondergaard, and M. Kristensen, “Direct mapping of light propagation in photonic crystal waveguides,” Opt. Commun. 212, 51–55 (2002). [CrossRef]

11.

S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, K. Asakawa, N. Kawai, and K. Inoue, “Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,” J. Appl. Phys. 89, 855–858 (2001). [CrossRef]

12.

T. Baba, N. Fukaya, and J. Yonekura, “Observation of light propagation in photonic crystal waveguides with bends,” Electron. Lett. 35, 654–655 (1999). [CrossRef]

13.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000). [CrossRef]

14.

E. Chow, S.Y. Lin, S.G. Johnson, P.R. Villeneuve, J.D. Joannopoulos, J.R. Wendt, G.A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000). [CrossRef] [PubMed]

15.

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]

16.

F. Bresson, C. C. Chen, G. C. Chi, and Y. W. Chen, “Inclusion of defects in opal-like photonic crystals layers with a stop-band in the visible range,” Appl. Surf. Science 17, 281–288 (2003). [CrossRef]

17.

V. Mizeikis, S. Juodkazis, A. Marcinkevicius, S. Matsuo, and H. Misawa, “Tailoring and characterization of photonic crystals,” J. of Photochemistry and Photobiology C 2, 35–69 (2001) [CrossRef]

18.

K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” (John Wiley & Sons Inc, New York, 2001), for the effective index method: pp. 20–35; for BPM: pp. 165–232; for FDTD: pp. 233–250.

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.7400) Optical devices : Waveguides, slab
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Research Papers

History
Original Manuscript: June 14, 2004
Revised Manuscript: August 2, 2004
Published: August 23, 2004

Citation
Chii-Chang Chen, Ya-Lun Tsai, Che-Lung Hsu, and Jenq-Yang Chang, "Propagation loss reduction of photonic crystal slab waveguides by microspheres," Opt. Express 12, 3934-3939 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-17-3934


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References

  1. A Chutinan, M. Okano, and S. Noda, �??Wider bandwidth with high transmission through waveguide bends, in two-dimensional photonic crystal slabs,�?? Appl. Phys. Lett. 80, 1698-1700 (2002). [CrossRef]
  2. S. J. McNab, N. Moll, Y. A. Vlasov, �??Ultra-low loss photonic integrated circuit withmembrane-type photonic crystal waveguides,�?? Opt. Express 11, 2927-2939 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927</a> [CrossRef] [PubMed]
  3. A. Talneau, L. Le Gouezigou and N. Bouadma, M. Kafesaki and C. M. Soukoulis, M. Agio, �??Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 µm,�?? Appl. Phys. Lett. 80, 547-549 (2002). [CrossRef]
  4. S. Y. Lin, E. Chow, S. G. Johnson, J. D. Joannopoulos,�?? Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,�?? Opt. Lett. 25, 1297-1299 (2000). [CrossRef]
  5. M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, A. Tunnermann, R. Iliew, C. Etrich, U. Peschel, F. Lederer, �?? High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,�?? Appl. Phys. Lett. 84, 663-665 (2004). [CrossRef]
  6. B. D�??Urso, O. Painter, J. O�??Brien, T. Tombrello, A. Yariv, A. Scherer, �??Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,�?? J. Opt. Soc. Am B 15, 1155-1159 (1998). [CrossRef]
  7. T. Baba, A.Motegi, T. Iwai, N. Fukaya, Y. Watanabe, A. Sakai, �??Light Propagation Characteristics of Straight Single-Line-Defect Waveguides in Photonic Crystal Slabs Fabricated Into a Silicon-on-Insulator Substrate,�?? IEEE J. Quantum Electron. 38, 743-752 (2002). [CrossRef]
  8. N. Kawai and K. Inoue, N. Carlsson, N. Ikeda, Y. Sugimoto, and K. Asakawa, T. Takemori, �??Confined Band Gap in an Air-Bridge Type of Two-Dimensional AlGaAs Photonic Crystal,�?? Phys. Rev. Lett. 86, 2289-2292 (2001). [CrossRef] [PubMed]
  9. W. Kuang, C. Kim, A. Stapleton, W. J. Kim, J. D. O�??Brien, �??Calculated out-of-plane transmission loss for photonic-crystal slab waveguides,�?? Opt. Lett. 28, 1781-1783 (2003). [CrossRef] [PubMed]
  10. S. I.Bozhevolnyi,V. S.Volkov,J. Arentoft,A. Boltasseva,T. Sondergaard,M. Kristensen,�?? Direct mapping of light propagation in photonic crystal waveguides,�?? Opt. Commun. 212, 51-55 (2002). [CrossRef]
  11. S. Yamada, T. Koyama, Y. Katayama, N. Ikeda, Y. Sugimoto, and K. Asakawa, N. Kawai and K. Inoue, �??Observation of light propagation in two-dimensional photonic crystal-based bent optical waveguides,�?? J. Appl. Phys. 89, 855-858 (2001). [CrossRef]
  12. T. Baba, N. Fukaya, J. Yonekura, �??Observation of light propagation in photonic crystal waveguides with bends,�?? Electron. Lett. 35, 654-655 (1999). [CrossRef]
  13. M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, T. P. Pearsall, �??Waveguiding in planar photonic crystals,�?? Appl. Phys. Lett. 77, 1937-1939 (2000). [CrossRef]
  14. E. Chow, S.Y. Lin, S.G. Johnson, P.R. Villeneuve, J.D. Joannopoulos, J.R. Wendt, G.A. Vawter, W. Zubrzycki, H. Hou, A. Alleman, �??Three-dimensional control of light in a two-dimensional photonic crystal slab,�?? Nature 407, 983-986 (2000). [CrossRef] [PubMed]
  15. D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, 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]
  16. F. Bresson, C. C. Chen, G. C. Chi, Y. W. Chen, �??Inclusion of defects in opal-like photonic crystals layers with a stop-band in the visible range,�?? Appl. Surf. Science 17, 281-288 (2003). [CrossRef]
  17. V. Mizeikis, S. Juodkazis, A. Marcinkevicius, S. Matsuo, H. Misawa, �??Tailoring and characterization of photonic crystals,�?? J. of Photochemistry and Photobiology C 2, 35�??69 (2001) [CrossRef]
  18. K. Kawano, T. Kitoh, �??Introduction to optical waveguide analysis,�?? (John Wiley & Sons Inc, New York, 2001), for the effective index method: pp. 20-35; for BPM: pp. 165-232; for FDTD: pp. 233-250.

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