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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 20 — Oct. 4, 2004
  • pp: 4781–4789
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Investigation of physical mechanisms in coupling photonic crystal waveguiding structures

Majid Badieirostami, Babak Momeni, Mohammad Soltani, Ali Adibi, Yong Xu, and Reginald K. Lee  »View Author Affiliations


Optics Express, Vol. 12, Issue 20, pp. 4781-4789 (2004)
http://dx.doi.org/10.1364/OPEX.12.004781


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Abstract

We explain the fundamental physical mechanisms involved in coupling triangular lattice photonic crystal waveguides to conventional dielectric slab waveguides. We show that the two waveguides can be efficiently coupled outside the mode gap frequencies. We especially focus on the coupling of the two structures within the mode gap frequencies and show for the first time that the diffraction from the main photonic crystal structure plays an important role on the reflection of power back into the slab waveguide. The practical importance of this effect and possible strategies to modify it are also discussed.

© 2004 Optical Society of America

1. Introduction

Photonic crystals (PCs) [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

,2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

] are novel optical structures with periodic variation of permittivity in space. Multiple scattering of electromagnetic waves from these periodic structures can result in the existence of a photonic bandgap (PBG), i.e., a range of frequency with no allowed electromagnetic modes. Ultrasmall optical devices such as cavities, waveguides, couplers, and lasers can be formed by adding defects to a perfect PC. Thus, PCs are attractive candidates for developing optical integrated circuits [3

3. A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. 90, 123901(1-4) (2003). [CrossRef] [PubMed]

]. Photonic crystal waveguides (PCWs) [4

4. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

,5

5. S. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

] are essential parts of such integrated structures for providing low-loss large-bandwidth connectivity between different optical devices. The PCs that are suitable for planar integrated optics are in the form of a lattice of circular holes in a dielectric material such as Si or GaAs. A conventional PCW in such structures is formed by adding a line defect in the form of either a line of removed holes [6

6. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with single mode propagation in the photonic bandgap,” Electron. Lett. 36, 1376–1378 (2000). [CrossRef]

] or a linear chain of coupled cavities (i.e., a line of holes with modified radius) [7

7. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998). [CrossRef]

,8

8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

]. The former is more widely used in integrated optics applications and will be used throughout this paper.

In addition to designing optimal single mode PCWs, efficient coupling of light into and out of these PCWs (and in general into all PC structures) is essential for designing practical PC integrated circuits. Using two tapered dielectric ridge waveguides that are coupled to PCWs in the input and output of the structure was proposed [12

12. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the Slab Modes in Photonic Crystal Optical Waveguides,” J. of Lightwave Tech. 18, 1554–1564 (2000). [CrossRef]

,13

13. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal waveguides,” Phys. Rev. B 63, 033308(1-4) (2001). [CrossRef]

,14

14. E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002). [CrossRef]

], and experimentally demonstrated [11

11. A. Jafarpour, E. Chow, C. M. Reinke, J. Huang, A. Adibi, A. Grot, L. W. Mirkarimi, G. Girolami, R. K. Lee, and Y. Xu, “Large-bandwidth ultra-low-loss guiding in bi-periodic photonic crystal waveguides,” App. Phys. B 79, 409–414 (2004). [CrossRef]

,15

15. S. J. Mc. Nab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express. 11, 2927–2939 (2003). [CrossRef]

]. It was shown that due to similarity between the mode patterns of a PCW and a slab (or ridge) waveguide with similar slab thickness, the two waveguides can be efficiently coupled over a large bandwidth [12

12. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the Slab Modes in Photonic Crystal Optical Waveguides,” J. of Lightwave Tech. 18, 1554–1564 (2000). [CrossRef]

]. The PCW used in the first demonstration of this idea had a reduced slab thickness to achieve single mode guiding [12

12. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the Slab Modes in Photonic Crystal Optical Waveguides,” J. of Lightwave Tech. 18, 1554–1564 (2000). [CrossRef]

], which makes it difficult to bend. More recent demonstrations of this coupling rely on conventional PCWs [14

14. E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002). [CrossRef]

]. In this paper, we use the proposed single-mode PCWs discussed in Ref. [6

6. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with single mode propagation in the photonic bandgap,” Electron. Lett. 36, 1376–1378 (2000). [CrossRef]

].

In this paper, we consider both transmission and reflection properties of the coupling between a slab waveguide and the single-mode PCW in the triangular lattice of air holes in Si. We explain all observed features based on fundamental physical mechanisms. The details of our simulations are summarized in Section 2. Transmission and reflection properties are summarized in Section 3. Major features of these coupled structures are explained in Section 4, and conclusions are made in Section 5.

2. Simulations

To calculate the power transmission spectrum from the slab waveguide to the PCW, we used a pulsed Huygens source to excite the fundamental TM mode in the slab waveguide at x=x 0 in Fig. 1. We calculated the spectrum of the power transmitted through the PCW by taking the Fourier transform of the fields and then integrating the Poynting vector over a surface of 60 calculation cells centered at the middle of the PCW at x=x 1 in Fig. 1. The power transmission coefficient is then calculated as the ratio of the transmitted power to the incident power. Also for the calculation of reflection spectrum, we put another surface of calculation cells with the same size centered in the middle of the slab waveguide before the Huygens source at x=x 2 in Fig. 1. The power reflection coefficient is calculated as the ratio of the reflected power to the incident power.

Fig. 1. Coupling of a slab waveguide to a triangular lattice PCW. The PCW is made by removing one row of air holes from a perfect PC. The thicknesses of both waveguides are the same and are equal to d=27 calculation cells. The PC lattice constant is a=24, and the radius of the air holes is r=0.3a. For designing single mode waveguide we increased the radii of the air holes next to the guiding region to r’=0.4a. The permittivity of dielectric used for the core of the slab waveguide and PCW is 7.9. Perfectly matched layers are used around the structure.

3. Theoretical results

Figure 2(a) shows one period of the PCW described in section 2. Magnetic field pattern of the fundamental even mode of this PCW at the normalized frequency a/λ=0.31 is also shown in Fig. 2(a). The dispersion diagram of the first two guided TM modes is shown in Fig. 2(b). It is clear from Fig. 2(b) that the odd mode is out of the PBG, thus, the PCW can be used for single–mode operation in the PBG. Note also that the fundamental even mode does not cover the entire PBG. The dispersion diagram of this mode is flattened in the middle of the PBG (around a/λ=0.295) due to the distributed Bragg reflection (DBR) in the guiding direction of the PCW [20

20. A. Adibi, Y. Xu, R. K. Lee, M. Loncar, A. Yariv, and A. Scherer, “Role of distributed Bragg reflection in photonic-crystal optical waveguides,” Phys. Rev. B 64, 041102(1-4) (2001). [CrossRef]

].

The frequency range of 0.266<a/λ<0.295 inside the PBG for which no guided mode exists is referred to as the mode gap [21

21. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113(1-5) 2001. [CrossRef]

] throughout this paper. While a guided mode with linear dispersion over a large bandwidth is desired for integrated optics applications, the existence (and especially the dynamic modification) of the mode gap is useful for switching applications. Also note that the magnetic field pattern of the fundamental TM even mode of the PCW is similar to that of the same mode in a slab waveguide with similar thickness. Thus, we expect to have good coupling between the two waveguides outside the mode gap.

To investigate the transmission and reflection properties of the coupling structure in Fig. 1, we placed a Huygens source at x=x 0=40 and calculated the power transmission and reflection coefficients at different frequencies (as described in section 2) at x=x 1=450 and x=x 2=20, respectively. The power transmission and reflection spectra are shown in Fig. 3(a). It is clear from Fig. 3(a) that the coupling between the two waveguides is very efficient outside the mode gap (or in the linear dispersion regime of the PCW). The small drop in the transmission around a/λ=0.343 is due to the excitation of the odd mode of the PCW caused by the small asymmetry in the FDTD-simulated structure. The width of this dip is equal to the frequency extent of the odd mode. Note that the odd mode for PCW studied in this paper exists at frequencies outside the PBG and thus, it is lossy. This efficient coupling is similar to what reported before for other types of PCWs and the result of similar properties of the fundamental guided modes of the two waveguides [12

12. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the Slab Modes in Photonic Crystal Optical Waveguides,” J. of Lightwave Tech. 18, 1554–1564 (2000). [CrossRef]

].

Fig. 2. (a) One period of the PCW with the magnetic field pattern of the fundamental even TM mode. (b) Dispersion diagram of the first two TM modes of the PCW in part (a). The odd mode exists outside the PBG and only the even mode remains inside the bandgap. The mode gap corresponds to the frequencies within the PBG with no allowed guided mode.

Despite extensive investigation of the efficient coupling between slab waveguides and PCWs outside the mode gap [21

21. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113(1-5) 2001. [CrossRef]

], the reflection properties of these structures (especially inside the mode gap) have not been investigated in detail. One might expect to get close to 100% reflection of power back into the slab waveguide in the mode gap since the PCW structure does not have any mode in the mode gap. However, the reflection spectrum in Fig. 3(a) shows a different behavior, i.e., only a small portion of the incident power is reflected back into the slab waveguide. This phenomenon is highly undesirable in switching applications resulting in a considerable loss of power.

To investigate the reason for the small reflection coefficient outside the mode gap, we calculated the field pattern in the structure (Fig. 1) when excited by a CW beam at a single normalized frequency (a/λ=0.28) inside the mode gap. The result is shown in Fig. 3(b). It is clear that the beam does not propagate in the PCW, and a large portion of the incident power is diffracted in the air above the dielectric slab and is absorbed by the PMLs. The fundamental physical reason for this behavior is discussed in section 4.

Fig. 3 (a) The transmission and reflection spectra of the power coupled from the slab waveguide to the PCW. A TM Huygens source is placed at x=x 0=40 and the transmitted and reflected powers at different frequencies are calculated at x=x 1=450 and x=x 2=20, respectively. Note that reflection coefficient within the mode gap is less than 100%. (b) The magnetic field profile at the normalized frequency a/λ = 0.28 inside the mode gap. The diffraction of power into the air above and below the dielectric slab is clear.

4. Discussion

The PC structure has no propagating electromagnetic mode inside the mode gap. Thus, only the first few periods of the structure in the guiding direction (i.e., x in Fig. 3(b)) contribute to the reflection and/or diffraction of the incident beam. Since the defect line (or the PCW) has no guided mode in the mode gap, we expect that the PCW structure acts similarly to a perfect PC structure at these frequencies. To investigate this hypothesis, we studied the properties of a slab waveguide coupled to a perfect PC structure shown in Fig. 4. We placed a CW TM Huygens source with normalized frequency (a/λ=0.28) at x=x 0=40 and monitored the magnetic field everywhere above the slab waveguide (in the air region) for both the coupled PCW structure (Fig. 1) and the coupled PC structure (Fig. 4). We then calculated the 2D spatial Fourier transform of these fields to obtain their plane-wave spectra. The results for the PCW structure and the PC structure are shown in Fig. 5(a) and 5(b), respectively. The similarity between the plane-wave spectra of the diffracted beams of the two structures is clear from Fig. 5. Most importantly, the strongest diffracted component occurs at (kx=0.0687, ky=0.0192) for both structures. Note that similar results hold for the air region below the slab waveguide due to the symmetry of the structure. The maximum diffraction components correspond to diffraction angles of ±15.9° with respect to the x-axis. Thus, the results shown in Fig. 5 confirm that the performance of the PCW structure in Fig. 1 inside the mode gap is primarily due to the PC structure and the line defect has a minor role. The diffraction from the periodic PC structure results in the low reflection of power back into the slab waveguide. In order to avoid this diffraction, the general properties of the PC structure at the boundary must be modified.

Fig. 4. A slab waveguide coupled to a perfect PC structure with r=0.3a (and a=24). Different diffracted orders are shown in the figure. Other properties of the structure are similar to those summarized in the captions of Fig. 1.

Since the PC structure in Fig. 4 is periodic, we can use an approximate technique based on grating diffraction to estimate its diffraction properties. In this approximation, the PC structure is considered as a 2D grating. The incident signal from the slab waveguide is expanded as a series of plane-wave components (using 2D spatial Fourier transformation) and the diffraction of each component is analyzed using coupled-wave theory [22

22. T. K. Gaylord and M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proceedings of the IEEE 73, 894–937 (1985). [CrossRef]

]. The individual diffracted components are then added together to obtain the final diffracted beam. For the analysis of the diffracted beam, we assume the region on the left side of the boundary between the PC structure and the slab waveguide to be air. Furthermore, we approximate the incident field of the fundamental magnetic mode (TM0) of the slab waveguide by the superposition of two plane waves with incident angles of ±48.8° with respect to x-axis as shown in Fig. 6(a). We calculated the incident angles by numerically solving the dispersion equation of the slab waveguide for the fundamental TM guided mode [23

23. T. Tamir, Guided-wave optoelectronics (Springer-Verlag, New York1990). [CrossRef]

]. The variation of the relative strength of different diffraction orders with the incident angle is shown in Fig. 6(b).

Fig. 5. The 2D spatial Fourier transform of the magnetic field above the slab waveguide for the case of a slab waveguide coupled to: (a) a PCW as shown in Fig. 1 and (b) a perfect PC structure with r=0.3a every where and no defect as shown in Fig. 4. All properties of the structures in (a) and (b) are the same as those in the caption of Fig. 1 and Fig. 4, respectively.

Figure 6(b) shows that for the approximate incident field from the slab waveguide (incident angle ±48.8°) most of the energy is diffracted into the first diffraction orders. Further we calculated the diffracted angle for these orders to be θ=±16.5° with respect to the x-axis. Considering the approximation involved in this calculation, the resulting diffracted angles agree with the angle calculated using FDTD simulations (i.e., θ=±15.9°). Figure 6 reconfirms that the diffraction from the PC structure plays a major role in coupling a slab waveguide to a PCW at frequencies inside the mode gap. This diffraction can reduce the reflection of power back into the slab waveguide at such frequencies making the design of optical switches based on the modification of the mode gap difficult.

Fig. 6. (a) The slab waveguide with TM0 mode as a superposition of two plane waves with incident angles of θ=±48.8°. (b)The variation of the approximate relative strength of different diffraction orders with the incident angle for the structure in Fig. 4 calculated using the grating theory with approximations described in the text. The dashed line corresponds to the incident angle of 48.8°

Fig. 7. (a) A slab waveguide coupled to a PCW with uniform perturbation of ±0.2a in the center of air holes in the y direction next to the interface. (b) The transmission and reflection spectra of the power coupled from slab waveguide to this perturbed structure with x=x 0=40, x=x 1=450, and x=x 2=20.

5. Conclusion

We presented here the main physical effects that affect the coupling of electromagnetic waves from a slab waveguide to a single-mode photonic crystal waveguide. We showed that the two waveguides can be efficiently coupled in the frequencies outside the mode gap (where the PCW has guided modes). Within the mode gap frequencies, power transmission is low, and the incident power from the slab waveguide is partially reflected back to the waveguide and partially diffracted into the air (or cladding of the slab waveguide) by the periodic PC structure. The diffraction effect results in power loss in switching applications of PCWs and can be reduced (and eventually eliminated) by proper design of the PC structure at the interface.

Acknowledgments

This work was supported by Air Force Office of Scientific Research (G. Pomrenke) and by National Science Foundation (F. Bartolli).

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

3.

A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. 90, 123901(1-4) (2003). [CrossRef] [PubMed]

4.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef] [PubMed]

5.

S. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

6.

A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with single mode propagation in the photonic bandgap,” Electron. Lett. 36, 1376–1378 (2000). [CrossRef]

7.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998). [CrossRef]

8.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

9.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princeton Univ. Press, Princeton, 1995).

10.

A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, “Mode dispersion in biperiodic photonic crystal waveguides,” Phys. Rev. B 68, 233102–233105 (2003). [CrossRef]

11.

A. Jafarpour, E. Chow, C. M. Reinke, J. Huang, A. Adibi, A. Grot, L. W. Mirkarimi, G. Girolami, R. K. Lee, and Y. Xu, “Large-bandwidth ultra-low-loss guiding in bi-periodic photonic crystal waveguides,” App. Phys. B 79, 409–414 (2004). [CrossRef]

12.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the Slab Modes in Photonic Crystal Optical Waveguides,” J. of Lightwave Tech. 18, 1554–1564 (2000). [CrossRef]

13.

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal waveguides,” Phys. Rev. B 63, 033308(1-4) (2001). [CrossRef]

14.

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729–3731 (2002). [CrossRef]

15.

S. J. Mc. Nab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express. 11, 2927–2939 (2003). [CrossRef]

16.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

17.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. AP-14, 302–307 (1966).

18.

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

19.

C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic-waves,” Phys. Rev. B 51, 16635–16642 (1995). [CrossRef]

20.

A. Adibi, Y. Xu, R. K. Lee, M. Loncar, A. Yariv, and A. Scherer, “Role of distributed Bragg reflection in photonic-crystal optical waveguides,” Phys. Rev. B 64, 041102(1-4) (2001). [CrossRef]

21.

M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113(1-5) 2001. [CrossRef]

22.

T. K. Gaylord and M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proceedings of the IEEE 73, 894–937 (1985). [CrossRef]

23.

T. Tamir, Guided-wave optoelectronics (Springer-Verlag, New York1990). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(230.7370) Optical devices : Waveguides
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Research Papers

History
Original Manuscript: August 24, 2004
Revised Manuscript: September 20, 2004
Published: October 4, 2004

Citation
Majid Badieirostami, Babak Momeni, Mohammad Soltani, Ali Adibi, Yong Xu, and Reginald Lee, "Investigation of physical mechanisms in coupling photonic crystal waveguiding structures," Opt. Express 12, 4781-4789 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4781


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References

  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  2. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  3. A. Chutinan, S. John, and O. Toader, �??Diffractionless flow of light in all-optical microchips,�?? Phys. Rev. Lett. 90, 123901(1-4) (2003). [CrossRef] [PubMed]
  4. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, �??High transmission through sharp bends in photonic crystal waveguides,�?? Phys. Rev. Lett. 77, 3787�??3790 (1996). [CrossRef] [PubMed]
  5. S. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, �??Experimental demonstration of guiding and bending electromagnetic waves in a photonic crystal,�?? Science 282, 274�??276 (1998). [CrossRef] [PubMed]
  6. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, �??Design of photonic crystal optical waveguides with single mode propagation in the photonic bandgap,�?? Electron. Lett. 36, 1376-1378 (2000). [CrossRef]
  7. N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127�??12133 (1998). [CrossRef]
  8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711-713 (1999). [CrossRef]
  9. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princeton Univ. Press, Princeton, 1995).
  10. A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, �??Mode dispersion in biperiodic photonic crystal waveguides,�?? Phys. Rev. B 68, 233102-233105 (2003). [CrossRef]
  11. A. Jafarpour, E. Chow, C. M. Reinke, J. Huang, A. Adibi, A. Grot, L. W. Mirkarimi, G. Girolami, R. K. Lee, and Y. Xu, �??Large-bandwidth ultra-low-loss guiding in bi-periodic photonic crystal waveguides,�?? App. Phys. B 79, 409-414 (2004). [CrossRef]
  12. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, �??Properties of the Slab Modes in Photonic Crystal Optical Waveguides,�?? J. of Lightwave Tech.18, 1554-1564 (2000). [CrossRef]
  13. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, �??Guiding mechanisms in dielectric-core photonic-crystal waveguides,�?? Phys. Rev. B 63, 033308(1-4) (2001). [CrossRef]
  14. E. Miyai, M. Okano, M. Mochizuki, and S. Noda, �??Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,�?? Appl. Phys. Lett. 81, 3729-3731 (2002). [CrossRef]
  15. S. J. Mc. Nab, N. Moll, and Y. A. Vlasov, �??Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,�?? Opt. Express.11, 2927-2939 (2003). [CrossRef]
  16. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  17. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. AP-14, 302�??307 (1966).
  18. J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185�??200 (1994). [CrossRef]
  19. C. T. Chan, Q. L. Yu, and K. M. Ho, �??Order-N spectral method for electromagnetic-waves,�?? Phys. Rev. B 51, 16 635�??16642 (1995). [CrossRef]
  20. A. Adibi, Y. Xu, R. K. Lee, M. Loncar, A. Yariv, and A. Scherer, �??Role of distributed Bragg reflection in photonic-crystal optical waveguides,�?? Phys. Rev. B 64, 041102(1-4) (2001). [CrossRef]
  21. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, �??Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,�?? Phys. Rev. B 64, 155113(1-5) 2001. [CrossRef]
  22. T. K. Gaylord, and M. G. Moharam, �??Analysis and Applications of Optical Diffraction by Gratings,�?? Proceedings of the IEEE 73, 894-937 (1985). [CrossRef]
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