## Investigation of physical mechanisms in coupling photonic crystal waveguiding structures

Optics Express, Vol. 12, Issue 20, pp. 4781-4789 (2004)

http://dx.doi.org/10.1364/OPEX.12.004781

Acrobat PDF (170 KB)

### 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

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]

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]

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]

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]

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. 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]

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]

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]

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]

**36**, 1376–1378 (2000). [CrossRef]

## 2. Simulations

*d*in Fig. 1). The lattice constant of the photonic crystal is represented by

*a*and the radius of the air holes is

*r*= 0.3

*a*unless otherwise stated. The effective permittivity of the dielectric material used for the core of the slab waveguide and PCW is

*ε*=7.9. Note that the dielectric material is Si with

*ε*=11.4. However, in order to take the effect of the finite out-of-plane thickness of the planar structure (in the z-direction in Fig. 1) into account, we calculated the effective index of

*ε*=7.9 using the method described in Ref. [16]. The polarization of the electromagnetic beams is TM (magnetic field normal to the plane of periodicity, i.e., in z direction in Fig. 1) throughout this paper due to the existence of a large PBG for this polarization. Also for having a single mode PCW we increased the radius of the air holes next to the guiding region from

*r*=0.3

*a*to

*r*’=0.4

*a*as described in Ref. [6

**36**, 1376–1378 (2000). [CrossRef]

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

*c*) is normalized to 1, and all spatial dimensions are in the units of FDTD calculation cells. To calculate the dispersion diagram of the guided modes of the PCW, we analyzed one period of the structure in the guiding direction and used enough layers of PC in the perpendicular direction as shown in Fig. 2(a). The dispersion diagram of the guided modes are then calculated using the order-N spectral technique [19

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]

**18**, 1554–1564 (2000). [CrossRef]

*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.

## 3. Theoretical results

*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]

*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]

*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

**18**, 1554–1564 (2000). [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]

*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.

## 4. Discussion

*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 (k

_{x}=0.0687, k

_{y}=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.

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]

_{0}) 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]

*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.

*a*with uniform probability distribution. The structure is shown in Fig. 7(a) and the variations of the normalized transmitted and reflected power with normalized frequency are shown in Fig. 7(b). It is clear that by changing the properties of only one column of the holes at the boundary, we can increase the reflection coefficient in the mode gap considerably. Note that the spatial Fourier transform of the permittivity in the column of holes next to the boundary has still a strong component at a spatial frequency of 2

*π/a*. Thus, the effect of diffraction still exists. Also, note that the modification of the PC structure at the interface modifies the coupling strength to the odd mode (by changing the symmetry). This explains the change in the strength of the transmission dip around

*a/λ*=0.34 in Fig. 7(b) compared to this in Fig. 3(a). A full optimization of the structure can be performed by considering both diffraction and coupling effects and finding the optimal PC structure at the interface. Using grating-based solution by modifying the periodicity of the PC structure at the boundary might also be considered.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical microchips,” Phys. Rev. Lett. |

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. |

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 |

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. |

7. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B |

8. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

9. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

10. | A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, “Mode dispersion in biperiodic photonic crystal waveguides,” Phys. Rev. B |

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 |

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. |

13. | A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Guiding mechanisms in dielectric-core photonic-crystal waveguides,” Phys. Rev. B |

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. |

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. |

16. | L. A. Coldren and S. W. Corzine, |

17. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. |

18. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

19. | C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic-waves,” Phys. Rev. B |

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 |

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 |

22. | T. K. Gaylord and M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proceedings of the IEEE |

23. | T. Tamir, |

**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

- E. Yablonovitch, �??Inhibited spontaneous emission in solid state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127�??12133 (1998). [CrossRef]
- 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]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princeton Univ. Press, Princeton, 1995).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
- 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).
- J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185�??200 (1994). [CrossRef]
- 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]
- 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]
- 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]
- T. K. Gaylord, and M. G. Moharam, �??Analysis and Applications of Optical Diffraction by Gratings,�?? Proceedings of the IEEE 73, 894-937 (1985). [CrossRef]
- T. Tamir, Guided-wave optoelectronics (Springer-Verlag, New York 1990). [CrossRef]

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