## Photonic crystal directional coupler switch with small switching length and wide bandwidth

Optics Express, Vol. 14, Issue 3, pp. 1223-1229 (2006)

http://dx.doi.org/10.1364/OE.14.001223

Acrobat PDF (326 KB)

### Abstract

A directional coupler switch structure capable of short switching length and wide bandwidth is proposed. The switching length and bandwidth have a trade-off relationship in conventional directional coupler switches. Dispersion curves that avoid this trade-off are derived, and a two-dimensional photonic crystal structure that achieves these dispersion curves is presented. Numerical calculations show that the switching length of the proposed structure is 7.1% of that for the conventional structure, while the bandwidth is 2.17 times larger.

© 2006 Optical Society of America

## 1. Introduction

1. Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Transaction on Electronics **E87C**, 316–327 (2004).

2. M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. M. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. **81**, 2680–2682 (2002). [CrossRef]

3. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature **407**, 608–610 (2000). [CrossRef] [PubMed]

4. N. Yamamoto, Y. Watanabe, and K. Komori, “Design of photonic crystal directional coupler with high extinction ratio and small coupling length,” Jpn. J. Appl. Phys. **44**2575–2578 (2005). [CrossRef]

5. Y. A. Vlasov, N. Moll, and S. J. McNab, “Mode mixing in asymmetric double-trench photonic crystal waveguides,” J. Appl. Phys. **95**, 4538–4544 (2004). [CrossRef]

6. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, ”Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express **12**, 1090–1096 (2004). [CrossRef] [PubMed]

7. P. I. Borel, A. Harpoth, L. H. Frandsen, and M. Kristensen, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express **12**1996–2001 (2004). [CrossRef] [PubMed]

8. J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D Photonic Crystals of Columns and Lightwave Devices Using the Scattering Matrix Method”, IEEE J. Lightwave Technol. **17**1500–1508 (1999). [CrossRef]

9. S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and Decoupling of Electromagnetic Waves in Parallel 2-D Photonic Crystal Waveguides,” IEEE J. Quantum Electron. **38**47–53 (2002). [CrossRef]

10. A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D Photonic Crystal Directional Couplers,” IEEE Photonics Technl. Lett. **15**694–696 (2003). [CrossRef]

11. Y. Tanaka, H. Nakamura, Y. Sugimito, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling Properties in a 2-D Photonic Crystal Slab Directional Coupler With a Triangular Lattice of Air Holes,” IEEE J. Quantum Electron. **41**76–84 (2005). [CrossRef]

4. N. Yamamoto, Y. Watanabe, and K. Komori, “Design of photonic crystal directional coupler with high extinction ratio and small coupling length,” Jpn. J. Appl. Phys. **44**2575–2578 (2005). [CrossRef]

## 2. Trade-off between switching length and bandwidth

*n*+ 1)

*π*after traveling the coupling length, the light is transfered to the opposite side waveguide. This process is described by

*k*

_{e}and

*k*

_{o}are the wavenumbers of the even and odd modes, respectively, and

*L*

_{c}is the coupling length. For switching operation, it is necessary to change the right-hand side of equation (1) to 2

*mπ*. The directional coupler is separated into two parts, as shown in Fig. 2. In one part, the wavenumbers of the even and odd modes are fixed at

*k*

_{e,fix}and

*k*

_{o,fix}, while in the other part, the wavenumbers change from

*k*

_{e,off}and

*k*

_{o,off}before switching to

*k*

_{e,on}and

*k*

_{o,on}after switching. When these parameters are satisfied, i.e.,

*L*

_{fix}is the length of the fixed parameter region,

*L*

_{sw}is the length of the variable wavenumber region, and

*m*and

*n*are arbitrary integers.

*L*

_{sw}is referred to as the switching length in this paper. From equations (2) and (3), the switching length can be expressed in two separate parts as

*L*

_{c}and the switching length

*L*

_{sw}. In both cases, it is necessary to reduce the length of the device in order to shorten the switching length. As the denominator of the right-hand side of the equation (4) increases, the switching length decreases. The switching operation here is performed by changing the refractive index of the material from

*n*

_{off}to

*n*

_{on}. Expressing the dispersion relation for even and odd modes by

*ω*

_{e}(

*k*,

*n*) and

*ω*

_{o}(

*k*,

*n*), the total differential of dispersion is given by

*dω*= 0, and the differential of wavenumber is obtained as

*ϕ*and the frequency fluctuation Δ

*ω*satisfies the following expression for the “off” switching condition:

*n*+ 1)

*π*,

*k*

_{e,off}and

*k*

_{o,off}with (2

*n*+ 1)

*π*+ Δ

*ϕ*,

*k*

_{e,off}+ Δ

*k*

_{e,off}, and

*k*

_{o,off}+ Δ

*k*

_{o,off}in equation (2), respectively. The relationship

*ω*becomes large when the difference between the group velocities of the even and odd modes is small. The allowable frequency fluctuation can be obtained for the “on” switch condition by a similar equation. The bandwidth for switching is determined as the common frequency region between the band-widths for the on and off switching conditions. It can thus be seen that the condition for wide bandwidth conflicts with the condition of small switching length for this device structure.

## 3. Structure allowing small switching length and wide bandwidth

*k*

_{e,on}-

*k*

_{e,off}in equation (4) cannot approximate by utilizing equation (5) in this case. It means that we should use equation (4) instead of equation (6) to express the switching length, and the small switching length can be realized even if the even and odd modes have same group velocity around the operating frequency. The fact that the both eigenmodes have same group velocity is important to avoid the intermodal dispersion between two eigenmodes which obstructs the short pulse propagation [12

12. F. Cuesta-Soto, B. García-Baños, and J. Martí, “Compensating intermodal dispersion in photonic crystal directional couplers,” Opt. Lett. **30**, 3156–3158 (2005). [CrossRef] [PubMed]

## 4. Numerical Design of a novel directional coupler switch

*a*, where

*a*is the lattice constants, and are arranged in a triangular lattice in a material with refractive index of 2.76 (i.e., a single-mode GaAs/air slab). Here, we consider the GaAs system because we will utilize InAs/GaAs quantum dots as nonlinear effect materials. The energy distributions of the even and odd modes are concentrated near the waveguide region for almost all wavelengths. However, the energy distribution of the even mode (marked by a circle in the figure) spreads into the central region between the waveguides. If the refractive index around the center of the waveguides is reduced, the dispersion of the even mode will be arrested and a fixed-frequency region will be obtained for the even mode.

*k*

_{e,fix}=

*k*

_{e,off}and

*k*

_{o,fix}=

*k*

_{o,off}in equation (2). In the on condition, only the refractive index of the variable-parameter region was changed (+0.1%). The index change will be achieved by the InAs/GaAs quantum dots system [13

13. H. Nakamura, S. Kohmoto, N. Carlsson, Y. Sugimoto, and K. Asakawa, “Large enhancemnet of optical non-linearity using quantum dots embedded in a photonic crystal structure for all-optical switch applications,” in *Proceedings of IEEE/LEOS the 13th Annual Conference* (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 488–489.

*L*

_{sw}and

*L*

_{fix}includes an optional integer (see eqs. (2) and (4)). The plotted switching lengths

*L*

_{sw}represent the minimum value of eq. (4).

*L*

_{fix}, which is necessary to calculate the bandwidth, is the minimum value satisfying eq. (2). This results in steps in the bandwidth curves. From the figure, the switching length of the proposed structure is extremely small in the frequency region 0.2905 – 0.2907. The lower frequency, 0.2905, is the flat frequency of the even mode in the initial state, and the upper frequency, 0.2907, is the flat frequency after the change in refractive index. The switching length therefore becomes small at the operating frequency (between the flat frequencies). The minimum switching length for the proposed structure is 12

*a*, which is 7% of the minimum switching length of the conventional structure. In the figure, the frequency range for novel structure is higher than the range for the conventional structure. It is necessary to shift the frequency range of the dispersion of novel structure keeping the flat dispersion shape for matching the frequency range with the conventional structure. For this purpose, we calculated several modified structure, and we have grasped the tendency of deformation of dispersion curves by changing the hole size and positions. The results will be shown in another paper [14].

*ω*/

*ω*Δ

*ϕ*is shown for each

*ω*in the figure. When the absolute value is large, the bandwidth is large. The bandwidth of the proposed structure becomes quite small at the flat frequencies of the even mode, but remains large between the flat frequencies. A small switching length and wide bandwidth can therefore be obtained simultaneously using this structure by setting the operating frequency between these flat frequencies. The switching lengths and bandwidths for various refractive index changes are listed in table 1. For each condition, the proposed structure has a smaller switching length and wider bandwidth than a conventional directional coupler switch with a simple triangular lattice.

## 5. Conclusion

*a*, which is 7.1% of that for the conventional structure, and the bandwidth is 0.391%, which is 2.17 times larger than that for the conventional structure. This is achieved using a refractive index change for switching of 0.1%. The means for changing the operating frequency will be presented in a subsequent paper [14].

## References and links

1. | Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Transaction on Electronics |

2. | M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. M. Qiu, “Low-threshold photonic crystal laser,” Appl. Phys. Lett. |

3. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

4. | N. Yamamoto, Y. Watanabe, and K. Komori, “Design of photonic crystal directional coupler with high extinction ratio and small coupling length,” Jpn. J. Appl. Phys. |

5. | Y. A. Vlasov, N. Moll, and S. J. McNab, “Mode mixing in asymmetric double-trench photonic crystal waveguides,” J. Appl. Phys. |

6. | Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, ”Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express |

7. | P. I. Borel, A. Harpoth, L. H. Frandsen, and M. Kristensen, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express |

8. | J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D Photonic Crystals of Columns and Lightwave Devices Using the Scattering Matrix Method”, IEEE J. Lightwave Technol. |

9. | S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and Decoupling of Electromagnetic Waves in Parallel 2-D Photonic Crystal Waveguides,” IEEE J. Quantum Electron. |

10. | A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D Photonic Crystal Directional Couplers,” IEEE Photonics Technl. Lett. |

11. | Y. Tanaka, H. Nakamura, Y. Sugimito, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling Properties in a 2-D Photonic Crystal Slab Directional Coupler With a Triangular Lattice of Air Holes,” IEEE J. Quantum Electron. |

12. | F. Cuesta-Soto, B. García-Baños, and J. Martí, “Compensating intermodal dispersion in photonic crystal directional couplers,” Opt. Lett. |

13. | H. Nakamura, S. Kohmoto, N. Carlsson, Y. Sugimoto, and K. Asakawa, “Large enhancemnet of optical non-linearity using quantum dots embedded in a photonic crystal structure for all-optical switch applications,” in |

14. | N. Yamamoto, J. Sugisaka, S. H. Jeong, and K. Komori, |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Optical Devices

**History**

Original Manuscript: November 29, 2005

Revised Manuscript: January 13, 2006

Manuscript Accepted: January 23, 2006

Published: February 6, 2006

**Citation**

Noritsugu Yamamoto, Toru Ogawa, and Kazuhiro Komori, "Photonic crystal directional coupler switch with small switching length and wide bandwidth," Opt. Express **14**, 1223-1229 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1223

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### References

- Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki,Y. Watanabe, K. Ishida, H. Ishikawa and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Transaction on Electronics E87C, 316-327 (2004).
- M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. M. Qiu, “ Low-threshold photonic crystal laser,” Appl. Phys. Lett. 81, 2680-2682 (2002). [CrossRef]
- S. Noda, A. Chutinan and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608-610 (2000). [CrossRef] [PubMed]
- N. Yamamoto, Y. Watanabe and K. Komori, “Design of photonic crystal directional coupler with high extinction ratio and small coupling length,” Jpn. J. Appl. Phys. 44 2575-2578 (2005). [CrossRef]
- Y. A. Vlasov, N. Moll and S. J. McNab, “Mode mixing in asymmetric double-trench photonic crystal waveguides,” J. Appl. Phys. 95, 4538-4544 (2004). [CrossRef]
- Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa and K. Inoue, ”Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express 12, 1090-1096 (2004). [CrossRef] [PubMed]
- P. I. Borel, A. Harpoth, L. H. Frandsen and M. Kristensen, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12 1996-2001 (2004). [CrossRef] [PubMed]
- J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D Photonic Crystals of Columns and Lightwave Devices Using the Scattering Matrix Method”, IEEE J. Lightwave Technol. 17 1500-1508 (1999). [CrossRef]
- S. Boscolo, M. Midrio and C. G. Someda, “Coupling and Decoupling of Electromagnetic Waves in Parallel 2-D Photonic Crystal Waveguides,” IEEE J. Quantum Electron. 38 47-53 (2002). [CrossRef]
- A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D Photonic Crystal Directional Couplers,” IEEE Photonics Technl. Lett. 15 694-696 (2003). [CrossRef]
- Y. Tanaka, H. Nakamura, Y. Sugimito, N. Ikeda, K. Asakawa, and K. Inoue, “Coupling Properties in a 2-D Photonic Crystal Slab Directional Coupler With a Triangular Lattice of Air Holes,” IEEE J. Quantum Electron. 41 76-84 (2005). [CrossRef]
- F. Cuesta-Soto, B. García-Baños, and J. Martí “Compensating intermodal dispersion in photonic crystal directional couplers,” Opt. Lett. 30, 3156-3158 (2005). [CrossRef] [PubMed]
- H. Nakamura, S. Kohmoto, N. Carlsson, Y. Sugimoto, and K. Asakawa, “Large enhancemnet of optical nonlinearity using quantum dots embedded in a photonic crystal structure for all-optical switch applications,” in Proceedings of IEEE/LEOS the 13th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 488-489.
- N. Yamamoto, J. Sugisaka, S. H. Jeong, and K. Komori, unpublished.

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