## Realistic photonic bandgap structures for TM-polarized light for all-optical switching

Optics Express, Vol. 14, Issue 26, pp. 12794-12802 (2006)

http://dx.doi.org/10.1364/OE.14.012794

Acrobat PDF (639 KB)

### Abstract

We investigate manufacturable photonic crystal (PhC) structures with a large photonic bandgap for TM-polarized light. Although such PhC structures have been the object of only a limited number of studies to date, they are of central importance for ultra fast all-optical switches relying on intersubband transitions in AlAsSb/InGaAs quantum wells, which support only TM polarization. In this paper, we numerically study substrate-type PhCs for which the two-dimensional approximation holds and three-dimensional photonic-crystal slabs, both with honeycomb lattice geometry. Large TM PBGs are obtained and optimized for both cases. Two types of PhC waveguides are proposed which are able to guide TM modes. Their unique properties show the potential to apply as waveguiding structures in all-optical switches.

© 2006 Optical Society of America

## 1. Introduction

1. S. Kawanishi, “Ultrahigh-speed optical time-division-multiplexed transmission technology based on optical signal processing,” IEEE J. Quantum Electron. **34**, 2604 (1998). [CrossRef]

^{2}) are required. Switches relying on intersubband transitions (ISBT) within the conduction band exhibit ultrafast relaxation time and large optical nonlinearities. Transitions in the telecommunication wavelength range are achievable, e.g., in InGaAs/AlAsSb quantum wells (QWs) exhibiting a large conduction band discontinuity of 1.6 eV [3

3. P. Cristea, Y. Fedoryshyn, and H. Jäckel, “Growth of AlAsSb/InGaAs MBE-layers for all optical switches,” J. Crystal Growth. **278**, 544–547 (2005). [CrossRef]

4. H. Yoshida, T. Mozume, A. Neogi, and O. Wada, “Ultrafast all-optcal switching at 1.3µm/1.55µm using novel InGaAs/AlAsSb/InP coupled double quantum well structure for intersubband transitions,” Electron. Lett. **35**, 1103 (1999). [CrossRef]

5. A V Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. **85**, 4866–4868 (2004). [CrossRef]

6. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. **87**, 151112 (2005). [CrossRef]

8. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**, 5751 (1999). [CrossRef]

9. W C L. Hopman, R M de Ridder, C. G. Bostan, S. Selvaraja, V. J. Gadgil, L. Kuipers, and A. Driessen, “Design and Fabrication of 2-Dimensional Silicon Photonic Crystal Membranes by Focused Ion Beam Processing,” presented at the ePiXnet winterschool on Optoelectronic Integration: Technology and Applications, ePiXnet Winter School, Pontresina, Switzerland, 13–17 Mar. 2006.

## 2. Photonic crystals with the honeycomb lattice geometry

### 2.1 Two-dimensional (2D) simulations of substrate-type PhCs

11. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B **46**, 4969 (1992). [CrossRef]

12. D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B **53**, 7134 (1996). [CrossRef]

13. S. Rowson, A. Chelnokov, J. M. Lourtioz, and F. Carcenac, “Reflection and transmission characterization of a hexagonal photonic crystal in the mid infrared,” J. Appl. Phys. **83**, 5061–5064 (1998). [CrossRef]

14. J. Ye, V. Mizeikis, Y. Xu, S. Matsuo, and H. Misawa, “Fabrication and optical characteristics of silicon-based two-dimensional photonic crystals with honeycomb lattice,” Opt. Commun. **211**, 205–213 (2002). [CrossRef]

4. H. Yoshida, T. Mozume, A. Neogi, and O. Wada, “Ultrafast all-optcal switching at 1.3µm/1.55µm using novel InGaAs/AlAsSb/InP coupled double quantum well structure for intersubband transitions,” Electron. Lett. **35**, 1103 (1999). [CrossRef]

15. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**173–190 (2001). [CrossRef] [PubMed]

**a**for triangular and square lattices, where

**a**is the lattice constant. For the honeycomb lattice geometry, attention should be paid to avoid an overlapping of nearest neighboring elements (the close-packed configuration). The holes-radii upper limit is 0.5

**a**/(3

^{1/2})≈0.288

**a**, which corresponds to a maximum air-filling factor of approximately 60%, compared to over 90% for the close-packed configuration in the triangular lattice. A relatively low filling factor nevertheless yielding a large PBG shows twofold benefits: (

*i*) it gives a large volume of high dielectric material, thus increasing the light-matter interactions and (

*ii*) decreases the propagation losses which are roughly proportional to the filling factor [16]. The honeycomb PhCs with manufacturable hole radii of r=0.24

**a**exhibits the largest TM PBG. The maximum mid-gap ratio is 11.1% and 8% for the first and second gaps, respectively, with center frequencies of 0.231 and 0.423 in normalized frequency (ω

**a**/2

*π*c), as shown in Fig. 2(a). This large TM PBG can be explained by the circular Brillouin zone of the honeycomb geometry. Indeed, we found a circular confinement (shape of the holes) to be beneficial for a large TM PBG, because it matches the shape of Brillouin zone [17]. Additionally, the remaining materials around the air holes isolate the high dielectric regions more efficiently than the square and triangular lattices.

*et.al.*, [18

18. C. Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B **81**, 235–244 (2005). [CrossRef]

### 2.2 Three-dimensional (3D) simulations for air-bridge type PhC slabs

19. Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. **91**, 922–929 (2002). [CrossRef]

**a**. For guided modes in the communication wavelength range, a slab thickness less than 340nm is required to maintain a single-mode operation with an approximate weighed average refractive index of air and dielectric material of the core layer [20]. The modes above the light-cone are not considered because unlike the Bloch guided modes, they should exponentially decay into the cladding layers. Therefore, although radiation modes are available within the frequency range of the PBG, no mode can be guided in the plane of the slab above the light cone. Due to the symmetry of our slab structure, the modes can be categorized into even and odd modes, according to their symmetries to the plane bisecting the slab. No mode mixing is possible in principle. Since only fundamental modes well exist in the PhC slab, the even modes are TE

_{0}-like and odd modes are TM0-like with respect to the mirror plane [21

21. M. Qiu, “Band gap effects in asymmetric photonic crystal slabs,” Phys. Rev. B **66**, 033103 (2002). [CrossRef]

**a**reproducibility is needed. In practice, using a hole radius r=0.24

**a**, for a lattice constant of 760nm the separation between holes is larger than 70nm, which is easily manufacturable using electron-beam lithography. It is also relevant to discuss the impact of the fabrication tolerance of the hole radius on the size and spectral position of the PBG. A good monitor of this influence can be obtained from the gap map diagram as shown in Fig. 3(a), which maps the dielectric- and air-band edge frequencies of PBGs as a function of the r/

**a**ratio. For the bands in the high-frequency region, the energy locates mainly in the high dielectric region, so these should be highly sensitive on the filling factor. This is confirmed by the large frequency shift of the high-frequency air-band edge compared to the low-frequency dielectric-band edge. By reducing the hole radius to 0.23

**a**, the gap covers the reduced frequency range from 0.452 to 0.509, yielding a mid-gap ratio of 11.9%, while the air and dielectric bands are shifted by 3% and 1%, respectively.

8. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**, 5751 (1999). [CrossRef]

**a**. Additionally, the simulations show that the dielectric- and air-band edges shift towards lower frequencies for an increasing slab thickness. This behavior can be explained by the fact that the fraction of the optical mode propagating through the dielectric material is larger for thicker slabs, thus shifting the photonic bandgap in a similar fashion as a reduction of the filling factor.

## 3. Waveguide designs in honeycomb lattice geometry

*Γ-M*direction is shorter than in the

*Γ-K*direction, the PBGs are likely to locate above the cladding light line. We propose two types of line defect PhCWs in the

*Γ-K*direction based on the honeycomb lattice geometry: the missing-hole waveguide formed by removing two rows of holes for a symmetric design as depicted in Fig. 4(a) and the additional-hole waveguide formed by introducing one row of holes in the center of the honeycomb lattice as illustrated in Fig. 4(b).

22. M. Qiu, “Effective index method for heterostructures-slab-waveguide-based two-dimensional photonic crystals” Appl. Phys. Lett. **81**, 1163–1165 (2002). [CrossRef]

*y*direction, a TM mode has the components

*H*,

_{x}*E*and

_{y}*E*. In two dimensions, the waveguide modes can also be classified as laterally even or odd parities according to the symmetry of the

_{z}*E*component with respect to the plane bisecting the waveguide. Based on the definition above, relying on the mode profiles shown in the Fig. 5(b), we identified the guided modes parities from the different bands with respect to the parity of the

_{z}*E*field. Dispersion curves of the modes with opposite parity can cross, regardless of the mode order, while the ones of modes with identical parities and different orders repel each other, giving rise to so-called mini-stopbands. 3D FDTD simulations can be used to determine how efficiently each propagating mode couple from access waveguides into the PhCWs, considering its modal and impedance mismatch, which is beyond the scope of this paper.

_{z}23. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B **62**, 8212 (2000). [CrossRef]

24. Y. Tanaka, Y. Sugimoto, N. Ikeda, H. Nakamura, Y. Watanabe, K. Asakawa, and K. Inoue, “Guided modes of a width-reduced photonic-crystal slab line-defect waveguide with asymmetric cladding,” J. Lightwave Technol. **23**, 2749–2755 (2005). [CrossRef]

25. A. V. Gopal, H. Yoshida, T. Simoyama, N. Georgiev, T. Mozume, and H. Ishikawa, “Understanding the ultra-low intersubband saturation intensity in InGaAs-AlAsSb quantum wells,” IEEE J. Quantum Electron. **39**, 299–305 (2003). [CrossRef]

*K*' instead of

*K*, which is the projected image of the

*M*point onto the

*Γ-K*direction. It is therefore important to consider a PBG having the air-band edge lower than 0.5 in normalized frequency, which is the intercept of the air line at the

*K'*point to obtain truly guided modes. For this reason and as discussed above, we increased the slab thickness to 0.58

**a**, which has a TM PBG from 0.415 to 0.462, corresponding to a mid-gap ratio of approximately 11%.

23. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B **62**, 8212 (2000). [CrossRef]

*i*) For instance the additional-hole PhCWs can be utilized for propagating two wavelengths at 1550nm and 1610nm relying on the same ISBT as such transitions are usually rather broadband with a 250nm full-width at half-maximum [26]. (

*ii*) Each single pulse can be propagated in separate waveguides by varying the lattice constants. With certain switching-topologies, e.g., a cross-waveguide geometry, one can achieve the switching functionality. As shown in Fig. 7, the flat bands reveal the low group velocity and low GVD properties which are helpful to increase the nonlinear optical interaction in all-optical switches. A compromise has however to be found to achieve low group velocity and large bandwidth simultaneously, for all-optical switches, a 0.5%~1% bandwidth at 1550nm is necessary to propagate a 1ps transform-limited Gaussian pulse. For both waveguide types, the largest bandwidth of a single guided band extracted from the 3D simulations is around 0.8%, which fulfills the requirements.

## 4. Conclusion

## Acknowledgment

## References and links

1. | S. Kawanishi, “Ultrahigh-speed optical time-division-multiplexed transmission technology based on optical signal processing,” IEEE J. Quantum Electron. |

2. | M. Nakazawa, “Tb/s OTDM technology,” Proc. 27 Eur. Conf. on Opt. Commun. |

3. | P. Cristea, Y. Fedoryshyn, and H. Jäckel, “Growth of AlAsSb/InGaAs MBE-layers for all optical switches,” J. Crystal Growth. |

4. | H. Yoshida, T. Mozume, A. Neogi, and O. Wada, “Ultrafast all-optcal switching at 1.3µm/1.55µm using novel InGaAs/AlAsSb/InP coupled double quantum well structure for intersubband transitions,” Electron. Lett. |

5. | A V Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. |

6. | T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. |

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

8. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

9. | W C L. Hopman, R M de Ridder, C. G. Bostan, S. Selvaraja, V. J. Gadgil, L. Kuipers, and A. Driessen, “Design and Fabrication of 2-Dimensional Silicon Photonic Crystal Membranes by Focused Ion Beam Processing,” presented at the ePiXnet winterschool on Optoelectronic Integration: Technology and Applications, ePiXnet Winter School, Pontresina, Switzerland, 13–17 Mar. 2006. |

10. | G. Stark, R. Wüest, F. Robin, D. Erni, H. Jäckel, A. Christ, and N. Kuster, “Extraction of the geometric parameters of photonics crystals using the effective-index method,” submitted to Opt. Lett. |

11. | P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B |

12. | D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B |

13. | S. Rowson, A. Chelnokov, J. M. Lourtioz, and F. Carcenac, “Reflection and transmission characterization of a hexagonal photonic crystal in the mid infrared,” J. Appl. Phys. |

14. | J. Ye, V. Mizeikis, Y. Xu, S. Matsuo, and H. Misawa, “Fabrication and optical characteristics of silicon-based two-dimensional photonic crystals with honeycomb lattice,” Opt. Commun. |

15. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

16. | M. Kafesaki, C. M. Soukoulis, and M. Agio, “Losses and transmission in two-dimensional slab photonic crystals,” Appl. Phys. |

17. | C. G. Bostan and R. M. de Ridder, “Design of photonic crystal slab structures with absolute gaps in guided modes,” J. Optoelectron. Adv Mater. |

18. | C. Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B |

19. | Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs,” J. Appl. Phys. |

20. | A. Yariv and P. Yeh, |

21. | M. Qiu, “Band gap effects in asymmetric photonic crystal slabs,” Phys. Rev. B |

22. | M. Qiu, “Effective index method for heterostructures-slab-waveguide-based two-dimensional photonic crystals” Appl. Phys. Lett. |

23. | S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B |

24. | Y. Tanaka, Y. Sugimoto, N. Ikeda, H. Nakamura, Y. Watanabe, K. Asakawa, and K. Inoue, “Guided modes of a width-reduced photonic-crystal slab line-defect waveguide with asymmetric cladding,” J. Lightwave Technol. |

25. | A. V. Gopal, H. Yoshida, T. Simoyama, N. Georgiev, T. Mozume, and H. Ishikawa, “Understanding the ultra-low intersubband saturation intensity in InGaAs-AlAsSb quantum wells,” IEEE J. Quantum Electron. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.1150) Optical devices : All-optical devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 6, 2006

Revised Manuscript: December 13, 2006

Manuscript Accepted: December 13, 2006

Published: December 22, 2006

**Citation**

Ping Ma, Franck Robin, and Heinz Jäckel, "Realistic photonic bandgap structures for TM-polarized light for all-optical switching," Opt. Express **14**, 12794-12802 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12794

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### References

- S. Kawanishi, "Ultrahigh-speed optical time-division-multiplexed transmission technology based on optical signal processing," IEEE J. Quantum Electron. 34, 2604 (1998). [CrossRef]
- M. Nakazawa, "Tb/s OTDM technology," Proc. 27 Eur. Conf. on Opt. Commun. 184 (2001).
- P. Cristea, Y. Fedoryshyn, and H. Jäckel, "Growth of AlAsSb/InGaAs MBE-layers for all optical switches," J. Crystal Growth. 278, 544-547 (2005). [CrossRef]
- H. Yoshida, T. Mozume, A. Neogi and O. Wada, "Ultrafast all-optical switching at 1.3μm/1.55μm using novel InGaAs/AlAsSb/InP coupled double quantum well structure for intersubband transitions," Electron. Lett. 35, 1103 (1999). [CrossRef]
- A V Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004). [CrossRef]
- T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751 (1999). [CrossRef]
- W C L. Hopman, R M de Ridder, C. G. Bostan, S. Selvaraja, V. J. Gadgil, L. Kuipers and A. Driessen, "Design and Fabrication of 2-Dimensional Silicon Photonic Crystal Membranes by Focused Ion Beam Processing," presented at the ePiXnet winterschool on Optoelectronic Integration: Technology and Applications, ePiXnet Winter School, Pontresina, Switzerland, 13-17 Mar. 2006.
- G. Stark, R. Wüest, F. Robin, D. Erni, H. Jäckel, A. Christ, N. Kuster, "Extraction of the geometric parameters of photonics crystals using the effective-index method," submitted to Opt. Lett.
- P. R. Villeneuve and M. Piché, "Photonic band gaps in two-dimensional square and hexagonal lattices," Phys. Rev. B 46, 4969 (1992). [CrossRef]
- D. Cassagne, C. Jouanin and D. Bertho, "Hexagonal photonic-band-gap structures," Phys. Rev. B 53, 7134 (1996). [CrossRef]
- S. Rowson, A. Chelnokov, J. M. Lourtioz and F. Carcenac, "Reflection and transmission characterization of a hexagonal photonic crystal in the mid infrared," J. Appl. Phys. 83, 5061-5064 (1998). [CrossRef]
- J. Ye, V. Mizeikis, Y. Xu, S. Matsuo and H. Misawa, "Fabrication and optical characteristics of silicon-based two-dimensional photonic crystals with honeycomb lattice," Opt. Commun. 211, 205-213 (2002). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8173-190 (2001). [CrossRef] [PubMed]
- M. Kafesaki, C. M. Soukoulis and M. Agio, "Losses and transmission in two-dimensional slab photonic crystals," Appl. Phys. 96, 4033-4038 (2004).
- C. G. Bostan and R. M. de Ridder, "Design of photonic crystal slab structures with absolute gaps in guided modes," J. Optoelectron.Adv Mater. 4,921-928 (2002).
- C. Y. Kao, S. Osher, and E. Yablonovitch, "Maximizing band gaps in two-dimensional photonic crystals by using level set methods," Appl. Phys. B 81, 235-244 (2005). [CrossRef]
- Y. Sugimoto, N, Ikeda, N. Carlsson, K. Asakawa, N. Kawai and K. Inoue, "Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slabs," J. Appl. Phys. 91, 922-929 (2002). [CrossRef]
- A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
- M. Qiu, "Band gap effects in asymmetric photonic crystal slabs," Phys. Rev. B 66, 033103 (2002). [CrossRef]
- M. Qiu, "Effective index method for heterostructures-slab-waveguide-based two-dimensional photonic crystals" Appl. Phys. Lett. 81, 1163-1165 (2002). [CrossRef]
- S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Linear waveguides in photonic-crystal slabs," Phys. Rev. B 62, 8212 (2000). [CrossRef]
- Y. Tanaka, Y. Sugimoto, N. Ikeda, H. Nakamura, Y. Watanabe, K. Asakawa and K. Inoue, "Guided modes of a width-reduced photonic-crystal slab line-defect waveguide with asymmetric cladding," J. Lightwave Technol. 23, 2749-2755 (2005). [CrossRef]
- A. V. Gopal, H. Yoshida, T. Simoyama, N. Georgiev, T. Mozume and H. Ishikawa, "Understanding the ultra-low intersubband saturation intensity in InGaAs-AlAsSb quantum wells," IEEE J. Quantum Electron. 39, 299-305 (2003). [CrossRef]

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