## Optical switches based on partial band gap and anomalous refraction in photonic crystals modulated by liquid crystals

Optics Express, Vol. 15, Issue 16, pp. 10033-10040 (2007)

http://dx.doi.org/10.1364/OE.15.010033

Acrobat PDF (309 KB)

### Abstract

Optical switches using two transmission properties in triangular photonic crystals infiltrated with liquid crystals (LCs) are investigated for incorporation in wave-guided structures for planar lightwave circuits. The two devices employ partial band gap and anomalous refraction, which are based on the anisotropic characteristics of LC reorientation under applied fields. These switches have been designed and their parameters have been analyzed by the plane wave and finite-difference time-domain calculations. In the on/off switching system, the partial band gap can be controlled when the normalized operation frequency is 0.27. The anomalous refraction can be modulated to deflect a light beam with a maximum deflection angle ~57° when the frequency is 0.3. The tunability induced by LCs can create a sharp switching in the photonic devices.

© 2007 Optical Society of America

## 1. Introduction

2. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. **3**, 211–219 (2004). [CrossRef]

3. S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, “All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures,” Phys. Rev. E **74**, 046603 (2006). [CrossRef]

4. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A **17**, 1012–1020 (2000). [CrossRef]

5. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. **78**, 3294–3297 (1997). [CrossRef]

6. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B **62**, 10696–10705 (2000). [CrossRef]

7. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two-dimensional photonic crystal,” Opt. Express **12**, 2919–2924 (2004). [CrossRef] [PubMed]

8. B. Momeni and A. Adibi, “Optimization of photonic crystal demultiplexers based on the superprism effect,” Appl. Phys. B **77**, 555–560 (2003). [CrossRef]

9. D. N. Chigrin, S. Enoch, C. M. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express **11**, 1203–1211 (2003). [CrossRef] [PubMed]

10. S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. **7**, 210 (2005). [CrossRef]

11. S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Tech. **17**, 1931–1943 (1999). [CrossRef]

13. H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E **67**, 056607 (2003). [CrossRef]

14. D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. **82**, 3176–3178 (2003). [CrossRef]

15. S. Xiong and H. Fukshima, “Analysis of light propagation in index-tunable photonic crystals,” J. Appl. Phys. **94**, 1286–1288 (2003). [CrossRef]

16. W. Park and J.-B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. **85**, 4845–4847 (2004). [CrossRef]

17. L. Feng, X.-P. Liu, J. Ren, Y.-F. Tang, Y.-B. Chen, Y.-F. Chen, and Y.-Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. **97**, 073104 (2005). [CrossRef]

## 2. Numerical procedures

**H**(

**r**) can be expressed as:

*ε*(

**r**) of a PhC structure is periodic with respect to the lattice vector and can be expanded in a Fourier series on the reciprocal lattice vector

**G**:

*n*

_{o}and extraordinary-refractive index

*n*

_{e}are for light with electric field polarization perpendicular and parallel to the director, respectively. When the nematic director rotates in the

*xy*plane, the components of the dielectric tensor can be represented as [13

13. H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E **67**, 056607 (2003). [CrossRef]

**n**=(cosΦ, sinΦ). Plane wave expansion method was employed to calculate photonic band diagram and CFCs [18]. We examine mainly TE modes (electric fields lie parallel to the

*xy*plane). Because of the directors parallel to the 2D plane, the electric fields with the TE mode can be strongly influenced by rotating the directors of LCs. Besides, the numerical problem in obtaining the eigenvalues from the Fourier coefficients of the inverse dielectric tensors can be solved by the method proposed by Ho

*et al*. [19

19. K. M. Ho, C. T. Chen, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

20. S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. **44**, 1630–1639 (1996). [CrossRef]

## 3. Photonic crystals infiltrated with liquid crystals

**k**can be modulated. A critical voltage is required to be small for the practical application of LCs in the tunable PhC. The LC 5CB is used since it has small critical voltage value and large optical anisotropy [12]. The value of refractive indices of LC 5CB is taken as

*n*

_{o}=1.522 and

*n*

_{e}=1.706 on the assumption that PhC operates at room temperature. When the external electric field is not applied, the LC becomes isotropic and its average refractive index is

*n*

_{av}=(2

*n*

_{o}+

*n*

_{e})/3=1.583. However, the optical anisotropy of the structure is not significant since the refractive index of Si (

*n*=3.45) compared with that of LC is higher. In order to obtain a higher anisotropy, a larger radius

*r*=0.35a is adopted to improve the anisotropy.

## 4. Tunable features and optical switches

_{x}/b

_{2}=0 (k

_{y}/b

_{2}=0), the intersection points can be obtained between the line and the contours of ω=0.27. In the case of Φ=0°, incident light will penetrate the PhC. As the directors are orientated at Φ=90°, these contours shown in Fig. 3(b) are shifted to the diagonal positions. Light will reflect back because the specified construction line without any intersection point falls into the partial band gap. A schematic drawing for a normal incident light is shown in Fig. 4 when an in-plane electric field is applied. The transmission of PhCs changes from the “on” to the “off” state when an external electric filed is applied at different directions. The FDTD method is used to simulate the light propagation in the PhC, and the results can be compared with the theoretical prediction by a CFC analysis. Corresponding to ω=0.27 for a=1µm, a Gaussian wave with the center wavelength λ=3.7µm is taken as an input. Figure 5 shows the transmission spectra of the observing point. In Fig. 5(a), the short wavelength gap edge for Φ=0° is at 3.8 µm, and the ratio of outgoing to incoming power at λ=3.7µm is about-6db for the on-state transmission. In Fig. 5(b), the short wavelength gap edge for Φ=90° is shifted to 3.6 µm and the off-state transmission ratio at λ=3.7µm is about - 50db. The effect due to the variation of LC directors is obvious. An optical switch based on the partial band gap effect can be obtained by rotating the directors with an in-plane electric field. For a=0.4185µm at the same normalized frequency, an optical switch for telecommunication wavelength (λ=1.55µm) can be designed to achieve a theoretical contrast ratio of 200:1(23dB) between switching on and off. The transmission normalized with respect to the incident amplitude is shown in Fig. 6.

*x*-axis). The energy propagation direction in 2D PhCs is oriented in the direction of the group velocity vector

**V**

_{g}=

**∂ω**/

**∂k**, which is always perpendicular to the CFC and points towards increasing values of frequency. The tangential components of wave vectors of incident waves and refractive waves are always conserved across the interface between two materials. A construction line which indicates the momentum conservation can be set up to determine the group velocity direction at the intersection with the CFCs. For a normal incident light, the arrows in Fig. 7 indicate the directions of light. In Fig. 7(a), LC directors are orientated at Φ=40°, and the light propagation direction is refracted at approximate -45° to the k

_{x}-axis direction. In Fig. 7(b) for Φ=-40°, and the refractive angle is about 45°. Changing the LC directors, which leads to the distortion of CFCs of the PhC, could shift the maximum deflection angle by ~90°. Beam-deflection-type switches could be accomplished via the large variation of deflection angle. In addition, the FDTD method is also used to investigate the tunable refraction effect. Figures 8(a) and (b) display the magnetic field maps of propagating waves at Φ=40° and -40°, respectively. The arrows indicate the light propagating directions. The field patterns show that the refracted angles for the TE polarization are about -29° and 28°, respectively. The FDTD simulations clearly verify that the light propagating direction can be altered by rotating the LC directors, even though the refracted angles determined from the FDTD method are not in very good agreement with those determined from the CFC analysis. The finiteness of the system and the effect of the Goos-Hänchen shift result in the mismatch between the employed numerical methods [21

21. A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B **71**, 235115 (2005). [CrossRef]

## 5. Conclusion

## Acknowledgements

## References and links

1. | J.-M. Lourtioz, H. Benisty, V. Berger, J.-M Gérard, D. Maystre, and A. Tchelnokov, |

2. | M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. |

3. | S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, “All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures,” Phys. Rev. E |

4. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A |

5. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. |

6. | M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B |

7. | X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two-dimensional photonic crystal,” Opt. Express |

8. | B. Momeni and A. Adibi, “Optimization of photonic crystal demultiplexers based on the superprism effect,” Appl. Phys. B |

9. | D. N. Chigrin, S. Enoch, C. M. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express |

10. | S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New J. Phys. |

11. | S. John and K. Busch, “Photonic bandgap formation and tunability in certain self-organizing systems,” J. Lightwave Tech. |

12. | I. C. Khoo and S. T. Wu, |

13. | H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E |

14. | D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. |

15. | S. Xiong and H. Fukshima, “Analysis of light propagation in index-tunable photonic crystals,” J. Appl. Phys. |

16. | W. Park and J.-B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. |

17. | L. Feng, X.-P. Liu, J. Ren, Y.-F. Tang, Y.-B. Chen, Y.-F. Chen, and Y.-Y. Zhu, “Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents,” J. Appl. Phys. |

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

19. | K. M. Ho, C. T. Chen, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

20. | S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. |

21. | A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B |

**OCIS Codes**

(130.1750) Integrated optics : Components

(230.3720) Optical devices : Liquid-crystal devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 30, 2007

Revised Manuscript: July 5, 2007

Manuscript Accepted: July 9, 2007

Published: July 25, 2007

**Citation**

Yao-Yu Wang, Jiun-Yeu Chen, and Lien-Wen Chen, "Optical switches based on partial band gap and anomalous refraction in photonic crystals modulated by liquid crystals," Opt. Express **15**, 10033-10040 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10033

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### References

- J.-M. Lourtioz, H. Benisty, V. Berger, J.-M Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, Berlin, 2005).
- M. Soljaèiæ and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Mater. 3, 211-219 (2004). [CrossRef]
- S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, and K. Busch, "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures," Phys. Rev. E 74, 046603 (2006). [CrossRef]
- B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000). [CrossRef]
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, "High extraction efficiency of spontaneous emission from slabs of photonic crystals," Phys. Rev. Lett. 78, 3294-3297 (1997). [CrossRef]
- M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000). [CrossRef]
- X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004). [CrossRef] [PubMed]
- B. Momeni and A. Adibi, "Optimization of photonic crystal demultiplexers based on the superprism effect," Appl. Phys. B 77, 555-560 (2003). [CrossRef]
- D. N. Chigrin, S. Enoch, C. M. Sotomayor Torres, and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11, 1203-1211 (2003). [CrossRef] [PubMed]
- S. He, Y. Jin, Z. Ruan and J. Kuang, "On subwavelength and open resonators involving metamaterials of negative refraction index," New J. Phys. 7, 210 (2005). [CrossRef]
- S. John and K. Busch, "Photonic bandgap formation and tunability in certain self-organizing systems," J. Lightwave Tech. 17, 1931-1943 (1999). [CrossRef]
- I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
- H. Takeda and K. Yoshino, "Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals," Phys. Rev. E 67, 056607 (2003). [CrossRef]
- D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, "Electro-optic control of the superprism effect in photonic crystals," Appl. Phys. Lett. 82, 3176-3178 (2003). [CrossRef]
- S. Xiong and H. Fukshima, "Analysis of light propagation in index-tunable photonic crystals," J. Appl. Phys. 94, 1286-1288 (2003). [CrossRef]
- W. Park and J.-B. Lee, "Mechanically tunable photonic crystal structure," Appl. Phys. Lett. 85, 4845-4847 (2004). [CrossRef]
- L. Feng, X.-P. Liu, J. Ren, Y.-F. Tang, Y.-B. Chen, Y.-F. Chen, and Y.-Y. Zhu, "Tunable negative refractions in two-dimensional photonic crystals with superconductor constituents," J. Appl. Phys. 97, 073104 (2005). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- K. M. Ho, C. T. Chen, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
- S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat. 44, 1630-1639 (1996). [CrossRef]
- A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005). [CrossRef]

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