## Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals

Optics Express, Vol. 15, Issue 15, pp. 9287-9292 (2007)

http://dx.doi.org/10.1364/OE.15.009287

Acrobat PDF (1598 KB)

### Abstract

A device for optical switches and logic gates is proposed in two-dimensional photonic crystals based on self-collimated beams. The main structure of the device is a line-defect-induced 3 dB splitter. Operating principle, as revealed by both theoretical calculation and finite-difference time-domain simulation, is based on the interference of reflected and transmitted self-collimated beams. This device is potentially applicable for photonic integrated circuits.

© 2007 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. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature (London) , **426**, 404 (2003). [CrossRef]

4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B , **58**, R10096–R10099 (1998). [CrossRef]

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**, 1212–1214 (1999). [CrossRef]

12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**, 231114-1 (2007). [CrossRef]

7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

10. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express **11**, 3153–3159 (2003). [CrossRef] [PubMed]

13. M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. **28**, 2506–2508 (2003). [CrossRef]

15. Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, “High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals,” Opt. Express **14**, 1783–1788 (2006). [CrossRef] [PubMed]

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**, 1212–1214 (1999). [CrossRef]

7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**, 1212–1214 (1999). [CrossRef]

10. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express **11**, 3153–3159 (2003). [CrossRef] [PubMed]

11. B. Miao, C. Chen, S. Shi, and D. W. Prather, “A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices,” IEEE Photon. Technol. Lett. **17**, 61–63 (2005). [CrossRef]

## 2. Operation principle and structure analysis

*θ*>

*θ*=arcsin (1/

_{c}*n*) because of the conservation of momentum components parallel to the interface, where a PC and air corresponds to an optically denser medium (high refractive index,

_{H}*n*) and an optically thinner medium, respectively [7

_{H}7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

*a*×12√2

*a*of 2D square lattice PC composed of silicon (Si) rods in air. The radius and dielectric constant of the host Si rods are

*r*=0.35

*a*and

*ε*=12.0, respectively, where

*a*is the lattice constant (the parameters refer to Ref. [8]). A line defect is created by reducing the radii

*r*=0.274

_{d}*a*of 25 rods aligned in the

**Γ-X**direction different from those of the host rods, as denoted by the green area in Fig. 1. The band diagram and the equifrequency contours (EFCs) of the first band for the

**-polarized mode (electric-field is parallel to the rod axes) are shown in Fig. 2. In the EFCs [see Fig. 2(b)], the curves of the frequencies around 0.194(**

*E**a*/λ), where λ is the wavelength of light in free space, can be identified as squares with round corners centered at the

**M**point. It is known that the light propagation direction in the PC is identical to the direction of group velocity given by

*v*=

_{g}*∇*, where

_{k}ω(k)*ω*is the optical frequency at the wave vector

*k*[16

16. R. S. Chu and T. Tamir, “Group velocity in space-time periodic media,” Electron. Lett. **7**, 410–412 (1971). [CrossRef]

**-polarized light of the frequencies around 0.194(**

*E**a/λ*) propagate along the

**Γ-M**direction. There are four faces in the device structure, two adjoining faces of them function as input faces (I

_{1}, I

_{2}), and the other two as output faces (O

_{1}, O

_{2}), as shown in Fig. 1. Before launching the self-collimated beams to investigate the switching and logical functions, the phase shift between the reflected and transmitted beams should be known.

*n*) PC prisms separated by a low refractive index (

_{H}*n*) gap with a spatially symmetric structure. Note that the dielectric constituents are nonabsorbing (the imaginary parts of the dielectric constants are set to zero), which means that the proposed device is a lossless (conservation of energy) system. According to Refs. [12

_{L}12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**, 231114-1 (2007). [CrossRef]

17. Z. Y. Ou and L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. **57**, 66–67 (1989). [CrossRef]

*π*/2 phase difference between the reflected and transmitted beams. This is a general result for a spatially symmetric, lossless beam splitting system. Especially for the splitter, by utilizing the line defect in PC to split the self-collimated beams, if the rod radii of the line defect are smaller than that of the host rods, there will be a π/2 phase lag of reflected beam compared to the transmitted beam [12

12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**, 231114-1 (2007). [CrossRef]

*π*/2 phase lag compared to the transmitted beam after the self-collimated beam goes through the line defect.

## 3. Calculations of the structure

*te*, so the reflection amplitude is

^{iφ}*re*for both the incident beams, where

^{i(φ+π/2)}*φ*,

*t*, and

*r*are real, and

*t=r*=1/√2. So the transmission and the reflection amplitudes are

*e*/√2 and

^{iφ}*e*/2)/√2, respectively. The computational simulation is carried out by using a finite-difference time-domain (FDTD) code with perfectly matched layer boundary condition and an

^{i(φ+π}**-polarized Gaussian wave with the full width at half maximum 5**

*E**a*is used. A monochromatic wave of the frequency 0.194(

*a*/λ) is launched into the PC device along the

**Γ-M**direction, and then a self-collimated beam will be excited.

_{1}and I

_{2}sees a beam field (Bloch wave) of

*φ*, and

_{1}*φ*are real,

_{2}*E*represents a plane wave and

*u*is a function with the same periodicity as the PC. There is only a phase difference between the two incident beams. Then the reflected and transmitted beams can be expressed as:

_{1}and O

_{2}can be written as a linear combination (interference) of the reflected and the transmitted beams:

*u*or

*E*is zero (when the phase difference becomes meaningless quantity), we can see that

*I*=2|

_{o1}*uE*|

^{2},

*I*=0 when

_{o2}*φ*-

_{1}*φ*=2

_{2}*kπ+*π/2 and

*I*=0,

_{o1}*I*=2|

_{02}*uE*|

^{2}when

*φ*/2, where

_{1}-φ_{2}=2kπ-π*k*is an integer.

## 4. Results and discussions of switches and logic gates

_{1}and I

_{2}), as shown schematically in Fig. 4(a). A phase modulator was introduced at input branch I

_{2}, the beam coupled into which we regard as the control beam. Figures 4(b) and 4(c) show the simulated steady-state field distributions of the switch. It shows that once the control beam is introduced, the output state of the device can be controlled. When the phase difference

*φ*sets as 2

_{1}−φ_{2}*kπ+π*/2, where

*k*is an integer, the input lights will be output from the face O

_{1}, and there is no output light in the face O

_{2}, as shown in Fig. 4(b). If the phase difference is changed to 2

*kπ-π*/2, the light power will be output from the face O

_{2}, and the output face O

_{1}will be cut off, as shown in Fig. 4(c). So the switching is achieved.

*φ*sets as 2

_{1}-φ_{2}*kπ*+

*π*/2, the output face O

_{1}and O

_{2}operates as OR and XOR logic gates, respectively, as shown in Fig. 3 and Fig. 4(b). Alternatively, if the phase difference

*φ*sets as 2

_{1}-φ_{2}*kπ*-

*π*/2, the output face O

_{1}and O

_{2}operates as XOR and OR logic gates, respectively, as shown in Fig. 3 and Fig. 4(c). The total device functions are shown in Table 1 for both cases. The logic 0 and 1 in the table indicate without and with output signal, respectively.

_{1}and O

_{2}for Fig. 3(a) and Fig. 4(b) are shown in Fig. 5. From Fig. 5(a), it is clear that, at frequency 0.194(

*a*/λ), the incident power is split equally into the two output faces. In the frequency range 0.188–0.199(

*a*/λ), the fluctuation of the output intensities are within 20% referred to the intensity at frequency 0.194(

*a*/λ), and the sum of the transmitted and reflected intensity is larger than 93%. Moreover, from Fig. 5(b), it can be found that, in the frequency range 0.188–0.199(

*a*/λ), there is nearly no fluctuation of the output intensities from the two output faces and the extinction ratio, defined as 10log(

*I*), is larger than 17 dB (maximum 20.1 dB). So we conclude that the switching and logical function of the structure is applicable in the frequency range 0.188–0.199(

_{O1}/I_{O2}*a*/λ).

## 5. Conclusion

*a*/λ). The extinction ratio for the switch within the applicable frequencies is larger than 17 dB (maximum 20.1 dB). The device has simple geometric structure and clear operating principle, which shows that this device could be a strong candidate for future PICs.

## 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. | P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature (London) , |

4. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B , |

5. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. |

6. | J. Witzens, M. Lončar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. |

7. | X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. |

8. | S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

9. | M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

10. | C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express |

11. | B. Miao, C. Chen, S. Shi, and D. W. Prather, “A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices,” IEEE Photon. Technol. Lett. |

12. | D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. |

13. | M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. |

14. | N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, “Integrated all-optical switch in a cross-waveguide geometry,” Appl. Phys. Lett. |

15. | Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, “High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals,” Opt. Express |

16. | R. S. Chu and T. Tamir, “Group velocity in space-time periodic media,” Electron. Lett. |

17. | Z. Y. Ou and L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. |

18. | R. Ramaswami and K. N. Sivarajan, |

**OCIS Codes**

(230.1150) Optical devices : All-optical devices

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: May 11, 2007

Revised Manuscript: June 27, 2007

Manuscript Accepted: July 5, 2007

Published: July 13, 2007

**Citation**

Yuanliang Zhang, Yao Zhang, and Baojun Li, "Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals," Opt. Express **15**, 9287-9292 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9287

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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]
- P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Photonic crystals: Imaging by flat lens using negative refraction," Nature (London), 426, 404 (2003). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B, 58, R10096-R10099 (1998). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999). [CrossRef]
- J. Witzens, M. Lončar, and A. Scherer, "Self-collimation in planar photonic crystals," IEEE J. Sel. Top. Quantum Electron. 8, 1246-1257 (2002). [CrossRef]
- X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
- S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 87, 1811061-3 (2005).
- M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 90, 1131211-3 (2007).
- C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, "Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides," Opt. Express 11, 3153-3159 (2003). [CrossRef] [PubMed]
- B. Miao, C. Chen, S. Shi, and D. W. Prather, "A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices," IEEE Photon. Technol. Lett. 17, 61-63 (2005). [CrossRef]
- D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, "Photonic crystal Mach-Zehnder interferometer based on self-collimation," Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]
- M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003). [CrossRef]
- N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, "Integrated all-optical switch in a cross-waveguide geometry," Appl. Phys. Lett. 88, 1711041-3 (2006). [CrossRef]
- Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, "High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals," Opt. Express 14, 1783-1788 (2006). [CrossRef] [PubMed]
- R. S. Chu and T. Tamir, "Group velocity in space-time periodic media," Electron. Lett. 7, 410-412 (1971). [CrossRef]
- Z. Y. Ou and L. Mandel, "Derivation of reciprocity relations for a beam splitter from energy balance," Am. J. Phys. 57, 66-67 (1989). [CrossRef]
- R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, San Francisco, 1998), chap. 3.1.

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