## Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity

Optics Express, Vol. 14, Issue 6, pp. 2446-2458 (2006)

http://dx.doi.org/10.1364/OE.14.002446

Acrobat PDF (247 KB)

### Abstract

In the paper, a novel three-port channel drop filter in two dimensional photonic crystals (2D PCs) with a wavelength-selective reflection micro-cavity is proposed. In the structure, two micro-cavities are used. One is used for a resonant tunneling-based channel drop filter. The other is used to realize wavelength-selective reflection feedback in the bus wave-guide, which consists of a point defect micro-cavity side-coupled to a line defect waveguide based on photonic crystals. Using coupled mode theory in time, the conditions to achieve 100% drop efficiency are derived thoroughly. The simulation results by using the finite-difference time-domain (FDTD) method imply that the design is feasible.

© 2006 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987). [CrossRef] [PubMed]

4. Steven G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Materialia. **51**, 5823 (2003). [CrossRef]

5. A. Sharkawy, S. Shi, and D.W. Prather, “Multichannel wavelength division multiplexing using photonic crystals,” Appl. Opt. **40**, 2247 (2001). [CrossRef]

6. H. Takahashi, S. Suzuki, and I. Nishi, “Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating,” IEEE J.Lightwave Technol. **12**,989 (1994). [CrossRef]

7. Zimmermann. J, Kamp. M, Forchel. A, and März R, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. **230**, 387 (2004). [CrossRef]

10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960 (1998). [CrossRef]

10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960 (1998). [CrossRef]

15. B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science , **300**, 1537 (2003). [CrossRef] [PubMed]

16. H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. **84**, 2226–2228 (2004). [CrossRef]

17. Bong-Shik Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B. **71**, 195101 (2005). [CrossRef]

18. Sangin Kim, Ikmo Park, Hanjo Lim, and Chul-Sik Kee, “Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback,” Opt. Express **12**, 5518 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5518. [CrossRef] [PubMed]

15. B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science , **300**, 1537 (2003). [CrossRef] [PubMed]

17. Bong-Shik Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B. **71**, 195101 (2005). [CrossRef]

19. Yoshihiro Akahane, Takashi Asano, Hitomichi Takano, Bong-Shik Song, Yoshinori Takana, and Susumu Noda, “Two-dimensional photonic-crystal-slab channel drop filter with flat-top response,” Opt. Express **13**, 2512 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2512. [CrossRef] [PubMed]

21. Akihiko Shinya, Satoshi Mitsugi, Eiichi Kuramochi, and Masaya Notomi, “Ultrasmall multi-channel resonant-tunneling filter using mode gap of width-tuned photonic-crystal waveguide,” Opt. Express **13**, 4202 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4202. [CrossRef] [PubMed]

## 2. Theoretical modeling

*S*

_{+1}(

*S*

_{-1}) and

*S*

_{+2}(

*S*

_{-2}), respectively. The time evolution of the cavity amplitude denoted by

*a*can be described as [11

11. C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322 (1999). [CrossRef]

22. Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. **62**, 7389 (2000). [CrossRef]

*S*

_{+2}=0) and

*S*

_{+1}has a

*e*

^{jωt}time dependence, by solving Eqs. (1)–(3), the back reflection from the input port

*R*and the transmission through the waveguide

*T*can be expressed as

*R*|

^{2}≈0 , |t|

^{2}≈1) at ω = ω

_{0}is realized if the condition

*Q*

_{o}≫

*Q*

_{e}is satisfied. The full-width at half-maximum (FWHM) of the reflection spectrum can be expressed as σ = 2ω

_{0}(1/

*Q*

_{o}+1/2

*Q*

_{e}). With the condition

*Q*

_{o}≫

*Q*

_{e}, the FWHM will be rewritten as σ ≈ ω

_{0}/

*Q*

_{e}, So at a given resonant frequency, the line-width is determined by the value of

*Q*

_{e}, which is dependent on the distance between the cavity and the waveguide to a great degree. In order to obtain the narrow FWHM, the distance between the cavity and waveguide should be chosen reasonably so that

*Q*

_{e}is as high as possible.

*a*is applied to the three-port channel drop filter (shown in Fig. 2), where the reflection cavity is put on the side of the bus waveguide, and the channel drop waveguide perpendicular to it is put on the other side to avoid the direct coupling between the reflection cavity and the channel drop cavity. The two cavities possess mirror reflection symmetry with respect to their own reference planes that are their center planes, respectively, and the waveguides with them support single mode in the designed spectrum range. The amplitudes of the incoming waves into the system are denoted by

*S*

_{+i}or

*S*′

_{+i}and

*S*

_{-i}or

*S*′

_{-i}are the amplitudes for the outgoing waves (

*i*= 1,2,3). The time evolution of the amplitudes of the cavity

*a*and

*b*, and the incoming and outgoing waves can be described as

*S*

_{+2}=0,

*S*

_{+3}= 0) and

*S*

_{+1}has a

*e*

^{Jωt}time dependence, by solving Eqs. (6) to (11),

*S*′

_{+1}can be written as

*b*can be obtained at steady state

*T*

_{1}is the transmission through the drop waveguide,

*T*

_{2}is the transmission through the bus waveguide, and

*R*′ is the back reflection from the input port.

*Q*

_{oa}≫

*Q*

_{3},

*Q*

_{ob}≫

*Q*

_{1}and

*ω*

_{0a}=

*ω*

_{0b}=

*ω*

_{0}, substituting Eq. (16) into Eq. (19), the drop efficiency η can be expressed by

*ω*

^{4}is highest in the numerator of Eq. (22), and the highest order term is

*ω*

^{6}in the denominator, so the filter response is similar to a Lorentzian line shape.

*Q*

_{1}/

*Q*

_{2}=

*k*, where

*k*is a plus value, η

_{0}can be rewritten as

*k*= 2 and ϕ = (2

*n*+ 1)π if

*Q*

_{oa}≫

*Q*

_{3}and

*Q*

_{ob}≫

*Q*

_{1}, where

*n*is the integer. At the resonant frequency, Fig. 3(a) shows the drop efficiency as a function of ϕ when

*k*is equal to 1, 2 and 3, respectively, and Fig. 3(b) shows the curve of drop efficiency as a function of

*k*at ϕ = (2

*n*+ 1)π . It is clear that the drop efficiency is less sensitive to the phase error near

*π*and has the maximum efficiency for a rather wide range of

*k*at

*k*= 2, which possibly brings flexibility with respect to the design of the filter. At

*k*= 2 and ϕ=2

*n*π, the drop efficiency is zero, and the light is totally reflected back to the input port.

*Q*

_{ob}≫

*Q*

_{1}and

*Q*

_{ob}≫

*Q*

_{2}, we assume

*Q*

_{1}/

*Q*

_{2}= 2, η

_{0}can be rewritten as

*l*, where

*l*is the reflection ratio of the cavity

*a*at the resonant frequency. It is proved that the enhanced drop efficiency is less sensitivity to the reflection ratio of the reflection cavity

*a*when the phase ϕ is around π, and the ultimate drop efficiency is always higher than the reflection ratio

*l*at ϕ = (2

*n*+ 1)>π.

*Q*

_{oa}≫

*Q*

_{3}, assuming

*n*+ 1)π and

*Q*

_{1}/

*Q*

_{2}= 2, at ω = ω

_{0a}= ω

_{0b}= ω

_{0}, from Eq. (19), the drop efficiency η0 can be rewritten as,

*Q*

_{oa}≫

*Q*

_{3}and

*ω*=

*ω*

_{0a}, when the resonant frequency of the channel drop cavity co satisfies the term as follows,

_{0}can be expressed as,

*k*is equal to

*Q*

_{1}/

*Q*

_{2}. Fig. 3(e) shows the curve of drop efficiency as a function of phase ϕ for

*k*= 1, 2 and 3. At

*k*= 2, the curve has a maximum value in the rather wide phase range around π, where close to 100% drop efficiency can be attained in the phase range between 0.8 π and 1.2π if the frequency term Eq. (40) is satisfied. The continuous phase range may be available to design the multi-channel drop filters conveniently while the phase term ϕ = (2

*n*+ 1)π is limited by the discrete number of lattice constant. If

*k*<2, it is also possible to achieve 100% drop efficiency, where the phase ϕ is not equal to(2

*n*+ 1)π.

*n*+ 1)π and quality factor ratio

*k*= 2, if

*Q*

_{oa}≫

*Q*

_{3}and

*Q*

_{ob}≫

*Q*

_{1}, supposing

*Q*

_{1}in the case. Fig. 3(f) plots the drop efficiency as a function of the frequency detuning coefficient

*m*for

*Q*

_{1}=10

^{3},10

^{4}and 10

^{5}. As

*Q*

_{1}is increased, the filter signal become narrow, and the drop efficiency sharply reduces with the detuning between the two resonant frequencies varied. In order to achieve the high drop efficiency, the detuning needs to be reduced sufficiently in the high Q three-port system, and this implies that the precise fabrication technology is necessary.

## 3. Design and numerical calculations

*a*, where a is the lattice constant of the square array, and it possesses a band-gap only for the transverse magnetic (TM) mode that has its electric field parallel to the rods. The smaller rod that defines the defect of the channel drop cavity has a radius of 0.042

*a*, and the two rods at the interface between the cavity and drop wave-guide or bus wave-guide have a radius of 0.211

*a*. They all have the same dielectric constant as the background rods. The structure has been simulated using the FDTD method with PML absorbing boundary conditions to calculate the transmission performance.

*Q*factor

*Q*

_{o}is larger than 10

^{6}, and the system

*Q*factor

*Q*

_{s}is calculated to be 1020 (The total computation domain is 30

*a*×30

*a*along the x, y directions in the plane, where

*a*is the lattice constant.). So it is evident that

*Q*

_{o}≫

*Q*

_{s}is satisfied. Using the coupled mode theory [18

18. Sangin Kim, Ikmo Park, Hanjo Lim, and Chul-Sik Kee, “Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback,” Opt. Express **12**, 5518 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5518. [CrossRef] [PubMed]

*Q*

_{1}/

*Q*

_{2}≈ 2 is easily obtained, where

*Q*

_{1}and

*Q*

_{2}represent the quality factors due to the rates of decay into the bus waveguide or drop waveguide, respectively.

*Q*factor of 1020. It is clear that the realization of extremely high-

*Q*cavities is necessary in the filter for DWDM system.

*a*and 7

*a*, respectively, the numerical results are also in very good agreement with the theory analysis perfectly due to the satisfaction of the phase condition. At

*d*=

*a*, the results do not match the theoretical analysis very well although the condition ϕ = π is satisfied. The reason is that the strong coupling between the two cavities via the bus waveguide occurs, which is on account of so little distance between them, and it is unsuitable to apply the theory analysis in section 2 to the case. To get a compact filter, we choose the distance as 3a or 5a in the example.

## 4. 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. | Kazuaki Sakoda, |

4. | Steven G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Materialia. |

5. | A. Sharkawy, S. Shi, and D.W. Prather, “Multichannel wavelength division multiplexing using photonic crystals,” Appl. Opt. |

6. | H. Takahashi, S. Suzuki, and I. Nishi, “Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating,” IEEE J.Lightwave Technol. |

7. | Zimmermann. J, Kamp. M, Forchel. A, and März R, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. |

8. | A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, “Mach Zehnder interferometer employing coupled-resonator optical waveguides,” Opt.Lett. |

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

10. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. |

11. | C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. |

12. | Ziyang Zhang and Min Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt.Express |

13. | B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. |

14. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical investigation of channel drop tunneling processes,” Phys. Rev. B. |

15. | B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science , |

16. | H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. |

17. | Bong-Shik Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B. |

18. | Sangin Kim, Ikmo Park, Hanjo Lim, and Chul-Sik Kee, “Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback,” Opt. Express |

19. | Yoshihiro Akahane, Takashi Asano, Hitomichi Takano, Bong-Shik Song, Yoshinori Takana, and Susumu Noda, “Two-dimensional photonic-crystal-slab channel drop filter with flat-top response,” Opt. Express |

20. | Kyu H. Hwang and G. Hugh Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express |

21. | Akihiko Shinya, Satoshi Mitsugi, Eiichi Kuramochi, and Masaya Notomi, “Ultrasmall multi-channel resonant-tunneling filter using mode gap of width-tuned photonic-crystal waveguide,” Opt. Express |

22. | Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. |

23. | O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am B |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 17, 2006

Revised Manuscript: March 6, 2006

Manuscript Accepted: March 8, 2006

Published: March 20, 2006

**Citation**

Hongliang Ren, Chun Jiang, Weisheng Hu, Mingyi Gao, and Jingyuan Wang, "Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity," Opt. Express **14**, 2446-2458 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2446

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### References

- E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58,2059 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58,2486 (1987). [CrossRef] [PubMed]
- K. Sakoda, Optical properties of photonic crystals (NY: Springer-Verlag Berlin Heidelberg, New York, 2004.)
- StevenG. Johnson and J. D. Joannopoulos, "Designing synthetic optical media: photonic crystals," Acta Materialia. 51,5823 (2003). [CrossRef]
- A. Sharkawy, S. Shi, and D.W. Prather, "Multichannel wavelength division multiplexing using photonic crystals," Appl. Opt. 40,2247 (2001). [CrossRef]
- H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994). [CrossRef]
- Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004). [CrossRef]
- A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, "Mach Zehnder interferometer employing coupled-resonator optical waveguides," Opt.Lett. 28,405 (2003). [CrossRef] [PubMed]
- 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 (1998). [CrossRef]
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998). [CrossRef]
- C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999). [CrossRef]
- Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt.Express 13,2596 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596 [CrossRef] [PubMed]
- B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004). [CrossRef]
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999). [CrossRef]
- B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science, 300,1537 (2003). [CrossRef] [PubMed]
- H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004). [CrossRef]
- Bong-Shik Song, T. Asano, Y. Akahane, and S.Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B. 71,195101 (2005). [CrossRef]
- S. Kim, I. Park, H. Lim, and C.-S. Kee, "Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback," Opt. Express 12,5518 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5518 [CrossRef] [PubMed]
- Y. Akahane, Takashi Asano, Hitomichi Takano, Bong-Shik Song, Yoshinori Takana, and Susumu Noda, "Two-dimensional photonic-crystal-slab channel drop filter with flat-top response," Opt. Express 13,2512 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2512 [CrossRef] [PubMed]
- K. H. Hwang and G. . Song, "Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure," Opt. Express 13, 1948 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1948 [CrossRef] [PubMed]
- A. Shinya, S. Mitsugi, E. Kuramochi, and M. Notomi, "Ultrasmall multi-channel resonant-tunneling filter using mode gap of width-tuned photonic-crystal waveguide," Opt. Express 13,4202 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4202 [CrossRef] [PubMed]
- Y. Xu, Y. Li, R. K. Lee, and A. Yariv, "Sattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E. 62, 7389 (2000). [CrossRef]
- O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999). [CrossRef]

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