## Design of a high-*Q* channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure

Optics Express, Vol. 13, Issue 6, pp. 1948-1957 (2005)

http://dx.doi.org/10.1364/OPEX.13.001948

Acrobat PDF (241 KB)

### Abstract

A significant improvement on the basic design of a channel add-drop multiplexer of the in-plane type, based on the two-dimensional photonic-crystal membrane structure of triangular-lattice holes, has been made to increase the channel-selectivity *Q* factor as high as 7300, which demonstrates the viability of the original basic design. The three-dimensional finite-difference time-domain simulation shows that theoretically it is possible to design a channel add-drop multiplexer with better than -0.7 dB of the forward-drop insertion loss, -29 dB of the pass-through cross-talk at the center frequency. A revised coupled-mode analysis with an augmented directional coupling gives a good agreement between its parametric analysis and the finite-difference time-domain analysis in regard to the detailed asymmetric forward-drop frequency response of the multiplexer.

© 2005 Optical Society of America

## 1. Introduction

1. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960–963 (1998). [CrossRef]

2. K. H. Hwang, S. Kim, and G. H. Song, “Design of a photonic-crystal channel-drop filter based on the two-dimensional triangular-lattice hole structure,” in *Photonic Crystal Materials and Devices II*,
A. Adibi, A. Scherer, and S.-Y. Lin, eds., Proc. SPIE **5360**, 405–410 (2004). [CrossRef]

3. 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]

1. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960–963 (1998). [CrossRef]

*Q*factor due to power leak from out-of-plane radiation of the electromagnetic fields, which was not mentioned in those early papers [1

1. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. **80**, 960–963 (1998). [CrossRef]

5. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B **59**, 15882–15892 (1999). [CrossRef]

*Q*is one of the important conditions for a channel ADM with correspondingly high channel selectivity. Additionally, such a high

*Q*design should be compatible with the following basic design rules for such a device [5

5. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B **59**, 15882–15892 (1999). [CrossRef]

7. 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**, 1322–1331 (1999). [CrossRef]

**80**, 960–963 (1998). [CrossRef]

*condition of attuned degeneracy*.

*Q*

_{out}should be orders of magnitude greater than the in-plane

*Q*

_{in}, which we will call the

*condition of low-insertion loss*.

*Q*

_{even}and

*Q*

_{odd}should match, which will be shown later to be the

*phase-matching condition*.

3. 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]

## 2. Coupled-mode theory

*𝓐*

_{L}(

*t*) and

*𝓐*

_{R}(

*t*) represent the complex amplitudes for the resonators in the left and right sides, respectively, oscillating with the ‘positive’ frequency only as

*e*

^{-i2πνt}in response to the incident time-harmonic wave at frequency

*ν*. Whereas

*𝒮*

_{+j}(

*t*) [

*𝒮̂*

_{+j}(

*t*)] and

*𝒮*

_{-j}(

*t*) [

*𝒮̂*

_{-j}(

*t*)] represent the similarly oscillating amplitudes of the incoming and the outgoing waves, respectively, at the entrance-points [mid-points] of port

*j*of two cascaded four-port devices in the back-to-back configuration of Fig. 3. Then, those amplitudes at the two resonators will evolve in time according to a set of coupled-wave equations [7

7. 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**, 1322–1331 (1999). [CrossRef]

*ν*

_{0}is the resonance frequency of a single resonator in isolation, τ

_{out}is the decay time by loss mostly due to out-of-plane radiation, τ

_{bus}is the decay time constant for coupling into one of the two waveguide buses,

*µ*is the coefficient for coupling between the localized modes of the two resonators, and σ

_{1}and σ

_{2}are the coupling coefficients between the resonator and the waves propagating in the opposite directions along the bus. We take σ

_{1}=σ

_{2}≡σ for our configuration, while [8]

*𝒮*

_{+j}(

*t*)}, we may write

*β*is the propagation constant of the guided mode;

*δβ*is the newly introduced coefficient for lossless directional coupling between the two symmetric parallel waveguides; and

*L*is the half-length of the entire structure.

*S*

_{+2},

*S*

_{+3}, and

*S*

_{+4}will vanish. Each of reflection, transmission, backward-drop, and forward-drop coefficients is obtained by calculating the ratio between the outgoing wave at the respective port and the incoming wave at port 1;

*condition for attuned degeneracy*was given a new interpretation from the viewpoint of coupled-mode theory in Ref. [7

7. 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**, 1322–1331 (1999). [CrossRef]

*d*is the effective distance between the two symmetric resonators. Additionally, the so-called

*phase-matching condition*of [1

**80**, 960–963 (1998). [CrossRef]

*m*being a positive integer must be satisfied to achieve a maximized power transfer to the forward-drop port.

*Q*factor of a

*single*resonance mode is defined as the ratio of the stored energy

*U*and the energy leak dissipation per unit radian of oscillation;

*ν*is the width of the full-width half-maximum (FWHM) of the power spectrum in the case of a well-isolated single resonance mode. Below, we use Eq. (17) for our estimation of the channel selectivity in the channel ADM by picking up Δ

*ν*as the FWHM bandwidth from the channel-drop spectrum after performing the digital Fourier transform (DFT) of the FDTD data, although, strictly speaking, two resonance modes are involved in the latter spectrum.

## 3. Simulation and results

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

*a*for each of the electromagnetic field components in the 3D FDTD simulation with 60 discretized time steps per one cycle [10

10. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

11. G. H. Song, S. Kim, and K. H. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” J. Opt. Soc. Korea **6**, 79–85 (2002). [CrossRef]

*a*is so strong that the air regions of the width of only 2

*a*below and above the dielectric membrane before the PML of five grid points, as shown schematically in Fig. 4, appear to be sufficient for the analysis of resonators of

*Q*up to 100, 000.

11. G. H. Song, S. Kim, and K. H. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” J. Opt. Soc. Korea **6**, 79–85 (2002). [CrossRef]

*r*=0.300

*a*creates a relatively large bandgap in the frequency domain for the incident waves of the chosen TE-like polarization with the electric field vibrating on the plane of the membrane of thickness 0.75

*a*with the electric-permittivity contrast of

*ε*=11.56

*ε*

_{0}versus

*ε*

_{0}. The frequency responses in Eq. (13) of the PhC structure can be obtained by performing DFT on the output FDTD data from the Gaussian-enveloped input wave packet. Analysis on the data from a similar procedure with even narrower an envelope has given reliable numbers for the bandgap range of 0.244<

*aν/c*≡

*a*/λ<0.312, in the scale of normalized frequencies, with

*c*being the light speed in vacuum.

### 3.1. Line-defect waveguide buses

12. S. Johnson and J. D. Joannopoulos, *Photonic crystals; the road from theory to practice* (Kluwer Academic Publishers, Boston/Dordrecht/London, 2002). [PubMed]

*r*

_{s}=0.265

*a*opposed to the default radius of

*r*=0.300

*a*, for which one can refer to Fig. 1(a) for the illustration. With such modifications, the two single-mode frequency ranges are found from 0.247 to 0.261 and from 0.273 to 0.301, which are shown in Fig. 5(b) with yellow shadings.

*aν/c*=0.25338 with the corresponding propagation constant of ±17/52×2

*π/a*in the first Brillouin zone. Being in the lower range, the wave at the above frequency has been confirmed to be free from out-of-plane radiation by the 3D FDTD simulation.

### 3.2. The resonator system

*r*

_{m}, is set commonly to 0.267

*a*. The two edge holes of the single resonator, denoted by

*l*in Fig. 1(a), have been displaced from their original places by 0.15

*a*from the default positions in order to enhance the

*Q*factor of the resonator. All the

*Q*factors of the resonator modes are obtained according to Eq. (16) by plotting the decaying oscillation of the field at the center of the resonator which was let loose from the initial excitation.

*Q*factor of a single stick-shape resonator in the absence of other defect structures is measured to be 141,000, that of the same resonator in the presence of the two waveguide buses is measured to be merely 6,600. Since

*Q*

^{-1}=

*Q*

_{in}=7,100 and

*Q*

_{out}=93,700.

*d*=13

*a*for the distance between the centers of the two stick-shape resonators. The individual

*Q*factors of the even- and the odd-symmetric modes of the two symmetric stick-shape resonators in the presence of the waveguide buses are separately calculated by applying appropriate boundary conditions on the half of the aforementioned FDTD computational domain. A sizable, thus undesirable difference in the two

*Q*factors have been measured;

*Q*

_{even}=8,120 and Q

_{odd}=5,560 for the even- and the odd-symmetric modes, respectively. It is a yet-to-be-corrected compromise in the prototype design failing rule 3 in the list of design rules in Sec. 1. As was implied in [7

**35**, 1322–1331 (1999). [CrossRef]

### 3.3. Channel add-drop multiplexer

*aν*

_{0}/

*c*=0.25338, where the mutual coupling

*µ*between the two symmetric resonators are slightly off-tuned by the factor of 1.005 from the recipe value in Eq. (14) for a better FDTD result. The channel-selectivity

*Q*factor by Eq. (17), estimated from the DFT analysis with the data from the 3D FDTD simulation is found to be around 7,320.

13. G. H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A **11**, 2027–2037 (1994). [CrossRef]

*π*in Eq. (20) instead of the exact eight and a half times

*π*from Eq. (15). It was once reported that the separation of the two center frequencies of the even- and the odd-symmetric resonance modes [14

14. J. Romero-Vivas, D. N. Chigrin, A. V. Lavrinenko, and C. M. S. Torres, “Resonant add-drop filter based on a photonic quasicrystal,” Opt. Express **13**, 826–835 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-826. [CrossRef] [PubMed]

*Q*

_{even}and 0.25338009/

*Q*

_{odd}. In such a circumstance, the phase matching condition of Eq. (15) has been sacrificed. We thus believe that the aforementioned phase mismatch is the primary reason for the sidelobes in the spectral response of the backward-drop port.

*Q*factor of the resonance system.

## 4. Conclusion

*Q*of 7,300 with -0.7 dB of the forward-drop power and the -29 dB of the pass-through crosstalk at the center of the filtering frequency.

*Q*number for a PhC channel-drop filter. With the on-going development in modern nano-fabrication technology [3

3. 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]

## Acknowledgments

## References and links

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

2. | K. H. Hwang, S. Kim, and G. H. Song, “Design of a photonic-crystal channel-drop filter based on the two-dimensional triangular-lattice hole structure,” in |

3. | 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. |

4. | K. H. Hwang and G. H. Song, “Design of a two-dimensional photonic-crystal channel-drop filter based on the triangular-lattice holes on the slab structure,” in |

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

6. | H. A. Haus, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, private communications (2002). |

7. | 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. |

8. | H. A. Haus, |

9. | S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. |

10. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

11. | G. H. Song, S. Kim, and K. H. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” J. Opt. Soc. Korea |

12. | S. Johnson and J. D. Joannopoulos, |

13. | G. H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A |

14. | J. Romero-Vivas, D. N. Chigrin, A. V. Lavrinenko, and C. M. S. Torres, “Resonant add-drop filter based on a photonic quasicrystal,” Opt. Express |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.3990) Optical devices : Micro-optical devices

(230.7400) Optical devices : Waveguides, slab

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 24, 2005

Revised Manuscript: March 6, 2005

Published: March 21, 2005

**Citation**

Kyu 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**, 1948-1957 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1948

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### References

- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, �??Channel drop tunneling through localized states,�?? Phys. Rev. Lett. 80, 960�??963 (1998). [CrossRef]
- K. H. Hwang, S. Kim, and G. H. Song, �??Design of a photonic-crystal channel-drop filter based on the two-dimensional triangular-lattice hole structure,�?? in Photonic Crystal Materials and Devices II, A. Adibi, A. Scherer, and S.-Y. Lin, eds., Proc. SPIE 5360, 405�??410 (2004). [CrossRef]
- 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]
- K. H. Hwang and G. H. Song, �??Design of a two-dimensional photonic-crystal channel-drop filter based on the triangular-lattice holes on the slab structure,�?? in Proc. 30th European Conference on Optical Communication (Stockholm, Sweden, 2004) 5, 76�??77.
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, �??Theoretical analysis of channel drop tunneling processes,�?? Phys. Rev. B 59, 15882�??15892 (1999). [CrossRef]
- H. A. Haus, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, private communications (2002).
- 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�??1331 (1999). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985).
- S. D. Gedney, �??An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,�?? IEEE Trans. Antennas Propag. 44, 1630�??1639 (1996). [CrossRef]
- K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propag. 14, 302�??307 (1966). [CrossRef]
- G. H. Song, S. Kim, and K. H. Hwang, �??FDTD simulation of photonic-crystal lasers and their relaxation oscillation,�?? J. Opt. Soc. Korea 6, 79�??85 (2002). [CrossRef]
- S. Johnson and J. D. Joannopoulos, Photonic crystals; the road from theory to practice (Kluwer Academic Publishers, Boston/Dordrecht/London, 2002). [PubMed]
- G. H. Song, �??Theory of symmetry in optical filter responses,�?? J. Opt. Soc. Am. A 11, 2027�??2037 (1994). [CrossRef]
- J. Romero-Vivas, D. N. Chigrin, A. V. Lavrinenko, and C. M. S. Torres, �??Resonant add-drop filter based on a photonic quasicrystal,�?? Opt. Express 13, 826�??835 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-826.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-826.</a> [CrossRef] [PubMed]

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