## Design of ultra-compact metallo-dielectric photonic crystal filters

Optics Express, Vol. 13, Issue 16, pp. 6175-6180 (2005)

http://dx.doi.org/10.1364/OPEX.13.006175

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### Abstract

Filter characteristics of metallo-dielectric photonic crystal slabs are analyzed using the Multiple Multipole Program combined with the Model-Based Parameter Estimation technique. This approach takes losses and material dispersion into account and provides highly accurate results at short computation time. Starting from this analysis, different ultra-compact band pass filters for telecommunication wavelengths are designed. The filters consist of five silver wires embedded in a waveguide structure. By applying stochastic and deterministic techniques the filter structures are optimized to obtain the desired characteristics.

© 2005 Optical Society of America

## 1. Introduction

1. Ch. Hafner, Cui Xudong, and R. Vahldiek, “Metallic Photonic Crystals at Optical Frequency,” J. Comp. Theor. Nanoscience **2**, No.2, 240–250 (2005). [CrossRef]

2. O. Takayama and M. Cada, “Two-dimensional hexagonal metallic photonic crystals embedded in anodic porous alumina for optical wavelengths,” Appl. Phys. Lett. , **Vol. 85**, No. 8, pp.1311–1313 (2004). [CrossRef]

5. P. B. John and R. W. Christie, Phys. Rev. B **6**, 4370 (1972). [CrossRef]

6. A. S. Jugessur, P. Pottier, and R. M. De La Rue, “Engineering the filter response of photonic crystal
microcavity filters,” Opt. Express **12**, 1304–1312 (2004),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1304. [CrossRef] [PubMed]

7. W. Nakagawa, Pang-Chen Sun, Chyong-Hua Chen, and Y. Fainman, “Wide-field-of -view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. **27**, 191–193 (2002). [CrossRef]

8. C. Ciminelli, F. Peluso, and M. N. Armenise, “Modeling and Design of Two-Dimensional Guided-Wave
photonic band-gap devices, ” J. Lightwave. Technol. **23**, 886–901 (2005). [CrossRef]

7. W. Nakagawa, Pang-Chen Sun, Chyong-Hua Chen, and Y. Fainman, “Wide-field-of -view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. **27**, 191–193 (2002). [CrossRef]

9. R. Costa, A. Melloni, and M. Martinelli, “Bandpass Resonant filters in Photonic-Crystal Waveguides”, IEEE Photo. Tech. Lett. **15**, 401–403 (2003). [CrossRef]

10. J. Smajic, Ch. Hafner, and D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express **11**, 566–571 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-566. [CrossRef] [PubMed]

*x*direction and consists of a few layers of rods or holes in

*y*direction [6

6. A. S. Jugessur, P. Pottier, and R. M. De La Rue, “Engineering the filter response of photonic crystal
microcavity filters,” Opt. Express **12**, 1304–1312 (2004),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1304. [CrossRef] [PubMed]

7. W. Nakagawa, Pang-Chen Sun, Chyong-Hua Chen, and Y. Fainman, “Wide-field-of -view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. **27**, 191–193 (2002). [CrossRef]

8. C. Ciminelli, F. Peluso, and M. N. Armenise, “Modeling and Design of Two-Dimensional Guided-Wave
photonic band-gap devices, ” J. Lightwave. Technol. **23**, 886–901 (2005). [CrossRef]

## 2. Metallic Photonic Crystal Analysis

*a*. It is important to note that such band diagrams are only valid for a single lattice constant

*a*because of the frequency dependence of the material properties (permittivity) and that the standard band diagrams do not contain any information on the material losses.

*T*(ω), similar to a higher order bandpass filter. The number of peaks with high transmission increases with the number of MPhC layers. In order to obtain a band pass filter one therefore might design a single MPhC in such a way that the band pass area is simply the frequency range of the first mode between the Γ and X points, i.e., one essentially designs a PhC operating between two band gaps. For obtaining a band pass around 1.55μm wavelength, i.e., 193.55THz, one could reduce the radii of the silver rods or reduce the lattice constant. However, from Fig. 1(b) one can see that such a band pass filter shows not a very good performance. In order to improve the filter performance, numerical optimizers are applied in the following.

## 3. Deterministic filter optimization

*N*strongly interacting 1-layer slabs. This may also be considered as a single

*N*-layer slab with arbitrary radii to be optimized. For reasons of simplicity and in order to keep the computation time short, only

*N*=5 is considered in the following. Increasing the number of layers is similar to increasing the filter order in classical filter design: With a relatively low order, no high-performance filter can be achieved. At first sight, one might therefore conclude that

*N*=5 is far from being enough for obtaining a reasonable band pass filter with desired center frequency and band width. It will be demonstrated below that the filters with

*N*=5 layers can be optimized in such a way that 1) the desired center frequency is obtained precisely, 2) the insertion loss is reasonably small, 3) and that the bandwidth may be modified within some limits.

*N*PhC layers but also the locations of the rods or even the shape and material properties of the rods. This increases the search space and offers the chance of finding better or more compact filters at the price of longer computation time. However, for a first exploratory study it is reasonable to start with low numbers of model parameters to be optimized. Therefore, we assume that the rods are not displaced and have circular cross sections, i.e., we only optimize

*N*=5 radii

*r*. Assuming symmetry of the structure, i.e.,

_{i}*r*=

_{4}*r*and

_{2}*r*=

_{5}*r*, further simplifies the optimization, that is the optimization is carried out in the 3-dimensional parameter space

_{1}*r*, where

_{i}*i*=1,2,3. Starting with a reasonable initial guess – obtained from the analysis of various 5 layer structures – the optimization procedure presented in [13

13. J. Smajic, Ch. Hafner, and D. Erni, “Optimiztion of photonic crystal structures,” J. Opt. Soc. Am. A **21**, 2223–2232 (2004). [CrossRef]

## 4. Stochastic optimization

## 5. Dielectric background – MDPhC filters

*N*and the search space are big enough. The dielectric background material essentially allows one to reduce the size of the filter structure. At the same time, the contrast of the permittivity |

*ϵ*|/|

_{metal}*ϵ*| is reduced which in turn can deteriorate the filter performance.

_{background}*500nm*because the wavelength in such a dielectric medium is shorter than in free space. Thus, the size of this filter is shorter than the MPhC filter considered before. One can see that the width of the pass band is increased. This has two reasons: First, it is more difficult to obtain narrow band pass filters when the contrast of the permittivity is reduced as mentioned above. Second, the fitness definition of the optimization procedure affects not only the center frequency but also the bandwidth. Here, the fitness definition widens the bandwidth, i.e., the goal of the optimization was not to find a minimum bandwidth solution. Note that such wide-band solutions are difficult to obtain with the concept of a cavity embedded between two PhC layers acting as mirrors.

## 6. Summary

## Acknowledgments

## References and links

1. | Ch. Hafner, Cui Xudong, and R. Vahldiek, “Metallic Photonic Crystals at Optical Frequency,” J. Comp. Theor. Nanoscience |

2. | O. Takayama and M. Cada, “Two-dimensional hexagonal metallic photonic crystals embedded in anodic porous alumina for optical wavelengths,” Appl. Phys. Lett. , |

3. | Ch. Hafner, “Drude Model Replacement by Symbolic Regression,” J. Comp. Theor. Nanoscience |

4. | |

5. | P. B. John and R. W. Christie, Phys. Rev. B |

6. | A. S. Jugessur, P. Pottier, and R. M. De La Rue, “Engineering the filter response of photonic crystal
microcavity filters,” Opt. Express |

7. | W. Nakagawa, Pang-Chen Sun, Chyong-Hua Chen, and Y. Fainman, “Wide-field-of -view narrow-band spectral filters based on photonic crystal nanocavities,” Opt. Lett. |

8. | C. Ciminelli, F. Peluso, and M. N. Armenise, “Modeling and Design of Two-Dimensional Guided-Wave
photonic band-gap devices, ” J. Lightwave. Technol. |

9. | R. Costa, A. Melloni, and M. Martinelli, “Bandpass Resonant filters in Photonic-Crystal Waveguides”, IEEE Photo. Tech. Lett. |

10. | J. Smajic, Ch. Hafner, and D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express |

11. | Ch. Hafner, J. Smajic, and D. Erni, “Simulation and Optimization of Composite Doped Metamaterials,” Chapter in M. Riedt and W. Schommers, “Handbook of Theoretical and Computational Nanotechnology,” American Scientific Publishers (2005). |

12. | E. Miller, “Model-Based Parameter Estimation in Electromagnetics,” IEEE AP |

13. | J. Smajic, Ch. Hafner, and D. Erni, “Optimiztion of photonic crystal structures,” J. Opt. Soc. Am. A |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.7400) Optical devices : Waveguides, slab

(350.2450) Other areas of optics : Filters, absorption

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 28, 2005

Revised Manuscript: July 28, 2005

Published: August 8, 2005

**Citation**

Cui Xudong, Christian Hafner, and Rüdiger Vahldieck, "Design of ultra-compact metallo-dielectric photonic crystal filters," Opt. Express **13**, 6175-6180 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6175

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### References

- Ch. Hafner, Cui Xudong, R. Vahldiek, �??Metallic Photonic Crystals at Optical Frequency,�?? J. Comp. Theor. Nanoscience 2, No.2, 240-250 (2005). [CrossRef]
- O. Takayama, and M. Cada, "Two-dimensional hexagonal metallic photonic crystals embedded in anodic porous alumina for optical wavelengths," Appl. Phys. Lett., Vol. 85, No. 8, pp. 1311-1313 (2004). [CrossRef]
- Ch. Hafner, �??Drude Model Replacement by Symbolic Regression,�?? J. Comp. Theor. Nanoscience 2, 88-98 (2005).
- <a href="http://alphard.ethz.ch/hafner/MaX/max1.htm">http://alphard.ethz.ch/hafner/MaX/max1.htm</a>.
- P. B. John, R. W. Christie, Phys. Rev. B 6, 4370 (1972). [CrossRef]
- A. S. Jugessur, P. Pottier and R. M. De La Rue, �??Engineering the filter response of photonic crystal microcavity filters,�?? Opt. Express 12, 1304-1312 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1304">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1304</a>. [CrossRef] [PubMed]
- W. Nakagawa, Pang-Chen Sun, Chyong-Hua Chen, and Y. Fainman, �??Wide-field-of �??view narrow-band spectral filters based on photonic crystal nanocavities,�?? Opt. Lett. 27, 191-193 (2002). [CrossRef]
- C. Ciminelli, F. Peluso, M. N. Armenise, �??Modeling and Design of Two-Dimensional Guided-Wave photonic band-gap devices,�?? J. Lightwave. Technol. 23, 886-901 (2005). [CrossRef]
- R. Costa, A. Melloni, M. Martinelli, �??Bandpass Resonant filters in Photonic-Crystal Waveguides�??, IEEE Photo. Tech. Lett. 15, 401-403 (2003). [CrossRef]
- J. Smajic, Ch. Hafner, and D. Erni, "On the design of photonic crystal multiplexers," Opt. Express 11, 566-571 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-566">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-566</a>. [CrossRef] [PubMed]
- Ch. Hafner, J. Smajic, D. Erni, "Simulation and Optimization of Composite Doped Metamaterials," Chapter in M. Riedt, W. Schommers, Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers (2005).
- E. Miller, "Model-Based Parameter Estimation in Electromagnetics,�?? IEEE AP Vol. 40, No.1 (1998).
- J. Smajic, Ch. Hafner, D. Erni, �??Optimiztion of photonic crystal structures,�?? J. Opt. Soc. Am. A 21, 2223-2232 (2004). [CrossRef]

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