## Investigation of optical properties of circular spiral photonic crystals

Optics Express, Vol. 15, Issue 20, pp. 13236-13243 (2007)

http://dx.doi.org/10.1364/OE.15.013236

Acrobat PDF (1678 KB)

### Abstract

The photonic bandgap of three-dimensional photonic crystals, formed by arranging circular spirals in face-centre-cubic lattice, was theoretically investigated. The structure was found to have a relative photonic bandgap of up to 25% in both direct and inversed configurations. The conditions under which the structure has a bandgap larger than 10% are described. Some considerations for optimizing such photonic crystal fabrication by two-photon polymerization are given. The theoretical results are implemented to fabricate polymeric structures that can be used as templates for photonic crystals with full photonic bandgap larger than 10% centered in the near-infrared region.

© 2007 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]

5. C.T. Chan, K.M. Ho, and C.M. Soukoulis, “Photonic bandgaps in experimentally realizable periodic dielectric structures,” Europhys. Lett. **16**, 563 (1991) [CrossRef]

6. C.T. Chan, S. Datta, K.M. Ho, and C.M. Soukoulis, “A-7 structure: A family of photonic crystals,” Phys. Rev. B **50**, 1988 (1994) [CrossRef]

*n*=3.6) might have up to 30% relative complete gap (the ratio between the gap size and the mid gap frequency). Since the central wavelength (the wavelength in the center of the photonic bandgap) is approximately equal half of the crystal period, the structure must be fabricated with submicron resolution in order to shift the bandgap to the optical communication wavelengths. Despite some progress in the last years, realization of this particular structure with a submicron resolution is beyond present technological capabilities. Fortunately, the design can be simplified at the expense of the bandgap size. The <100> diamond like structure, also known as the woodpile structure, is a good example of a simplified architecture [7

7. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic bandgaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. **89**, 413 (1994). [CrossRef]

8. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science **289**, 604 (2000) [CrossRef] [PubMed]

9. K. Kaneko, H. B. Sun, X. M. Duan, and S. Kawata,“Submicron diamond-lattice photonic crystals produced by two-photon laser nanofabrication,” Appl. Phys. Lett. **83**, 2091 (2003). [CrossRef]

10. A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev B **57**, R2006 (1998) [CrossRef]

*C*, a diameter

*D*and is made of rods with a cross section width

*w*and length

*l*. The fcc configuration is achieved by shifting adjacent spirals in by half a period as they wind in the vertical <001> direction. The lateral lattice constant

*a*is twice the distance between adjacent spirals and the vertical lattice constant is the pitch length

*C*.

*n*=3.5) [10

10. A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev B **57**, R2006 (1998) [CrossRef]

*a*(here “

*a*” is the lateral lattice constant), and the spirals have a diameter of 0.32

*a*and a pitch of

*a*. The realization of this structure, is quite complex- mainly due to the half a period shift between the adjacent objects. As a result, despite its attractiveness, the structure received minor attention since its proposal.

11. K. K. Seet, V. Mizeikis, S. Matsuo, S. Juodkazis, and H. Misawa, “Three-Dimensional Spiral-Architecture Photonic Crystals Obtained By Direct Laser Writing,” Adv. Mater. **17**, No.5, 541 (2005) [CrossRef]

*a*= 3.6 μm), the spiral diameter of 2.7 μm (0.75

*a*) and a pitch of 3.6 μm (

*a*). The rods had elliptical cross section, which is an inherent feature of the fabrication technique (typically with an aspect ratio between 2 and 3). The structure did not have complete bandgap since its normalized parameters were significantly different from the ones suggested in ref [10

10. A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev B **57**, R2006 (1998) [CrossRef]

12. K. K. Seet, V. Mizeikis, S. Juodkazis, and H. Misawa, “Three-dimensional horizontal circular spiral photonic crystals with stop gaps below 1 *μ*m,” Appl. Phys. Lett. **88**, 221101 (2006) [CrossRef]

*a*-0.3

*a*). Here, the structure had a local pseudo gap in the Γ-Z direction. According to our calculations, no complete band gap could have been obtained with these structure proportions even if it would have been realized in silicon.

## 2. Band structure calculations

### 2.1. Methods

*a*, which is twice the distance between adjacent spirals. The band structures were calculated using the MIT’s Photonic Bands (MPB) tool that was developed by Johnson et al. [13

13. S. G. Johnson “MIT Photonic-Bands,” (Massachusetts Institute of Technology2002) http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands

14. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173 (2001) [CrossRef] [PubMed]

*n*=3.5. The ellipsoids, closely packed along the spiral track, produced a smooth structure. This way the realistic fabrication approach of 2PP process was imitated. Each structure was represented by its unit-cell in which a computational grid with resolution between 16 and 32 pixels/unit-cell-direction was defined. The effective dielectric tensor in each grid point was found by averaging over a mesh of 3 points. We used a convergence tolerance of 1.0e-7.

### 2.2 Results and discussion

*a*, length of 0.25

*a*and had a diameter of 0.4

*a*and a pitch length of

*a*, Fig. 2(a). In the parametric space the structure is represented by the 4 element vector [

*w*,

*l*,

*D*,

*C*] = [0.2

*a*0.25

*a*, 0.4

*a*,

*a*]. The structure has a fill factor of 20%, i.e. the dielectric material constitutes 20% of the total volume. It has a complete photonic bandgap between the second and the third bands with the relative width of 24.6%. The gap is centered at 0.58 normalized frequency (see Fig. 2(b)). The lower gap edge is located near the X direction, at (0.4, 0, 0) and the upper edge at R (0.5, 0.5, 0.5). In the inverted configuration, i.e. dielectric background, the maximal relative complete bandgap was found when [

*w*,

*l*,

*D*,

*C*] = [0.5

*a*, 0.55

*a*, 0.35

*a*,

*a*], corresponding to a dielectric fill factor of 22%. The relative complete photonic bandgap was 25.1%. The gap in this case is centered at 0.56 normalized frequency. Despite the differences between the optimal structure parameters reported here and the ones that were previously reported by ref [10

**57**, R2006 (1998) [CrossRef]

**57**, R2006 (1998) [CrossRef]

*w*,

*l*,

*D*,

*C*] = [0.17μm, 0.22μm, 0.35μm, 0.9μm].

20. S. Juodkazis1, V. Mizeikis1, K. Seet1, M. Miwa, and H. Misawa, “Two-photon lithography of nanorods in SU-8 photoresist,” Nanotechnology **16**, 846–849 (2005) [CrossRef]

21. D. Tan et al., “Reduction in feature size of two-photon polymerization using SCR500,” Appl Phys Lett **90**, 071106 (2007) [CrossRef]

*a*is the lattice constant). According to our calculations, each parametric combination in this range will reveal a photonic bandgap larger than 10%. Fig. 3(b) depicts the photonic bandgap map of the inverse spiral architecture. Fig. 3(c) presents the dependence of the photonic bandgap on the elongation factor. In order to minimize the elongation effect, the optimal configurations for each elongation factor were searched and presented in Fig. 3(d). In the case of the direct architecture, the optimal values of the rod’s width and spiral’s diameter (0.25

*a*and 0.4

*a*, respectively) are preserved for wide range of elongation factors. For refractive indexes of about 3 the photonic band gap center frequency can be approximated for the direct and inverse cases by Eq. (3) and Eq. (4), respectively.

## 3. Structure realization

### 3.1. Methods

*Micro Resist Technology*). The samples were set on a piezo stage with 3nm resolution and 100μm travel distance (Piezo Jena). The experimental setup is described in detail elsewhere [16

16. J. Serbin, A. Ovsianikov, and B. Chichkov, “Fabrication of woodpile structures by two-photon polymerization and investigation of their optical properties,” Opt. Express **12**, 5221 (2004) [CrossRef] [PubMed]

17. H. B. Sun, T. Suwa, K. Takada, R. P. Zaccaria, M. S. Kim, K. S. Lee, and S. Kawata, “Shape precompensation in two-photon nanowriting of photonic lattices,” Appl Phys Lett **85**, 3708 (2004) [CrossRef]

### 3.2 Results and discussion

*a*= 3μm), and each spiral has 1.2 μm diameter (0.4

*a*) and 3 μm pitch (

*a*) and they complete 6 vertical revolutions. The rods have width of 0.45 μm (0.15

*a*) and length of 0.9 μm (0.3

*a*). The polymeric structure satisfies the 10% complete gap boundary conditions that were presented above and can be used as a template for high refractive index material (a replication approach is described for example in ref [19

19. N. Tétreault, et al., “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. **18**, 457–460 (2006) [CrossRef]

## 4. Conclusion

## Reference 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. | Busch Kurt, Lölkes Stefan, Wehrspohn Ralf, and B. Föll Helmut, |

4. | Jean-Michel Lourtioz, Henri Benisty, Vincent Berger, Jean-Michel Gerard, Daniel Maystre, and Alexei Tchelnokov, |

5. | C.T. Chan, K.M. Ho, and C.M. Soukoulis, “Photonic bandgaps in experimentally realizable periodic dielectric structures,” Europhys. Lett. |

6. | C.T. Chan, S. Datta, K.M. Ho, and C.M. Soukoulis, “A-7 structure: A family of photonic crystals,” Phys. Rev. B |

7. | K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic bandgaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. |

8. | S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science |

9. | K. Kaneko, H. B. Sun, X. M. Duan, and S. Kawata,“Submicron diamond-lattice photonic crystals produced by two-photon laser nanofabrication,” Appl. Phys. Lett. |

10. | A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev B |

11. | K. K. Seet, V. Mizeikis, S. Matsuo, S. Juodkazis, and H. Misawa, “Three-Dimensional Spiral-Architecture Photonic Crystals Obtained By Direct Laser Writing,” Adv. Mater. |

12. | K. K. Seet, V. Mizeikis, S. Juodkazis, and H. Misawa, “Three-dimensional horizontal circular spiral photonic crystals with stop gaps below 1 |

13. | S. G. Johnson “MIT Photonic-Bands,” (Massachusetts Institute of Technology2002) http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands |

14. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

15. | H. B. Sun, S. Matsuo, and H. Misawa, “Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin,” Appl. Phys. Lett. |

16. | J. Serbin, A. Ovsianikov, and B. Chichkov, “Fabrication of woodpile structures by two-photon polymerization and investigation of their optical properties,” Opt. Express |

17. | H. B. Sun, T. Suwa, K. Takada, R. P. Zaccaria, M. S. Kim, K. S. Lee, and S. Kawata, “Shape precompensation in two-photon nanowriting of photonic lattices,” Appl Phys Lett |

18. | M Deubel, G. V. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. |

19. | N. Tétreault, et al., “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. |

20. | S. Juodkazis1, V. Mizeikis1, K. Seet1, M. Miwa, and H. Misawa, “Two-photon lithography of nanorods in SU-8 photoresist,” Nanotechnology |

21. | D. Tan et al., “Reduction in feature size of two-photon polymerization using SCR500,” Appl Phys Lett |

**OCIS Codes**

(140.3390) Lasers and laser optics : Laser materials processing

(220.4000) Optical design and fabrication : Microstructure fabrication

(230.5298) Optical devices : Photonic crystals

(160.5335) Materials : Photosensitive materials

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 15, 2007

Revised Manuscript: September 21, 2007

Manuscript Accepted: September 24, 2007

Published: September 27, 2007

**Citation**

Nir Grossman, Aleksandr Ovsianikov, Alexander Petrov, Manfred Eich, and Boris Chichkov, "Investigation of optical properties of circular spiral photonic crystals," Opt. Express **15**, 13236-13243 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13236

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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]
- B. Kurt, L. Stefan, W. Ralf, and B. Föll Helmut, Photonic Crystals (Wiley-VCH, Berlin, 2004).
- J.-M. Lourtioz, H.i Benisty, V. Berger, J.-M. Gerard, D. Maystre, A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer-Verlag Berlin and Heidelberg 2005).
- C. T. Chan, K. M. Ho and C. M. Soukoulis, "Photonic bandgaps in experimentally realizable periodic dielectric structures," Europhys. Lett. 16, 563 (1991). [CrossRef]
- C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, "A-7 structure: A family of photonic crystals," Phys. Rev. B 50, 1988 (1994). [CrossRef]
- K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic bandgaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413 (1994). [CrossRef]
- S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604 (2000). [CrossRef] [PubMed]
- K. Kaneko, H. B. Sun, X. M. Duan, and S. Kawata," Submicron diamond-lattice photonic crystals produced by two-photon laser nanofabrication," Appl. Phys. Lett. 83, 2091 (2003). [CrossRef]
- A. Chutinan, and S. Noda, "Spiral three-dimensional photonic-band-gap structure," Phys. Rev B 57, R2006 (1998). [CrossRef]
- K. K. Seet, V. Mizeikis, S. Matsuo, S. Juodkazis, and H. Misawa, "Three-Dimensional Spiral-Architecture Photonic Crystals Obtained By Direct Laser Writing," Adv. Mater. 17, 541 (2005). [CrossRef]
- K. K. Seet, V. Mizeikis, S. Juodkazis, and H. Misawa, "Three-dimensional horizontal circular spiral photonic crystals with stop gaps below 1 µm," Appl. Phys. Lett. 88, 221101 (2006). [CrossRef]
- S. G. Johnson "MIT Photonic-Bands," (Massachusetts Institute of Technology 2002), http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands>
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173 (2001). [CrossRef] [PubMed]
- H. B. Sun, S. Matsuo, and H. Misawa, "Three-dimensional photonic crystal structures achieved with two-photon-absorption photopolymerization of resin," Appl. Phys. Lett. 74, 786 (1999). [CrossRef]
- J. Serbin, A. Ovsianikov, and B. Chichkov, "Fabrication of woodpile structures by two-photon polymerization and investigation of their optical properties," Opt. Express 12, 5221 (2004). [CrossRef] [PubMed]
- H. B. Sun, T. Suwa, K. Takada, R. P. Zaccaria, M. S. Kim, K. S. Lee, S. Kawata, "Shape precompensation in two-photon nanowriting of photonic lattices," Appl. Phys. Lett. 85, 3708 (2004). [CrossRef]
- M. Deubel, G. V. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 3, 444 (2004). [CrossRef] [PubMed]
- N. Tétreault, et al., "New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates," Adv. Mater. 18, 457-460 (2006). [CrossRef]
- S. Juodkazis1, V. Mizeikis1, K. Seet1, M. Miwa, and H. Misawa, "Two-photon lithography of nanorods in SU-8 photoresist," Nanotechnology 16, 846-849 (2005). [CrossRef]
- D. Tan, et al., "Reduction in feature size of two-photon polymerization using SCR500," Appl Phys Lett 90, 071106 (2007). [CrossRef]

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