## Multi-step multi-beam laser interference patterning of three-dimensional photonic lattices

Optics Express, Vol. 14, Issue 6, pp. 2309-2316 (2006)

http://dx.doi.org/10.1364/OE.14.002309

Acrobat PDF (703 KB)

### Abstract

We present laser interference patterning of three-dimensional photonic lattice structures with three-step three-beam irradiation. In contrast to one-step four-beam interference patterning, the proposed method makes it possible to continuously tune the lattice constant and the photonic band gap without distortion of the lattice shape. We analytically show that all fourteen Bravais lattices are possible to be produced by choosing proper incident vectors of laser beams. A simple routine to seek the geometrical configuration of the incident beams for producing arbitrary Bravais lattices is shown. Furthermore, We experimentally demonstrate the fabrication of three-dimensional photonic lattices in the photoresist SU-8. Significant photonic band gap effects have been observed from the well-defined photonic lattices.

© 2006 Optical Society of America

## 1. Introduction

1. J. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) **386**, 143–149 (1997). [CrossRef]

2. L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. **27**, 900–902 (2002). [CrossRef]

3. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographiclithography,” Nature (London) **404**, 53–56 (2000). [CrossRef]

4. S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. **76**, 2668–2670 (2000). [CrossRef]

5. D. N. Sharp, A. J. Turberfield, and R. G. Denning, “Holographic photonic crystals with diamond symmetry,” Phys. Rev. B **68**, 205102 (2003). [CrossRef]

6. S. Shoji, H.-B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. **83**, 608–610 (2003). [CrossRef]

7. G. J. Schneider, E. D. Wetzel, J. A. Murakowski, and D. W. Prather, “Fabrication of three-dimensional Yablonovite photonic crystals by multiple-exposure UV interference lithography,” Proc. SPIE **5720**, 9 (2005). [CrossRef]

4. S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. **76**, 2668–2670 (2000). [CrossRef]

6. S. Shoji, H.-B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. **83**, 608–610 (2003). [CrossRef]

## 2. Theoretical

9. L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. **26**, 1858–1860 (2001). [CrossRef]

*k*is the wave vectors of

_{ij}*j*th beams in

*i*th irradiation,

*k*is the wave number

*k*= |

*k*| = 2π/λ, and θ is the inclination angle of the laser beams from the plane normal to rods. In the same way, other two groups of three laser beams for the rest two rod arrays in (0,1,0) and (1,0,0) can be easily found by rotation operation as;

_{ij}**k**

_{21}= (

*k*sin θ/√2,

*k*cosθ,

*k*sinθ/√2),

**k**

_{22}= (-

*k*sinθ/√2,

*k*cos θ,

*k*sinθ/√2),

**k**

_{23}= (

**k**sinθ/√2,

*k*cosθ,-

*k*sinθ/√2), and

**k**

_{31}= (

*k*cosθ,

*k*sinθ/√2,

*k*sinθ/√2),

**k**

_{32}=(

*k*cosθ,-

*k*sinθ/√2,

*k*sinθ/√2),

**k**

_{33}= (

*k*cosθ,

*k*sinθ/√2,-

*k*sinθ/√2).

**E**

_{ij}, ϕ

_{ij}are amplitude and phase delay of the electric field, and the maximum light intensity is at (

**k**

_{ij}-

**k**

_{ij'})

**r**= 2

*n*π(

*n*is an integer number). Specifically, the differences

**k**

_{ij}-

**k**

_{ij'}determines the reciprocal lattice vectors of the resultant light pattern. If all the laser beams have the same intensity and phase delay, the rod arrays intersect each other on the points, which are ordered in three-dimensional simple square lattice. Fig. 1 shows the calculated light intensity distribution formed by three-step irradiation three-beam laser interference with the aforementioned beam geometries, which gives scaffold shaped simple-cubic lattice. The lattice constant (i.e., the distance between the nearest neighbor rods)

*d*= λ/√2sinθ is freely chosen by θ without distortion of the lattice element and lattice symmetry.

*u*is integer number and

_{i}**a**

_{i}is the primitive translation vector which defines the lattice type. The most simple lattice is simple-cubic, in which three primitive translation vectors are given as

**a**

_{01}= (a,0,0),

**a**

_{02}= (0,a,0),

**a**

_{03}= (0,0,a), where

*a*is the lattice constant. Since in all kinds of Bravais lattices the position vectors of lattice points are expressed by linear combination of three translation components, any other types of Bravais lattice are expressed by the linear transformation of simple-cubic lattice. Namely, the operator matrix for the linear transformation can be defined as,

**A**and

**A**

_{0}are matrices of three translation vectors for the desired lattice and simplecubic lattice, respectively. As one example, now we discuss about body-centered cubic (bcc) lattice. The primitive translation vectors for bcc lattice are (

*a*/2,

*a*/2,-

*a*/2), (

*a*/2,-

*a*/2,

*a*/2), and (-

*a*/2,

*a*/2,

*a*/2). Therefore, from equation (5) the operator matrix Δ is given as follows,

2. L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. **27**, 900–902 (2002). [CrossRef]

10. L. Yuan, G. P. Wang, and X. Huang, “Arrangements of four beams for any Bravais lattice,” Opt. Lett. **28**, 1769–1771 (2003). [CrossRef] [PubMed]

12. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case
Employing Nonspherical Atoms,” Phys. Rev. Lett. **67**, 2295–2298 (1991). [CrossRef] [PubMed]

## 3. Experimental demonstration

11. Yu. V. Miklyaev, D. C. Meisel, A. Blanco, G. von Freymann, K. Busch, W. Koch, C. Enkrich, M. Deubel, and M. Wegener, “Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations,” Appl. Phys. Lett. **82**, 1284 (2003). [CrossRef]

12. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case
Employing Nonspherical Atoms,” Phys. Rev. Lett. **67**, 2295–2298 (1991). [CrossRef] [PubMed]

^{2}, and the exposure time was 1/60 second for one irradiation.

^{-1}and 3500 cm

^{-1}in both of transmittance and reflectance are attributed to the absorption of the SU-8 polymer. We observed another broad dip in transmission and a simultaneous reflection peak at around 4300 cm

^{-1}, which corresponds to the wavelength of 2.3 μm. The lattice periodicity in the < 1,1,1 > direction in the fabricated structure is about 0.8 μm. The refractive index of SU-8 is 1.57 at near infrared region. By judging the lattice constant and the diameter of the rods the filling fraction and the mean refractive index of the structure is estimated as 80 % and 1.46, respectively. From these value the wavelength of photonic band gap in the < 1,1,1 > direction is approximately estimated as 2.33 μm. We also calculated the photonic band diagram by three-dimensional plane wave approximation method[12

12. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case
Employing Nonspherical Atoms,” Phys. Rev. Lett. **67**, 2295–2298 (1991). [CrossRef] [PubMed]

*L*3 is equivalent to the < 1,1,1 > direction. Since the refractive index of SU-8 polymer is low there is no absolute photonic band gap in the structure, and in the Γ-

*L*3 direction the gap opens at a frequency of 4500 cm

^{-1}. This diagram also shows a good agreement with our experimental results. We concluded that the enhancement of reflection and depression in transmission around 4300 cm

^{-1}appeared by Bragg diffraction of light by the lattice structure.

## 4. Conclusion

## References and links

1. | J. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) |

2. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. |

3. | M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographiclithography,” Nature (London) |

4. | S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. |

5. | D. N. Sharp, A. J. Turberfield, and R. G. Denning, “Holographic photonic crystals with diamond symmetry,” Phys. Rev. B |

6. | S. Shoji, H.-B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. |

7. | G. J. Schneider, E. D. Wetzel, J. A. Murakowski, and D. W. Prather, “Fabrication of three-dimensional Yablonovite photonic crystals by multiple-exposure UV interference lithography,” Proc. SPIE |

8. | S. Shoji, H.-B. Sun, and S. Kawata, “Multi-Beam Interference Laser Fabrication of an Inverse Structure of Yablonovite Photonic Crystal,” Technical Digest of International Symposium on Photonic and Electromagnetic Crystal Structures V (PECS-V), 34 (2004). |

9. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of a microfiber bundle by interference of three noncoplanar beams,” Opt. Lett. |

10. | L. Yuan, G. P. Wang, and X. Huang, “Arrangements of four beams for any Bravais lattice,” Opt. Lett. |

11. | Yu. V. Miklyaev, D. C. Meisel, A. Blanco, G. von Freymann, K. Busch, W. Koch, C. Enkrich, M. Deubel, and M. Wegener, “Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations,” Appl. Phys. Lett. |

12. | E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case
Employing Nonspherical Atoms,” Phys. Rev. Lett. |

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(220.4000) Optical design and fabrication : Microstructure fabrication

(220.4610) Optical design and fabrication : Optical fabrication

(230.0250) Optical devices : Optoelectronics

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: January 17, 2006

Revised Manuscript: March 7, 2006

Manuscript Accepted: March 13, 2006

Published: March 20, 2006

**Citation**

Satoru Shoji, Remo Zaccaria, Hong-Bo Sun, and Satoshi Kawata, "Multi-step multi-beam laser interference patterning of three-dimensional photonic lattices," Opt. Express **14**, 2309-2316 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2309

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### References

- J. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997). [CrossRef]
- L. Z. Cai, X. L. Yang, and Y. R. Wang, "All fourteen Bravais lattices can be formed by interference of four noncoplanar beams," Opt. Lett. 27, 900-902 (2002). [CrossRef]
- M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographiclithography," Nature (London) 404, 53-56 (2000). [CrossRef]
- S. Shoji and S. Kawata, "Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin," Appl. Phys. Lett. 76, 2668-2670 (2000). [CrossRef]
- D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003). [CrossRef]
- S. Shoji, H.-B. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003). [CrossRef]
- G. J. Schneider, E. D. Wetzel, J. A. Murakowski, and D. W. Prather, "Fabrication of three-dimensional Yablonovite photonic crystals by multiple-exposure UV interference lithography," Proc. SPIE 5720, 9 (2005). [CrossRef]
- S. Shoji, H.-B. Sun, and S. Kawata, "Multi-Beam Interference Laser Fabrication of an Inverse Structure of Yablonovite Photonic Crystal," Technical Digest of International Symposium on Photonic and Electromagnetic Crystal Structures V (PECS-V), 34 (2004).
- L. Z. Cai, X. L. Yang, and Y. R. Wang, "Formation of a microfiber bundle by interference of three noncoplanar beams," Opt. Lett. 26, 1858-1860 (2001). [CrossRef]
- L. Yuan, G. P. Wang, and X. Huang, "Arrangements of four beams for any Bravais lattice," Opt. Lett. 28, 1769-1771 (2003). [CrossRef] [PubMed]
- Yu. V. Miklyaev, D. C. Meisel, A. Blanco, G. von Freymann, K. Busch, W. Koch, C. Enkrich, M. Deubel, and M. Wegener, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication, optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284 (2003). [CrossRef]
- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic Band Structure: The Face-Centered-Cubic Case Employing Nonspherical Atoms," Phys. Rev. Lett. 67, 2295-2298 (1991). [CrossRef] [PubMed]

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