Translation of interference pattern by phase shift for diamond photonic crystals

We demonstrate the construction of diamond photonic crystal structures by the translation of a multi-beam interference pattern. Using phase shift of each beam, the double-exposed interference patterns can be aligned in the [111] direction for a face-centered cubic (FCC) and [210] direction for a body-centered cubic (BCC), respectively, producing diamond D from FCC and BCC-diamond like structure from BCC. The present result shows that the complete bandgap has been retained with slight deviation from ideal diamond symmetry. ©2005 Optical Society of America OCIS codes: (090.0090) Holography; (260.3160) Interference; (220.4000) Microstructure fabrication References and links 1. S. H. Park, D. Qin, and Y. Xia, "Crystallization of mesoscale particles over large areas," Adv. Mater. 10, 1028-1032 (1998). 2. G. M. Gratson, M. J. Xu, and J. A. Lewis, "Microporiodis structures Direct writing of three-dimensional webs," Nature 428, 386-386 (2004). 3. M. Deubel, G. Von 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, 444-447 (2004). 4. S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, "Finer features for functional microdevices Micromachines can be created with higher resolution using two-photon absorption," Nature 412, 697-698 (2001). 5. S. Noda, N. Yamamoto, and A. Sasaki, "New realization method for three-dimensional photonic crystal in optical wavelength region," Jpn. J. Appl. Phys. 35 (7B), L909-L912 (1996). 6. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000). 7. M. Maldovan and E. L. Thomas, "Diamond-structured photonic crystals," Nat. Mater. 3 (9), 593-600 (2004). 8. 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). 9. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band-gaps in 3-dimensions New layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994). 10. S. H. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of 3-dimensional photonic crystals at submicron length scales," Appl. Phys. Lett. 65, 1466-1468 (1994). 11. O. Toader and S. John, "Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals," Science 292, 1133-1135 (2001). 12. M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, "Photonic properties of bicontinuous cubic microphases," Phys. Rev. B 65, 165123 (2002). (C) 2005 OSA 28 November 2005 / Vol. 13, No. 24 / OPTICS EXPRESS 9841 #9138 $15.00 USD Received 24 October 2005; revised 16 November 2005; accepted 16 November 2005 13. C. K. Ullal, M. Maldovan, M. Wohlgemuth, and E. L. Thomas, "Triply periodic bicontinuous structures through interference lithography: a level-set approach," J. Opt. Soc. Am. A 20, 948-954 (2003). 14. D. N. Sharp, A. J. Turberfield, and R. G. Denning, "Holographic photonic crystals with diamond symmetry," Phys. Rev. B 68, 205102 (2003). 15. O. Toader, T. Y. M. Chan, and S. John, "Photonic band gap architectures for holographic lithography," Phys. Rev. Lett. 92 (4), 043905 (2004). 16. M. Maldovan, C. K. Ullal, W. C. Carter, and E. L. Thomas, "Exploring for 3D photonic bandgap structures in the 11 f.c.c. space groups," Nat. Mater. 2 (10), 664-667 (2003). 17. J. H. Moon, S.-M. Yang, D. J. Pine, and W.-S. Chang, "Multiple-exposure holographic lithography with phase shift," Appl. Phys. Lett. 85, 4184-4186 (2004). 18. A. Chelnokov, S. Rowson, J. M. Lourtioz, V. Berger, and J. Y. Courtois, "An optical drill for the fabrication of photonic crystals," J. Opt. A-Pure Appl. Op. 1 (5), L3-L6 (1999). 19. J. Qi, M. E. Sousa, A. K. Fontecchio, and G. P. Crawford, "Temporally multiplexed holographic polymerdispersed liquid crystals," Appl. Phys. Lett. 82, 1652-1654 (2003). 20. S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, "Creating periodic threedimensional structures by multibeam interference of visible laser," Chem. Mater. 14, 2831-2833 (2002). 21. J. H. Moon, A. Small, G.-R. Yi, S.-K. Lee, W.-S. Chang, D. J. Pine, and S.-M. Yang, "Patterned polymer photonic crystals using soft lithography and holographic lithography," Synth. Met. 148, 99-102 (2005). 22. M. J. Escuti, J. Qi, and G. P. Crawford, "Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals," Opt. Lett. 28 (7), 522-524 (2003). 23. N. Tereault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Perez-Willard, S. John, M. Wegener, and G. A. Ozin, "New route to three-dimensional photonic bandgap materials: silicon double inversion of polymer templates.," Adv. Mater. in press (2005).


Introduction
Photonic crystals (PCs) are called a semiconductor of light and can be used to confine, manipulate and guide photons, which are desired for the next-generation optical circuits.Recently, multibeam interference lithography has been developed and holds promise in the PC fabrication over colloidal assembly, 1 direct writing, [2][3][4] and multi-layer process 5 because it produces 3D structure with controlled lattice and unit atoms in a large area by single exposure and is compatible with conventional photolithography. 6o date, among the proposed 3D PC structures, diamond networks have been known as the most optimized structure for having a large and complete photonic bandgaps (PBGs).Moreover, the PBG is robust enough to allow some structural modification within diamond symmetry, 7 This makes diamond structures especially attractive over other symmetries since it allows application of a wide range of approaches to fabricate them conveniently, including Yablonobite by direct drilling, 8 woodpile (<100>-diamond) by layer-by-layer fabrication, 9 <110>-diamond by layer-by-layer followed by reactive ion etching, 10 spiral diamond made by GLAD. 113][14][15] All of these structures possess a PBG with larger than 20% between the 2nd and 3rd bands for a refractive index of n=3.6, close to the largest yet calculated. 16n this letter, we demonstrate the translation of 3D interference patterns to construct two types of diamond structures.Since the diamond symmetry can be alternatively achieved by two simple FCC or BCC sublattices, displaced along [111] direction or [210] direction, respectively, the superposition of the interference pattern and the shifted pattern could approximate the corresponding diamond structures.Specifically, we demonstrate phaseinduced translation of simple FCC and BCC interference pattern.Then, we investigate the PBGs of two superimposed interference patterns of FCC and BCC, displaced in aforementioned directions.In practice, this method can be achieved by using double-exposure holographic lithography with phase shift to produce polymeric structures. 17,18The doubleexposure with phase shift is preferred over multiplexed holography because of the alignment process. 19Compared to the previously reported results based on single-exposure of four-beam interference by controlling the polarization of each beam, 13,14 our method provide much improved contrast of the diamond interference.The high pattern contrast is crucial for practical fabrication of photoresist structures with clearly defined and completely opened pores. 20,21Furthermore, this method can be flexible by simply changing the intensity of each exposure, resulting in F43m from FCC interference pattern, which has two different PBGs. 16

Translation of FCC or BCC interference patterns by phase shift
To construct 3D structures, four-beam interference has been widely used.The intensity distribution of the interference field of four mutually coherent laser beams can be described by Fourier superposition and can be written as ( ) ( ) where E n , ε n , k n and φ n are the amplitude, the unit polarization vector, the wave vector, and the phase of the nth beam, respectively.Here, we control the phase of each beam (i.e., relative phase difference) to shift the interference pattern in a certain direction.In experiment, the phase shift can be produced by phase retarders such as wave-plates, 17 Kerr cells and Pockels cells.When the relative phase difference in Eq. ( 1) is changed by ( ) Meanwhile, since the interference term translated by ′ r in space can be described by, The relationship between the spatial translation in a certain direction and the phase shift can be obtained by comparing Eqs. ( 2) and ( 3) and given by ( Then, the phase shift of each beam, which induces the translation of the interference pattern, can be determined by Eq. ( 4) of each interference term.First, we consider the translation of the simple FCC interference pattern with Fm3m space group. 22This structure has been demonstrated with four beams: one set of two coplanar beams interfere with another set whose plane is perpendicular to the first.This requires a substrate that is transparent at the light's wavelength and a prism to compensate for the refraction at the air-photoresist interface. 22When the initial phase of each beam is set at zero in Eq. ( 2), the interference pattern can be expressed as: where d is the lattice constant.In order to translate the interference pattern in [111] direction by βd/4π as shown in Fig. 1(a), phase differences are calculated from Eq. ( 4) and given by For BCC interference pattern, the beam configurations for m3m symmetry of the unit atoms.aredescribed as:

Superimposition of interference patterns with phase shift
With these pattern shifts, the double-exposed interference patterns can be described by the superposition of all exposed patterns.Figs.Remarkably, overall exposures make a local connection between the atoms formed in each exposure.For the FCC, the superposition of interference patterns with the translation by 0.25d describes the level surface of diamond D structure (see Fig. 2(e)) and given by It is noteworthy that this equation contains 8 terms instead of 4 terms reported in the literature in describing the same type of diamond D. These additional terms provide higher contrast of the interference pattern. 7,14Considering the intensity profile along [111] direction (see Fig. 3), the contrast defined by difference between maximum and minimum intensities is Meanwhile, instead of diamond D, this method can simply construct F43m , with the different intensity of each exposure, which possesses dual complete PBGs. 14n Figs.4(a), 4(b) and 4(c), the position of two BCC lattices (red for the initial pattern and blue for the shifted one), described by Eq. ( 6) are shown for β = 0.40π, 0.50π and 1.16π, respectively.From Eq. ( 6), the level surface of this structure can be expressed by

Bandgap of superimposed FCC and BCC interference patterns
In Fig. 5, we calculate the PBG of the double-exposed FCC as a function of the degree of phase shift.The polymeric structure by interference lithography can be used as a template to deposit high refractive index materials. 23The PCs with a refractive index of 3.45 produced The bandgap width has a local maximum at a translation of 0.25d(1,1,1) with diamond D symmetry defined in Eq. ( 6).They show a complete PBG with the gap-midgap ratio as large as 25%, which is comparable to the previous results. 12Meanwhile, the complete PBG appears when the degree of the pattern translation is varied from 0.18 to 0.30.The relatively large PBG is retained within the deviation from diamond D (See Fig 3, these structures possess the PBG width above 10%).Therefore, this method is feasible even with the experimental error during translation.Likewise the FCC case, we calculate PBG of BCC-based structure as shown in Fig. 5.The optimized volume fraction for a maximum PBG is found 0.20 for the normalized pattern shift of 0.25, and the optimal volume fraction is increased to 0.30 for the upper and lower bounds of the phase shift in Fig. 5.The bandgap width reaches a maximum, at which the gap- midgap ratio is 22%, for a translation of 0.25d(2,1,0) with BCC diamondlike structure defined in Eq. ( 8). 15

Conclusion
In summary, we propose a double exposure technique that translates interference patterns by phase shift.The superposition of two interference patterns constructs partially overlapped structures, whereas the two unit atoms in the basis are connected with each other.We demonstrate that translation in both the [111] direction for FCC and the [210] direction for BCC, respectively, can produce diamond D and BCC diamond-like structure, respectively.Such structures show a large complete PBG between 2nd and 3rd bands, in agreement with the literature results.The large bandgap is retained within the accepted variation in the degree of translation.Compared to the single-exposed diamond D structures, the present doubleexposed structure has a better contrast of the interference pattern, which is necessary to their practical realization in interference lithography.
2(a), 2(b) and 2(c) show the relative position of two FCC lattices (red for the initial pattern and blue for the shifted one), described by Eq. (5) for β = 0.80π, 1.00π and 1.16π, respectively.Then, the translations are 0.20d, 0.25d and 0.29d, all in [111] direction.The level surfaces of the double-exposed FCC interference patterns are reproduced in Figs.2(d)-2(f) for their respective cases.

Fig. 4 .
Fig. 4. The relative position of the BCC interference pattern (red) and the shifted pattern (blue) in the [210] direction by (a) 0.20d(1,1,1), (b) 0.25d(1,1,1), and (c) 0.29d(1,1,1).The level surface of each double-exposed BCC interference pattern with a filling fraction of 0.20 is shown in (d -f) for their respective cases.The level surfaces of the double-exposed BCC interference patterns are reproduced in Figs.4(d)-4(f) for their respective cases.For the BCC-based structure, shift of interference pattern into [210] by 0.25d constructs a BCC diamond-like structure with diamond D symmetry in the body-centered tetragonal unit cell (see Fig 4(e)).7 From Eq. (6), the level surface of this structure can be expressed by 2005 OSA 28 November 2005 / Vol. 13, No. 24 / OPTICS EXPRESS 9845 from FCC-based structure show a complete PBG between the 2nd and 3rd bands.The volume fraction is optimized to achieve a maximum PBG at each point, ranging from 0.22 to 0.26.

Fig. 5 .
Fig. 5. Complete PBG of double-exposed (a) FCC and (b) BCC interference patterns as a function of the normalized pattern shift in [111] and [210], respectively.