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We propose a holographic design of five-beam symmetric umbrella configuration, where there are a central beam and four ambient beams symmetrically scattered around the central one with the same apex angle, for fabrication of three-dimensional photonic crystals with tetragonal or cubic symmetries, and systematically analyzed the band gap properties of resultant photonic crystals when the apex angle is continuously increased. Our calculations reveal that large complete photonic band gaps exist in a wide range of apex angle for a relatively low refractive index contrast. Specifically, the face-centered cubic structure with a relative band gap of 25.1% for ε = 11.9 can be obtained with this recording geometry conveniently where all the beams are incident from the same half-space. These results will provide us with more understanding of this important recording geometry and give guidelines to its use in experiments.
©2006 Optical Society of America
1. Introduction
Photonic crystals (PhCs) are structures in which the dielectric constant is periodically modulated on a length scale comparable to the desired wavelength of operation [1, 2], and the resultant photonic dispersion may exhibit photonic band gaps (PBGs) which are useful in controlling light behavior. In the last decade much attention has been attracted to the fabrication of PhCs with complete PBGs. Various techniques such as the electron-beam lithography, self-assembly, multiphoton polymerization, and holographic lithography (HL) have been proposed and demonstrated with different levels of success [3–7]. Among them the method of HL has some unique features such as inexpensive volume recording.
In HL a desired geometrical structure is formed by multibeam interference with single or multiple exposures [8–11]. We have shown that all 14 three-dimensional (3D) Bravais lattices can be produced this way [12], and different beam designs will result in different PBG properties of the resultant structures [13, 14]. For example, the face-centered-cubic (fcc) lattice can be obtained by the interference of one central beam and three ambient beams symmetrically scattered around the former [15, 16], but the structure made in this geometry has only quite a narrow PBG [16, 17]. An alternative beam design was proposed to fabricate fcc lattice with a large complete PBG, but it requires four beams incident from two opposite surfaces of a sample [18–20], making it difficult to realize in practice. In addition, in real experiments the directions of interference beams may be slightly deviated from their theoretical values for many practical reasons [16]. Therefore a more extensive investigation into the effect of beam angle derivation on the resulting structures and then on the PBGs will be helpful for a clearer understanding of the structure formation with this geometry and for the control of experimental parameters. Recently, a research group fabricated some tetragonal and cubic PhCs with the use of phase masks and analyzed the PBGs of these structures; however, what was employed in their calculations is the woodpile model instead of the real holographic structure [21], and the two cases are usually not the same since in HL the actual structures including the cell’s shape and size and consequently the corresponding PBG are strongly dependent on the specific beam design and light intensity threshold selection.
In this paper we propose a five-beam symmetric umbrella configuration which is more convenient to make 3D PhCs with large complete PBGs. For instance, an fcc lattice with 25.1% relative band gap for a dielectric constant contrast 11.9:1 can be obtained with all the beams incident from the same side of the sample. Furthermore, a systematical study on the structure and band gap evolution for a continuously varying apex angle in this geometry is provided. These analyses and results may serve as guidelines in both theory and practice.
2. Recording geometry and resultant structures
The recording geometry with five-beam symmetric umbrella configuration we proposed here is shown in Fig. 1, where the central beam (C-beam) with wave vector K c is set along the z direction, while the four ambient beams (A-beams) of wave vectors K 1 to K 4 are in the plane yoz or xoz, respectively, with the same apex angle θ with z axis. The five wave vectors can be expressed as functions of the apex angle,
This geometry can be realized conveniently with the use of a diffraction beam splitter (DBS) where a zero-order diffracted beam and four symmetric first-order diffracted beams are employed. For polarization, we choose the central beam to be circularly polarized and all the four A-beams linearly polarized. The corresponding unit polarization vectors are
If we use the same amplitude EA for all the A-beams and the amplitude EC for C-beam, and adopt double-exposure approach with the C-beam, beams 1 and 2 open for the first exposure while the C-beam, beams 3 and 4 open for the second exposure, the two exposures have the same exposure time, the total intensity will be
To get the best contrast we should maximize the ratio of √2EC EA /(2+4), and this requirement leads to the optimized beam amplitude ratio EC /EA = √2. For brevity we may write the relative intensity of the spatially varying part in Eq. (3) as
A common approach in HL is the use of a negative photoresist material such as SU-8. In this case the “underexposed” region can be selectively removed using a developer substance which leaves the “overexposed” region intact. The developed photoresist is then infiltrated with SiO2 and burned away, leaving a daughter inverse template. Finally the daughter template is inverted by high temperature infiltration with silicon [20]. In the following analysis we will consider this case. If we denote the light intensity threshold in Eq. (4) as I t and assume that the region of ΔI > I t is filled with a material of high refractive index while the other region is air, we may obtain a periodic microstructure whose concrete shape is determined by the specific apex angle θ and I t.
In general the lattice structures resulting from Eq. (3) or Eq. (4) are tetragonal symmetric structures. The continuous increase of apex angle θ leads to continuous variation of primitive vectors, reciprocal vectors and the irreducible Brillouin zone of the resultant structure. Our calculations show that the Brillouin zone changes from a small tetragonal cake spreading out on the xy plane when θ is very small to a long tetragonal pillar along the z axis when θ is close to 180° (see Fig. 2). When the apex angle θ reaches 70.53° (corresponding to c/a = √2, where a is the period of the interference pattern along x or y direction and c is that in z direction), the structure has fcc symmetry, similar to the diamond structure [22]. If the θ is near 70.53°, a lattice with face-centered-tetragonal (fct) symmetry is obtained. Figure 3 (a) and 3(b) show the real fcc structure and its primitive cell constructed by five-beam symmetric umbrella configuration when θ = 70.53°, which obviously differ from the rhombohedral structure with fcc symmetry and its primitive cell [Fig. 3(c) and 3(d)] formed by four-beam symmetric umbrella configuration [12] when the apex angle is 38.94°. If the apex angle achieves the value of θ = 90° (corresponding to c/a = 1), a body-centered-cubic (bcc) lattice is obtained. When the value of θ is near 90°, the structure has body-centered-tetragonal (bct) symmetry.
Fig. 2. Evolution of irreducible Brillouin zone (Movie, 155K). The irreducible Brillouin zones of tetragonal structures for θ = 80° (a), θ = 90° (b), and θ = 100° (c).
Fig. 3. (a). The real fcc structure formed by five-beam symmetric umbrella configuration when θ = 70.53° and I t = 1.39 in Eq. (4), the corresponding filling ration is 21.7%; (b) the primitive cell of the fcc structure shown in (a); (c) the rhombohedral structure with fcc symmetry fabricated by four-beam symmetric umbrella configuration when the apex angle is 38.94°; and (d) the primitive cell of the structure shown in (c).
3. Band gap calculations
The plane-wave expansion method [23, 24] has been used to study PBG properties of the structures of this kind and search for the corresponding optimum volume filling ratio f yielding maximum relative PBG for each apex angleθ. Calculations reveal that full photonic band gaps exist over a very wide range of apex angle with a relatively low refractive index contrast needed to open them. Figure 4 represents the relative gap sizes of optimized structures for the apex angle range of 50° < θ < 115° with a dielectric constant contrast of 11.9 to 1 corresponding to silicon in air. From Fig. 4 we can see clearly that there are complete PBGs above 10 % in the range of 52° < θ < 112°, and even larger PBGs above 20% for 59° < θ < 92°. The maximum relative gap size of 25.1% appears at θ = 70.53° for the fcc structure, and the relative gap size for the bcc structure formed when θ = 90° is 21.3%. As an example, the band structure of the fcc lattice is given in Fig. 5, obviously a large complete PBG, from 0.330ωa/2πc to 0.425ωa/2πc (where c is the light velocity), exists between the second and third bands. Comparing the band gap result with that of woodpile structure fabricated by woodpile model [21], one can find two results have similar trend that the band gap size is a function of c/a defined in last section or θ, and the fcc structure has the biggest gap size.
Fig. 4. Relative band gap of optimized structures as a function of apex angle for 50°< θ < 115° when ε = 11.9. The solid symbols are the date for fcc and bcc structures when θ = 70.53° and θ = 90°, respectively. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained.
Fig. 5. Photonic band structure for the fcc structure with θ = 70.53°. The position of the high symmetry points together with the irreducible Brillouin zone are shown in the inset.
The effect of the dielectric refractive index n of the structure formed by this five-beam symmetric umbrella configuration on the band gap size has also been investigated. As a special case, in Fig. 6, we illustrate the variation of relative band gaps with different filling ratios and different refractive index contrasts for the fcc symmetric structure formed when θ = 70.53°. In this case the minimum refractive index required to open a complete photonic band gap is slightly less than 1.95, lower than 2.05 obtained in one work of band gap calculation for the woodpile model as the lowest requirement to open the gap [25]. When the refractive index is 3.6 and the filling ratio is 21.5%, the relative band gap reaches as high as 27.3%. In addition, it is worth noting that the range of filling ratio yielding a complete band gap for each given refractive index is fairly wide, especially for the high refractive index. For example, when the dielectric refractive index is 3.4 or 3.6, a complete relative band gap larger than 10% can be obtained over a wide range of filling ratio, from 10% to 53%.
Fig. 6. Variation of relative band gaps with different filling ratios for different refractive index contrasts for the fcc symmetric structure with the apex angle θ = 70.53°.
4. Conclusions
In summary, we have designed a five-beam symmetric umbrella configuration for the fabrication of 3D photonic crystals using holographic lithography method. The photonic structures with different symmetries, including face-centered-tetragonal, face-centered-cubic, body-centered-tetragonal and body-centered-cubic, can be produced with this geometry of different apex angles. The theoretical analysis indicates that complete band gaps exist for the structures formed this way over a very wide range of apex angle. Particularly, the fcc and bcc symmetric structures can be obtained when θ = 70.53° and 90°, respectively; and their corresponding relative band gap sizes are as high as 25.1% and 21.3%, respectively, for a dielectric constant contrast 11.9:1. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained, respectively. Furthermore, the dielectric constant contrast or equally the refractive index contrast of the structures of this kind required to open complete PBGs is quite low (n ≥ 1.95), and the filling ratio range for a certain structure to assure complete PBGs is fairly large. These discussions give us more understanding of this configuration and its advantages in applications. For example, we can fabricate an fcc lattice with a large complete PBG with all the interference beams arranged in the same half-space of the sample, which is more convenient compared with the geometry where the beams are incident from two opposite sides; the requirement for an exact theoretical apex angle may be relaxed to a certain extent in practical fabrication from the viewpoint of PBG formation; and the filling ratio or equivalently the light threshold selection may also be done more easily. We believe that these results are helpful in both theory and practice of HL.
Acknowledgment
This work is supported by the National Natural Science Foundation (64077005) and the Doctoral Program Foundation of Ministry of Education (20020422047), China.
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