## Abstract

From the view of crystallography, a systematic theoretical study on one-step formation of two-dimensional compound photonic lattices by four noncoplanar elliptical waves is presented. A general formula for the interference intensity of *N* elliptically polarized waves, and relevant phase shifts that compensate for the initial phases and control the relative position and size of the motifs, have been deduced. Using appropriate polarization configurations, four kinds of beam geometries can be used to form various compound lattices. This provides an ideal new experimental platform for fabricating large-area compound photonic lattices.

© 2005 Optical Society of America

## 1. Introduction

Photonic crystals, which are periodically ordered dielectric or metallic microstructures [1,2], can be used to control the propagation of electromagnetic waves. The efficient fabrication of large-area visible optical or infrared photonic crystals has been a major challenge to material science over the last decade. Compared to semiconductor lithography [3,4] and self-assembly techniques [5] by which achievement of long-range order in large-scale has been difficult, holographic lithography (HL) can be used to quickly fabricate large-area photonic structures with high precision, such as photonic crystals [6,7] and quasi-crystals [8]. Moreover, by this approach, it is possible to discuss relatively complete crystallography on the submicrometre scale, since HL can provide more multitudinous structures than the atom scale which is restricted to chemical bonds.

It is known that the periodical lattices can be divided into the Bravais lattices and the compound lattices. Up to now, one-step three-beam and four-beam HL can create all two-dimensional (2D) Bravais lattices and three-dimensional (3D) Bravais lattices, respectively, but nearly impossible for the corresponding compound lattices; it suggests that an additional beam be needed. Meanwhile, it is generally regarded that the HL is difficult in fabrication of photonic lattices with periodical defects, most of which actually can be regarded as the compound structures. Therefore, systematic researches on the formation of compound lattices are inevitable. As a particular part of crystallography, formation of compound photonic lattices by HL has gradually gained interest [9–11]. For example, a kind of diamond-like structure by four-beam HL has been reported [9], but how to form most of 3D compound lattices is unclear. Multiple-exposure technique [10,11], which requires the sophisticated alignment of period and relative shift, has also been used to fabricate the compound photonic lattices, however the resultant pattern is far from the ideal structures [10]. It implies an approach of one-step HL adapted to the compound lattice instead of the simple lattice is necessary.

In this research, as a fundamental work for holographic compound lattices, we propose formation principles of 2D compound photonic lattices by one-step HL with four noncoplanar beams. By use of superposition of different simple lattices and control of the phase shifts introduced by elliptical polarization, various 2D compound lattices can be constructed.

## 2. Formation of 2D compound photonic lattices

To form 2D compound photonic lattices by one-step HL, there are two main clues. One is based on the traditional choices of **G**
_{ij} (=**k**
_{i}-**k**
_{j}, where **k**s are wave vectors) within a unit cell in reciprocal space; another clue relies on the choices of **G**
_{ij} vectors unlimited to a unit cell, so numerous choices can be made. In both cases, compensation for the initial phases and control of the relative size and position of the motifs require the introduction of elliptical polarization.

#### 2.1. Interference intensity of N elliptically polarized waves

Let *E*_{aj}*, δ*_{j}
, and **e**
_{aj} denote the *j*th wave’s amplitude, initial phase and unit polarization vector in the major axis, respectively; and let *E*_{bj}
and **e**
_{bj} be the amplitude and unit polarization vector in the minor axis, respectively. Since there is a phase difference of π/2 between *E*_{aj}
and *E*_{bj}
[12], the intensity equation for the interference of *N* elliptically polarized monochromatic plane waves can be given by

$$={{\sum _{j}{E}_{\mathit{aj}}}^{2}+\sum _{j}{E}_{\mathit{bj}}}^{2}+\sum _{i<j}2{{E}_{\mathit{ij}}}^{2}\mathrm{cos}[({\mathbf{k}}_{j}-{\mathbf{k}}_{i})\xb7\mathbf{r}+{\delta}_{j}-{\delta}_{i}-{t}_{\mathit{shift}\_\mathit{ij}}]$$

$$i,j=1,2,3,\cdots ,N\phantom{\rule{.1em}{0ex}},$$

where

and *θ*_{ij}
is the angle between the polarization directions of the *i*th and the *j*th waves. There are, in total, ${C}_{N}^{\mathit{2}}$
interference terms in the above equation. In the following, only the case *N*=4 which provides the minimum number of beams required to produce 2D compound lattices will be considered.

From Eq. (1), it is clear that a phase shift *t*_{shift _ ij}
introduced by the elliptical polarization can be used to compensate the effect of the undetermined initial phases. Moreover, the phase shift can control the relative position and size of the motifs between simple lattices. Of all the combination of the initial phases of *N* beams, only (*N*-1) are independent; however, all *C*^{2}_{N} of the phase shifts are relatively independent to each other. This is the reason why only the phase shifts, instead of two independent combinations of the initial phases, can reshape the motifs in the interference pattern of three beams [13]; when the number of beams is added up to four, there is only one additional independent combination of initial phases, which can be easily compensated by the phase shifts. Therefore, for the convenience of discussion, we can specify the initial phases in the following examples.

#### 2.2. Traditional choices of G_{ij}

In the case of the traditional choices of **G**
_{ij} vectors, two kinds of compound lattices for the primitive rectangular lattice and the square lattice can be obtained.

Four beams (e.g., ${\mathbf{k}}_{1}=\sqrt{2}\pi \u2044\lambda [\mathrm{cos}{\phi}_{1},\mathrm{sin}{\phi}_{1},1]$, ${\mathbf{k}}_{2}=\sqrt{2}\pi \u2044\lambda [-\mathrm{cos}{\phi}_{1},\mathrm{sin}{\phi}_{1},1]$, ${\mathbf{k}}_{3}=\sqrt{2}\pi \u2044\lambda [-\mathrm{cos}{\phi}_{1},-\mathrm{sin}{\phi}_{1},1]$, and ${\mathbf{k}}_{4}=\sqrt{2}\pi \u2044\lambda [\mathrm{cos}{\phi}_{1},-\mathrm{sin}{\phi}_{1},1]$) are chosen to yield a compound rectangular p-lattice. The beam configuration is shown in Fig. 1 (a). By appropriate polarization configuration and specified values of the threshold and the initial phases, the resultant pattern with a relatively large contrast is illustrated in Fig. 1 (b). After optimizing the polarizations and specifying the initial phases, the 3D structure of the square compound lattice as shown in Fig. 1 (c) can be obtained when *φ*
_{1}=*π*/4. It should be pointed out that, for these special **G**
_{ij} configurations, since two combinations of the initial phases in the diagonals can be directly deduced from two combinations in the two parallel directions, the relationship of the initial phases leads to the similar extremum conditions as that of three beams. However, the motifs in the lattices are not always centrosymmetric because of the positional influence of another lattice determined by the diagonal **G**s. For simplicity, suppose four beams are linearly polarized, their electric vectors are configured in the same projected directions on the xy-plane as their own **k**s and have the same magnitude *E*_{a}
. For the parallel **G**
_{12} and **G**
_{43} vectors, a relevant factor in the interference term can be easily deduced,

A similar expression can be obtained for the parallel **G**
_{32} and **G**
_{41} vectors,

From Eqs. (2) and (3), we can obtain two combinations of the initial phases (*δ*
_{3}-*δ*
_{1})and 2 4 (*δ*
_{2}-*δ*
_{4}) in the two diagonals. Obviously, the constant terms, which are determined by the initial phases in both Eqs. (2) and (3), can influence the amplitudes of the interference terms, consequently changing the distributed intensity and reshaping the motifs. It is not difficult to see that this result is also correct for the case of different intensity beams with elliptical polarizations.

We can comprehend the formation principles from the classes of **G**
_{ij} vectors. In the case of three beams, only one class of **G**
_{ij} vectors appears. When the number of beams is four, another class of **G**
_{ij} vectors along the diagonal emerges; two classes of **G**
_{ij} vectors for five beams and three classes for six beams, etc. In fact, the compound lattice can be constructed from two different simple lattices determined by two classes of **G**
_{ij} vectors in consideration of the relative intensity and position between two simple lattices. The relative intensity can be adjusted by an appropriate polarization configuration and the constant terms introduced by the initial phases; the relative position can be controlled by the initial phase combinations and the phase shifts introduced by elliptical polarization. For examples, unit cells for two compound lattices with different relative size and position of the motifs are shown in Fig. 1 (d).

#### 2.3. G_{ij} vector configurations under new choices

There are three kinds of **G**
_{ij} vector configurations under new choices: types of trapezium, arrow, and special tetragon.

### 2.3.1. Trapezium-type configuration

In the trapezium-type configuration, the formed lattices have property of orthogonality, and the compound lattices for the hexagonal, square, and rectangular p- and c-lattices can be obtained.

Fig. 2(a) shows a hexagonal compound lattice, the corresponding choices of the **G**
_{ij} vectors and the determination of corresponding **k**
_{j} vectors projected onto the xy-plane are schematically illustrated in Fig. 2(b). The compound lattice can be considered as the combination of four different simple lattices. Since there exists an additional independent combination of the initial phases, in general, a deviation of intensity maximum between different lattices will lead to inconsistent superposition of motifs, resulting in the construction of a compound lattice. Similarly, the phase shifts introduced by the elliptical polarizations can cause change in the relative size and shape of the motifs (see the inset of Fig. 2(b)). Another example from the trapezium-type **G**
_{ij} vector choice (Fig. 2(d)) is shown in Fig. 2(c), a compound rectangular p-lattice emerges in the resultant pattern.

### 2.3.2. Arrow-type configuration

The compound lattices for the hexagonal and special rectangular c-lattices can be obtained. Fig. 3(a) shows a new hexagonal compound lattice when the initial phases are specified and two linear polarizations at both ends of the arrow are perpendicular to each other. It is shown that the compound lattice contains three simple hexagonal lattices. From the choice method of the **G**
_{ij} vectors (Fig. 3(b)), it is known that four different simple lattices can be formed by the combination of any three beams. For simplicity of comprehension, the compound lattice can be approximately regarded as the superposition of two mainly different simple lattices as shown in the set of Fig. 3(b). Either an additional independent combination of the initial phases or the phase shifts introduced by the elliptical polarizations, can cause the superposition deviation of motifs between different simple lattices, consequently controlling the relative size and position of the motifs in the compound lattice according to the intensity pattern-superposition method [13]. Let *β* be half an obtuse angle of the rhomb, if *β* satisfies cos^{2}
*β*=1 2*m*, where *m* is an integer, the interference pattern reveals a centered rectangular compound lattice. The **G**
_{ij} vector choices in Fig. 3(d) are used to create the compound rectangular c-lattice as shown in Fig. 3(c).

### 2.3.3. Special tetragon-type configuration

The compound lattices for the hexagonal, square, and rectangular p- and c-lattices can be obtained. Fig. 4(a) shows a new square compound lattice realized by four linearly polarized beams when the initial phases are specified. The **G**
_{ij} vector choices and the determination of corresponding **k**
_{j} vectors projected onto the xy-plane are illustrated in Fig. 4(b). When beam 2 is elliptically polarized, the relative size of the motifs is changed (see the inset of Fig. 4(b)). It is shown clearly each unit cell contains four “atoms.” In fact, the higher the magnitude ratio of the **G**
_{ij} vectors is, usually, the more “atoms” each unit cell contains.

## 3. Conclusion

The formation principles of 2D compound photonic lattices by single step holographic lithography have been proposed. By superposition of different simple lattices, combined with the intensity pattern-superposition method, various 2D compound lattices can be constructed. The relative position and size of the motifs can be controlled by the phase shifts, which can be directly derived from a general formula for the spatial intensity distribution of *N* elliptically polarized waves. This analysis may lay the foundation for fabrication of 3D compound photonic lattices.

## Acknowledgments

This research is supported by the National Natural Science Foundation of China (grants 10274108), 973 (2003CB314901, and 2004CB719804) and 863 (2003AA311022) of China, and the Natural Science Foundation of Guangdong Province of China.

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