## Abstract

In holographic fabrication of photonic crystals the shape and size of the dielectric columns or particles (“atoms”) are determined by the isointensity surfaces of the interference field. Therefore their photonic band gap (PBG) properties are closely related to their fabrication design. As an example, we have investigated the PBGs of a kind of holographically formed two-dimensional (2-D) square lattice with pincushion columns rotated by 45°, and shown that this structure has complete PBGs in a wide range of dielectric contrast comparable to or even larger than those of the same lattice with square columns reported before. The optical design for making this structure is also given. This work may demonstrate the unique feature and advantages of photonic crystals made by holographic method and provide a guideline for their design and experimental fabrication.

© 2005 Optical Society of America

## 1. Introduction

Photonic crystals (PhCs) have attracted much interest for its potential in controlling and manipulating the propagation of light due to the existence of photonic band gap (PBG) [1, 2]. Although three-dimensional (3-D) PhCs seem more versatile, two-dimensional (2-D) PhCs have also been extensively studied, since the latter are usually easier to fabricate and may be employed in many applications [3–10]. The important task of band gap engineering for 2-D PhCs is to find proper structures having PBG for both TE (or *p*) and TM (or *s*) polarization modes and design methods to produce them.

A complete PBG in two dimensions was first demonstrated for a triangular lattice of air columns in dielectric material [3, 4] and for a square lattice of air columns [4, 5]. Since then an extensive investigation has been made in this area [6–10]. It is found that generally a PBG for s polarization is favored in the case of dielectric columns, while a PBG for *p* polarization is favored if the dielectric regions are connected; and that the PBG property of a lattice is also dependent on the shape and size of the columns. Therefore it is not easy to give simple arguments predicting the existence of a complete PBG for a given lattice. Usually the trial and error method has to be used to find “good” structures [11, 12]. An interesting example is a 2-D so-called chessboard lattice, a square lattice with square columns rotated by 45°, which may give quite large a complete PBG for a relatively low dielectric contrast from theoretical simulations, for instance, Δ*ω/ω*=8.5% for *ε*=8.9 (alumina) [12]. This is an encouraging result considering that the traditional square lattice of air columns with square cross section requires *ε*>12.3 for a complete PBG to exist [12].

For the fabrication of PhCs, a variety of techniques have been proposed. Among them the holographic method [13, 14] has its unique advantages such as one-step recording and the ability of obtaining inverse lattice by using a template [15]. Since in this method the shape and size of the lattice columns (in 2-D case) or “atoms” (in 3-D case) are actually determined by the isointensity surfaces of the interference field, the PBG property of resultant structure is closely related to the concrete fabrication process as we have indicated recently [16, 17]. Often the columns or atoms formed this way are of no regular shape like circular and square columns or spheres studied before. What effect may this fact have on the PhC’s band gap? Can the holographic method provide PhCs with as large a PBG as that of regular-shape columns specially designed? How should we design the interference beams to produce the desired structure? In this work we will discuss these questions with an example of the holographic counterpart of the 2-D chessboard lattice mentioned above [12].

## 2. Band gap analysis: relation between PBG and concrete structure

To produce a 2-D chessboard-like lattice holographically, we may use a light intensity distribution

where *a* is the lattice constant, *c* is a positive constant related to the shape of the dielectric columns resulted from this equation. If we choose an intensity threshold *I _{t}* and wash away the part of

*I*<

*I*, we can get a certain lattice. Furthermore, by filling this structure with a material of high refractive index and then removing the template, an inverse structure of high dielectric contrast will be obtained. Considering that theoretically the threshold intensity

_{t}*I*may be chosen arbitrarily, the constant 2 in Eq. (1) is of no importance. Actually we may use other value of it to get the same structure by changing

_{t}*I*. Therefore we will employ Eq. (1) for PBG analysis below.

_{t}Obviously, for the special case of *c*=0 and *I _{t}*=0, we can get a chessboard lattice with filling ratio (FR)

*f*=0.5. This structure can produce a complete band gap for a range of dielectric constant

*ε*>7 [12]. But more comprehensive study shows that this is not the optimized case for generating large PBG. In holographic approach there are two adjustable parameters,

*c*in Eq. (1) and

*I*in fabrication process. Both of them have critical influences on PBG and therefore give us more freedom to improve band gap property. In the following we will make a systematic investigation on their effects. In the calculations we use plane wave method [18], the number of plane waves used here are 797 for both

_{t}*s*and

*p*polarizations, and the accuracy within 1% is expected.

*First, we show the effect of I
_{t} on the filling ratio, the column shape, and the PBG for a given c, here c is assumed to be 0.31 for calculations (the reason will be explained below). In Fig. 1 we draw the curves of FR versus I
_{t} for both the normal and the inverse lattices. Fig. 2 gives the concrete shapes of several inverse lattices formed at different I
_{t}. Evidently the shape and FR of the dielectric columns (black in Fig. 2) vary gradually with I
_{t}. Generally a smaller It yields smaller dielectric columns with smaller FR, while a greater It gives rise to connected columns with a greater FR. The apparent difference here in holographic fabrication from the ideal chessboard design [12] is that usually the shape of dielectric columns now is no longer square but like a pincushion with the middle part of each side shrunken towards the center of the columns. This is characteristic of holographic method in which the shape of columns or atoms is decided by equal-intensity surfaces.*

*Naturally the different shapes and filling ratios resulted from different selection of intensity threshold I
_{t} will lead to different PBG for a given intensity distribution. In Fig. 3 we give the curve of relative PBG versus It for the inverse structure in the case of c=0.31 and ε=8.9. The corresponding gap map is plotted in Fig. 4. From these two figures we can clearly see the general trend that the s gap is easier to occur for the case of non-overlapping dielectric columns (smaller I
_{t} and f), while the p gap is favored in the case of air columns (greater I
_{t} and f); and the largest PBG occurs as a compromise of the two factors. However, different from the case of exact square columns, now the columns begin to be overlapped at a smaller f, about 0.355 instead of 0.5, due to the pincushion shape of the columns. Figs. 3 and 4 indicate that the complete PBG can be obtained in the range of I
_{t}=1.31 to 1.45 or equally f=0.307 to 0.405; and that the largest relative band gap width appears in I_{t}=1.37 when the corresponding FR is f=0.347. In this optimal condition we have Δω/ω=8.8 %, a little larger than the result of best design for square columns [12]. The band structure in this condition is shown in Fig. 5.*

*Next we investigate the effect of factor c in Fig. (1). Because the shape and size of dielectric columns and consequently the PBG of the resultant structure are dependent on both c and I
_{t}, we must scan I
_{t} for each given c to find the maximum PBG. In the scanning process the step size we used is 0.01 for both c and I
_{t}. In Fig. 6 we give some lattice pictures to show the column shape and size for several values of c and its corresponding optimized I
_{t} yielding maximum PBG. From these pictures a general rule may be concluded: The maximum PBG always appears for the structure in which the dielectric columns are just close to be but have not yet connected; and when the factor c increases the required optimized I
_{t} usually decreases leading to a smaller optimized FR f.*

*The relation between the value of c and the corresponding maximum relative band width Δω/ω for ε=8.9 is depicted by the curve in Fig. 7. It is clear from this figure that the complete PBG can be obtained in a quite wide range of c, the maximum relative PBG (8.8 %) is realized for c=0.31 (that is the reason we use it in above calculations), and the maximum Δω/ω keeps a large value in a considerable range of c, specifically, Δω/ω>7% for c=0.05~0.36. But we must indicate that the maximum PBG in Fig. 7 is reached for the optimal intensity threshold I
_{t}, and generally different c has different optimal It. As mentioned above, the optimal I
_{t} decreases when c increases, and the maximum PBG always occurs for the columns of pincushion shape rather than exact square shape. For example, the case of c=0 and I
_{t}=2 yields an exact chessboard lattice with square columns of f=0.5, its PBG is 4.3 %, coincident with previous result [12]; however, the optimal I
_{t}=1.98 for the same light intensity distribution will generate non-square columns of f=0.485 and the corresponding Δω/ω=6.6%, considerably larger than the former. This fact has again convincingly verified the unique feature and usefulness of holographic method in band gap improvement.*

*We have also studied the PBG properties of this kind of inverse structures for other dielectric constants and compared our results with those obtained for square columns [12]. Our calculations show that the complete PBG occurs when ε is greater than about 6.4 (nearly 1% PBG for c=0.27 and I
_{t}=1.51), lower than 7 in the case of square columns [12], and the comparison of maximum PBGs for several dielectric constants, namely, 8.9, 12 (GaAs) and 16 (Ge) as used in Ref. [12] is listed in table 1. The corresponding optimized column shapes for the latter two dielectric constants are given in Fig. 8. The similar look of the pictures in Fig. 8 has again confirmed the superiority of pincushion columns to the square columns in PBG generation, and the results in table 1 give us an assurance that the holographic structures are usually better than or at least comparable to the square column structures in PBG property in all cases.*

*3. Optical design of holographic fabrication*

*Considering the symmetry of Eq. (1), we may use four linearly polarized beams of the same intensity (assumed to be unit here) with the same angle θ with z axis shown in Fig. 9 to fabricate this structure. In this geometry the four wave vectors are*

*$${\mathit{K}}_{1}=K(\mathrm{sin}\theta \u2044\sqrt{2},\phantom{\rule{.2em}{0ex}}\mathrm{sin}\theta \u2044\sqrt{2},\phantom{\rule{.2em}{0ex}}\mathrm{cos}\theta ),\phantom{\rule{.9em}{0ex}}{\mathit{K}}_{2}=K\left(\mathrm{sin}\theta \u2044\sqrt{2},-\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{cos}\theta \right),$$*

$${\mathit{K}}_{3}=K\left(-\mathrm{sin}\theta \u2044\sqrt{2},-\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{cos}\theta \right),\phantom{\rule{.9em}{0ex}}{\mathit{K}}_{4}=K\left(-\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{cos}\theta \right),$$

$${\mathit{K}}_{3}=K\left(-\mathrm{sin}\theta \u2044\sqrt{2},-\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{cos}\theta \right),\phantom{\rule{.9em}{0ex}}{\mathit{K}}_{4}=K\left(-\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{sin}\theta \u2044\sqrt{2},\mathrm{cos}\theta \right),$$

*where K=2π/λ, λ is the wavelength. This symmetric arrangement, which can be realized with the use of a diffraction beam splitter (DBS) [19], may assure equal optical path of each beam in z axis, and then the intensity distribution of the interference field can be written as*

*$$I(x,y)=4+2\{\left({e}_{14}+{e}_{23}\right)\mathrm{cos}(2\pi x\u2044a)+\left({e}_{12}+{e}_{34}\right)\mathrm{cos}(2\pi y\u2044a)$$*

$$+{e}_{13}\mathrm{cos}[2\pi \left(x+y\right)\u2044a]+{e}_{24}\mathrm{cos}\left[2\pi \left(x-y\right)\u2044a\right],$$

$$+{e}_{13}\mathrm{cos}[2\pi \left(x+y\right)\u2044a]+{e}_{24}\mathrm{cos}\left[2\pi \left(x-y\right)\u2044a\right],$$

*where a=λ/(2^{1/2}sinθ), e
_{ij}=e
_{i}·e
_{j}, e
_{j} is the unit polarization vector of the jth plane wave. Comparing the cosine terms in Eqs. (3) and (1), we can find that the varying parts in these two equations have the same relation on the condition*

*From this equation and the symmetry of beam geometry, we may assume*

*$${e}_{1}=(l,m,n),\phantom{\rule{.9em}{0ex}}{e}_{2}=(l,-m,n),\phantom{\rule{.9em}{0ex}}{e}_{3}=(p,q,r),\phantom{\rule{.9em}{0ex}}{e}_{4}=(p,-q,r),$$*

*where l, m, n, p, q and r are all real numbers. Using normalization condition of each polarization vector, orthogonal condition of e
_{j} and K
_{j}, and Eq. (4), we have*

*$$\left(l+m\right)\mathrm{sin}\theta +\sqrt{2}n\phantom{\rule{.2em}{0ex}}\mathrm{cos}\theta =0,\phantom{\rule{.9em}{0ex}}\left(p+q\right)\mathrm{sin}\theta -\sqrt{2}r\phantom{\rule{.2em}{0ex}}\mathrm{cos}\theta =0,$$*

*$$\mathit{lp}+\mathit{nr}+\left(1+2c\right)\mathit{mq}\u2044\left(1-2c\right)=0,\phantom{\rule{.9em}{0ex}}{m}^{2}+{q}^{2}-2\mathit{mq}/\left(1-2c\right)=1.$$*

*There are six independent equations in Eqs. (6)–(8). Solving them for given c and θ we can obtain all the six parameters. A possible solution for c=0.31 and θ=30° is*

*$${\mathit{e}}_{1}=(0.76253,0.42730,-0.48575),\phantom{\rule{.9em}{0ex}}{\mathit{e}}_{2}=\left(0.76253,-0.42730,-0.48575\right),$$*

$${\mathit{e}}_{3}=\left(0.91602,-0.31839,0.24398\right),\phantom{\rule{.9em}{0ex}}{\mathit{e}}_{4}=(0.91602,0.31839,0.24398).$$

$${\mathit{e}}_{3}=\left(0.91602,-0.31839,0.24398\right),\phantom{\rule{.9em}{0ex}}{\mathit{e}}_{4}=(0.91602,0.31839,0.24398).$$

*Substituting Eq. (9) into Eq. (3), we have*

*$$I(x,y)=2.864\{1.397+\mathrm{cos}(2\pi x\u2044a)+\mathrm{cos}(2\pi y\u2044a)$$*

$$+0.31\mathrm{cos}[2\pi \left(x+y\right)\u2044a]+0.31\mathrm{cos}\left[2\pi \left(x-y\right)\u2044a\right]\}.$$

$$+0.31\mathrm{cos}[2\pi \left(x+y\right)\u2044a]+0.31\mathrm{cos}\left[2\pi \left(x-y\right)\u2044a\right]\}.$$

*Comparing this equation with Eq. (1), we can see the present intensity threshold I
_{t}’ corresponding to the I
_{t} for Eq. (1) to generate the same structure should be modulated as*

*For example, the optimized I
_{t}=1.37 for ε=8.9 should now become I
_{t}’=2.197, readers can verify the same FR f=0.347 as mentioned before.*

*4. Conclusions*

*We have made a theoretical study of a new kind of 2-D square lattice fabricated holographically with pincushion columns rotated by 45°. Calculations show that this structure can generate complete PBG comparable to or even larger than the corresponding optimized chessboard structure with square columns in a wider range of dielectric constant. The beam design for making this structure is also derived. These results have revealed the unique feature of holographically formed PhCs that their PBG properties are critically related to their fabrication process and then may be improved by proper design to get optimized column (atom in 3-D case) shape and size. This idea and the method of analysis here can be extended to other types of lattices and open a new freedom for PBG engineering.*

*Acknowledgments*

*This work is supported by the National Natural Science Foundation (64077005), Doctoral Program Foundation of Ministry of Education (20020422047), China, and National Key Lab Foundation, Institute of Crystal Materials, Shandong University, China.*

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