## Abstract

Our numerical simulation results demonstrate that 2D lattices compounding can create either a broad single complete photonic band gap or both first and second order complete band gaps. The results also show that photonic band gap properties are dependent on both the parameters of the single lattices and the relative position of the two compound lattices. Furthermore, if a compound structure is composed of two sets of lattices, the one with a larger periodic constant (** a_{2}**) will serve as defects. While the defect modes are direction independent as

**> 5**

*a*_{2}**, they are direction dependent as**

*a***< 5**

*a*_{2}**. Moreover, by optimizing of the rod size of the lattice with**

*a***, many kinds of defect modes can be obtained to satisfy the different applications. The transmitted spectra and reflected spectra of this kind of structures demonstrate that the transmittances of the defect modes are dependent**

*a*_{2}**.**

*a*_{2}© 2005 Optical Society of America

## 1. Introduction

Photonic crystals (PCs) are new materials with periodic structures [1,2]. One of their optical properties is the existence of photonic band gap. Within the photonic band gap electromagnetic modes are forbidden. They can control the spontaneous emission and the propagation of light. Therefore, PCs are suitable for a variety of applications, such as waveguides, micro-cavities and etc. Because a great variety of PC applications needs broad complete band gaps, the design and fabrication of PCs with broad complete band gaps have received much attention [3–6]. Several methods to broaden the complete band gaps have been proposed. For instance, Li et al achieved substantial improvement of absolute photonic band gaps in two-dimensional square and triangular lattices by introducing anisotropy material [7]. Trifonov et al suggested that symmetry reduction in square structures could effectively enlarge the absolute PBG [8]. Among these methods, lattice compounding is proved to be an effective and flexible method [9–12]. It is well known that no complete photonic band gap appears in the opal type square or triangular lattices which consist of rods or network of thin slabs. It is shown, in this work, that a broad complete photonic band gap (larger than 18%) can be obtained by the compounding of these two square lattices without a complete photonic band gap; and both single and double absolute band gaps can be obtained by the compounding of these two triangular lattices with no complete photonic band gap.

Secondly, one kind of the most widely used photonic crystal structures is defective photonic crystals. In the previous studies, defective two- or three-dimensional photonic crystals were mainly obtained by selective adding, removing or replacing the sites with high dielectric constant (or named photonic atom). In this work, it is shown that in a compound structure composed of a lattice with a larger periodic constant and a lattice with a small periodic constant, the lattice with a larger periodic constant will serve as defects. The properties of the defect modes of these structures are discussed in this paper. By the optimization of the periodic distance and the size of the rods of the larger periodic lattice, some defect modes with useful properties can be obtained.

## 2. Band gap properties of compound lattices and band engineering by lattices compounding

For demonstration, two structures of compound photonic lattices, as shown in Fig. 1, are discussed in this work. Figure 1(a) is a compound square lattice, which is composed of a square lattice formed by cylinder rods and a square lattice formed by thin sheets like a network. Figure 1(b) is a compound triangular lattice formed by a cylinder rods triangular lattice and a thin sheets triangular lattice. In the simulation, we suggest that they are made of silicon (dielectric constant *ε*= 12.096 with respect to *λ*= 1.55 μm).

Using Plane Wave Expansion method [13] and Order-N method [14], we calculated the photonic band gaps of these compound structures and the bandwidths of the complete band gaps of these structures as a function of the radius of the rods (** r**) and the width of the thin sheets (

**). The results are shown in Fig. 2 and Fig. 3. For the compound square lattice, the dependence of complete band gap on the parameter**

*d***and**

*r***are shown in Fig. 2 (c) and 2(d). As shown in Fig. 2(a), the complete band gap has a broad bandwidth of Δω/ω**

*d*_{g}=18.3 % when

**= 0.32**

*r***and**

*a***= 0.05**

*d***. Here, the parameters Δω and ω**

*a*_{g}denote the frequency width of the gap and the frequency at the middle of the gap respectively. And the value of periodic constant is

**= 1μm. We also calculated the transmittance spectrum and reflectance spectrum of this structure by using the finite-difference time-domain (FDTD) simulation method, as shown in Fig. 2(b). By comparing Fig. 2(a) and 2(b), the frequency of complete band gap obtained from these two methods are almost the same. In Fig. 2(d), the result shows that as**

*a***> 0.05**

*d***, the low frequency band edge remains unchanged, and as**

*a***< 0.05**

*d***, the high frequency band edge also remains unchanged.**

*a*In terms of the compound triangular lattice, the dependence of complete band gap on the parameter ** r** and

**are shown in Fig. 3(a) and (b). They indicate that either a single complete band gap or both first and second order complete band gaps will appear to be depending upon the parameters.**

*d*The representative band structures of the compound triangular lattice are illustrated in Fig. 4, including a single complete band gap and double complete band gaps in Fig. 4 (a) and (b) respectively. Figure 4(a) shows that the bandwidth of the single complete band gap is Δω/ω_{g}=8.15 %, and Fig. 4(b) shows the bandwidths of the first and second order complete band gaps are Δω/ω_{g}=5.81 % and 7.09 % accordingly.

Moreover, in Fig. 1, the center of the rod circles is overlapped with the intersection points of the thin sheets. In the case that the center of the circles has a shift with respect to the points of the intersections, numerical simulation results demonstrate that the shift has influence on the width of band gaps. For compound square lattice, the broadest complete band gap appears as the center of the rod circles is overlapped with the intersection point of the thin sheets. However, for compound triangular lattice, the broadest complete band gap of the lower frequency band appears as that the two centers shift 10% in the X-direction and 7.5% in the Y-direction, and the largest bandwidth can reach 9.12%.

## 3. Periodic defects doping by lattices compounding

As typical examples, we composed two square lattices to form a compound structure, in which a square lattice is exactly the same as that shown in Fig. 1 (a), as shown with dark color in Fig. 5 (dark color denotes the material with high dielectric constant, such as silicon), and its periodic constant is denoted by ** a**. It is shown in Fig. 2 (a) and (b) of section 2 that this square lattice has a complete band gap with a broad bandwidth of 18.3 %. In Fig. 5, violet color shows another square lattice with a larger periodic constant. The periodic constant of the square lattice with a larger periodic constant is denoted by

**.**

*a*_{2}Figure 5(a), (b) and (c) show ** a_{2}** = 3

**,**

*a***= 5**

*a*_{2}**, and**

*a***= 7**

*a*_{2}**respectively. Our results show that as these lattices with different periodic constants compound to form the compound structures, defect modes will appear in the complete band gap instead of band broadening. It means that the lattices with a larger periodic constant will serve as defects.**

*a*Our numerical simulation results show that the frequencies of these defect modes are dependent on the rod radius (** r_{2}**) of the square lattice with a larger periodic constant, as shown in Fig. 6. Our numerical simulation results also show that the defect modes-

**relationships are almost independent on**

*r*_{2}**, while the line-width of the defect modes are dependent on**

*a*_{2}**. The smaller the**

*a*_{2}**is, the broader the line-widths of the defect modes are.**

*a*_{2}By the optimization of the size of the rods (** r_{2}**) of the larger periodic lattice, a single defect mode (as shown by the green hatched area in Fig. 6), multiple defect modes with equal frequency spacing, and coexistence of both a single polarization defect mode and TE and TM overlap defect mode (as shown by the green line in Fig. 6) can be obtained respectively. Furthermore, the incident angle independent defect modes only appear when

**is larger than 5**

*a*_{2}**; otherwise, defect modes are incident angle dependent. Figure 7 shows the representative results. In Fig. 7,**

*a***is the value shown by the green lines in Fig. 6, in which the defect mode is a TE and TM overlap defect mode. The defect modes are incident angle dependent for**

*r*_{2}**= 3**

*a*_{2}**while they are incident angle independent for**

*a***= 7**

*a*_{2}**. Our calculations also demonstrate that the transmittances of the defect modes are not so high for**

*a***≥ 7**

*a*_{2}**as they are for**

*a***< 5**

*a*_{2}**. Furthermore, our calculation shows that cavity modes are easier to couple with each other when the periodic constant of defect structure (**

*a***) is smaller. When**

*a*_{2}**become bigger, the defect modes tend to be effectively closed cavity modes.**

*a*_{2}About the feasibility of fabrication of the structures studied in the paper, firstly, the two kinds structures above can be fabricate by semiconductor lithography method. Recently, a structure of Fig. 1(a) has been fabricated by holographic lithography in our laboratory.

## 4. Conclusion

Our numerical simulation results demonstrate that lattices compounding is a useful method to create broad complete photonic band gap and construct some new band gap structures, such as double complete photonic band gaps. This work demonstrates that the properties of photonic band gap can be controlled by both the parameters of the single lattices and the relative position of the two compound lattices. Furthermore, if a compound structure is composed of two sets of lattices with different periodic constants, the one with a larger periodic constant (** a_{2}**) will serve as periodic defects. The frequencies of these defect modes are dependent on the rod radius (

*r*_{2}) of square lattice with a larger periodic constant, and the defect modes-

**relationships are almost independent on**

*r*_{2}**. In addition, by the optimization of the size of the rods (**

*a*_{2}

*r*_{2}) of the larger periodic lattice, a single defect mode, multiple defect modes with equal frequency spacing, and coexistence of both a single polarization mode and TE and TM overlap defect mode can be obtained respectively. Unexpectedly, the incident angle independent defect modes only appear when

**is larger than 5**

*a*_{2}**; otherwise, defect modes are incident angle dependent. The transmittance spectra and reflectance spectra of this kind of structures demonstrate that the transmittances of the defect modes are dependent**

*a***.**

*a*_{2}## Acknowledgments

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

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