## Abstract

We present occurrence of the effective Bragg conditions with wide gapwidth and high reflectance in a Fibonacci superlattice, which is a typical one-dimensional quasicrystal. In the Fibonacci material, the number of effective Bragg conditions is two rather than one which appears in traditional periodic structures. Based on the effective Bragg conditions, this study proposes existence of omnidirectional, wideband and high reflectance in the quasiperiodic materials analogous to that in traditional materials.

© 2012 OSA

Since the discovery of quasicrystals, several quasiperiodic materials have been formulated and have attracted intense interest not only in regard to fundamental physical phenomena but also to technological applications [1–6]. Research has shown that the phenomena of photon propagating in quasiperiodic materials are different from those among periodic and disordered ones [7–12]. Light propagation can be forbidden in disorder structures by Anderson localization [13–15]. In periodic media, high reflectance can also be achieved by Bragg interference. Until now, Bragg’s law has played a major role in the measurement of crystalline solids and the design of several optical devices [16–19]. During implementation, a broad bandgap and high reflectance is required to reduce the performance sensitivity due to manufactory error. Based on the Bragg condition, maximum gapwidth can be achieved by multiplying of the quarter-wave length in each layer for the traditional bilayers [20,21]. In particular, with a maximum omnidirectional gap, the highest performance reflectors are found within the first and zero orders Bragg conditions for bilayer dielectric materials [20–23] and metamaterials [24–28], respectively.

However, for quasiperiodic materials, the broad bandgap and high reflectance of the quarter-wave thickness is destroyed and replaced by self-similar and cycling fracture gaps in the transmission spectra [7–9]. It seems that the high reflectance condition varies for different generation orders, layer thicknesses and materials in the dielectric Fibonacci superlattice (FSL) [9–12], which is a typical one-dimensional quasiperiodic material. Thus, it is not convenient to apply quasiperiodic materials in the design of optical devices, which have been easily implemented by using traditional periodical materials based on Bragg’s law. Until now, critical questions remain unanswered. Does broad bandgap and high reflectance exist for quasiperiodic materials similar to the Bragg condition or the quarter-wave stakes for the periodic materials? If the high reflectance exists in quasiperiodic materials, can we obtain the critical condition and express it as a concise law analogous to Bragg’s law for periodical materials.

Here, we show that the broad bandgap and high reflectance does exist in the FSL via gap map diagrams. The high reflectance condition for the FSL can be determined by the midline of the major gaps (MMGs) and described by two concise expressions, called effective Bragg conditions, which are analogous to the Bragg condition for periodic materials. Based on the effective Bragg condition, existence of the highest reflectance in the FSL are determined and described by two critical conditions analogous to the quarter-wave stacks in the traditional bilayers. Moreover, we show that an omnidirectional high reflector for the quasiperiodic materials can be achieved by the effective Bragg condition.

We first consider a FSL composed of two layers, A and B, with the following scheme: ${S}_{\nu}=\left\{{S}_{\nu -1}{S}_{\nu -2}\right\}$ for *v*≥2, with *S*_{0} = {*B*} and *S*_{1} = {*A*}. The numbers of layer *A* and *B* in the *S _{v}* superlattice are

*N*

_{v-}_{1}and

*N*

_{v-}_{2}, respectively. Transmission of light through the FSL is dependent on the light polarization and incident direction and the structure, including the layer thickness, effective refractive index of each layer, and the generation order. The transmission spectra of FSL with arbitrary generation order for arbitrary incident directions of TE and TM polarizations can be calculated by the transfer matrix method or the graph method [26]. The gap map is formed by the bandedges, which are determined by the bandedge formalism to avoid numerical instability [23,26]. Figure 1(a) shows the effect of the generation order on the variants of the bandgaps for the normal incidence, TE 45°, and TM 45° of the FSL, in which the periodic boundary is considered. We see that the number offorbidden gaps within the frequency range increases with increasing generation order. The widths of almost all of the forbidden gaps for each incidence direction and polarization are reduced as the generation order of the system increases. However, the widths of the major gaps indicated by arrow signs in the figure do not reduce significantly with higher generation order.

As a single FSL bonded by the media of air is considered, the transmission spectra of the normal incident of TE polarization through the FSL for different generation orders is shown in Fig. 1(b). When the generation order is low, such as *v* = 4, the frequency range of low transmissions cannot be located within the major gaps. However, as the generation order increases, the transmittance within the major gaps becomes lower. For generations greater than 6, the low transmission regions almost fit the forbidden gaps, as in the case of the traditional bilayers, {*AB*}^{∞}.Next, we consider the gap map diagram in the S_{8} FSL for normal incidence, versus the thickness filling factor *F*, defined by *F* = *d _{A}*/(

*d*+

_{A}*d*) as shown in Fig. 2(a) . The gap map can be divided by zero-gap lines,

_{B}*L*and

_{A,m}*L*, obtained by

_{B,n}*k*=

_{A}d_{A}*m*π and

*k*=

_{B}d_{B}*n*π, to several regions, each of which has the same pattern. The number of forbidden bands in each region increases as the generation order increases. Thus, the widths of almost all of the forbidden gaps will be reduced for an FSL with a higher generation order. However, the widths of some forbidden gaps within some filling factor ranges do not significantly reduce the gap width for the higher generation order. We define the part of the gap map bounded by

*L*

_{A}_{,}

*,*

_{m}*L*

_{A}_{,}

_{m}_{+1},

*L*

_{B}_{,}

*, and*

_{n}*L*

_{B}_{,}

_{n+}_{1}as region (

*m*,

*n*), where

*m*and

*n*are nonnegative integers. With a higher generation order, there are two major forbidden gap holes within each region [11]. We refer to the gap hole with higher frequency as the higher major hole and the other one as the lower major hole.

For the *S _{v}* FSL, the right end of the lower major hole in region (

*m*,

*n*) is located at the cross point ${C}_{m+{\tau}_{v-2,v-1},n}$ and the left one is located at

*C*

_{m,n}_{+1}, where ${\tau}_{v-2,v-1}={N}_{v-2}/{N}_{v-1}.$

*F*and Ω at the close point

*C*are, respectively, given by

_{p,q}*L*

_{B}_{,}

*to a different generation order.*

_{n}*F*and Ω at the right end of the lower major hole in region (

*m*,

*n*) are $(m+{\tau}_{v-2,v-1})\sigma /[n+(m+{\tau}_{v-2,v-1})\sigma ],$ $[n+(m+{\tau}_{v-2,v-1})\sigma ]/(2{n}_{B}\mathrm{cos}{\theta}_{B}).$ For the higher major hole, the right end is located at the cross points

*C*

_{m+}_{1}

*, which keep the same position for arbitrary generation orders. The left end of the higher major hole is located at ${C}_{m+{N}_{v-3}/{N}_{v-1},n+1},$ which moves for different generation orders. The values of*

_{,n}*F*and Ω at the left end of the top major hole in region (

*m*,

*n*) are $(m+{\tau}_{v-3,v-1})\sigma /[(n+1+(m+{\tau}_{v-3,v-1})\sigma ],$$[n+1+(m+{\tau}_{v-3,v-1})\sigma ]/(2{n}_{B}\mathrm{cos}{\theta}_{B}),$respectively. As the generation order of the FSL approaches infinity, we have the relations $\underset{v\to \infty}{\mathrm{lim}}{\tau}_{v-2,v-1}={\tau}^{-1}$ and $\underset{v\to \infty}{\mathrm{lim}}{\tau}_{v-3,v-1}={\tau}^{-2},$ where τ is the golden ratio, which is defined by $\tau =(1+\sqrt{5})/2\approx \mathrm{1.618.}$ The moving end of each major hole will asymptotically approach a fixed point on the zero-gap line for a higher generation order.

As we know, for the traditional bilayers, there is only one MMG in each region of the gap map [23]. The MMG in region (*m*,*n*) corresponds to the Bragg condition described by ${k}_{A}{d}_{A}+{k}_{B}{d}_{B}=(m+n+1)\pi ,$which almost corresponds to the center of the lowest transmission region [23]. Similarly, the center of the lowest transmission region also nearly matches the MMG for the FSL. Thus, we can determine the condition of the highest reflectance from the MMGs. The midline of the higher and lower major gap holes in region (*m*,*n*) are given by ${k}_{A}{d}_{A}+{\tau}_{v-2,v-1}{k}_{B}{d}_{B}=(m+1+{\tau}_{v-2,v-1}n)\pi $ and ${k}_{A}{d}_{A}+{\tau}_{v-2,v-1}{k}_{B}{d}_{B}=$ $[m+{\tau}_{v-2,v-1}(n+1)]\pi $ and, respectively, rewritten by

_{8}FSL. One of the special cases is

*v*→∞, by which$\underset{v\to \infty}{\mathrm{lim}}{\tau}_{v-2,v-1}=\mathrm{0.618.}$ For the FSL with a higher generation order, these equations can be approximately described by the infinite case. The other special case is

*v*= 2, which is identical to a bilayer superlattice. For the case of

*v*= 2, The midline of the higher and lower major gap holes in each region are overlapped and described by one line with the same equation as that of the Bragg condition for the bilayers.

In each major hole, we find that the extreme situation of the maximum gapwidth is near the center of the gap hole. Here, the center of the lower major gap hole in region (*m*,*n*), CLG_{m}_{,}* _{n}*, is defined by the intersection of the lines

*k*= (

_{A}d_{A}*m*+ 0.5

*τ*

_{v-}_{2,}

_{v-}_{1})π and

*k*= (

_{B}d_{B}*n*+ 0.5)π, which can be rewritten by ${d}_{A}{n}_{A}\mathrm{cos}{\theta}_{A}=(2m+{\tau}_{v-2,v-1}){\lambda}_{0}/4$ and ${d}_{B}{n}_{B}\mathrm{cos}{\theta}_{B}=(2n+1){\lambda}_{0}/4,$ respectively. Thus, the filling factor and the normalized frequency at the CLG

_{m}_{,}

*are given by*

_{n}

_{m}_{,}

*, is located on the intersection of the lines*

_{n}*k*= [

_{A}d_{A}*m*+ 0.5(1

*+ τ*

_{v-}_{3,}

_{v-}_{1})]π and

*k*= (

_{B}d_{B}*n*+ 0.5)π and is rewritten by ${d}_{A}{n}_{A}\mathrm{cos}{\theta}_{A}=(2m+1+{\tau}_{v-3,v-1}){\lambda}_{0}/4$ and ${d}_{B}{n}_{B}\mathrm{cos}{\theta}_{B}=(2n+1){\lambda}_{0}/4.$The filling factor and normalized frequency at the CHG

_{m}_{,}

*are*

_{n}*v*= 2, CHG

_{m}_{,}

*and CLG*

_{n}

_{m}_{,}

*are overlapped. The layer thickness of CLG*

_{n}

_{m}_{,}

*is equal to a multiple of the quarter wave, the same as the Bragg condition for the bilayers. In the fundamental region, the layer thickness of CLG*

_{n}_{0,0}is identical to the quarter wave stacks [21]. Figure 2(b) shows that the thickness filling factor for the center of the gap holes almost has a linear relation to the refractive index ratio,

*n*/(

_{A}*n*+

_{A}*n*). The difference between the filling factors of the center of the gap holes and maximum gaps gradually enlarges as the refractive indices of layers A and B get closer. The difference of the thickness filling factors may gradually increase as the refractive index ratio increases. However, the frequency and the gapwidth to gap center ratio for the center of the gap holes are very close to those of the maximum gaps for an arbitrary refractive index ratio, as shown in Figs. 2(c) and 2(d). We find that the gapwidth to gap center ratio almost has a linear relation to the refractive index ratio. As the difference between

_{B}*n*and

_{A}*n*increase, the gapwidth to gap center ratios increase. Of these centers of the gap holes, the maximum gapwidth ratio occurs at CLG

_{B}_{0,0}. Thus the omni-gap can be determined by only the overlap region of the major holes of the NI and TM1 states. The thickness filling factor and the frequency of the center of the omni-gap can be approximately expressed by the center of the NI and TM1 gap holes. The maximum gapwidth to gap center of the omin-gap is located in the lower major gap of region (0,0), which is called the fundamental omin-gap. In Fig. 3(b) , we see that transmissions for different incident directions and polarizations can be reduced significantly in the range of the omni-gap. Figure 3(c) shows that the gapwidth to gap center ratio of the center of the fundamental omni-gap (CFOG) of the FSL is very close to that of the maximum omnidirectional gap. The almost linear gapwidth to gap center of the CFOG decreases with the increase of the refractive index ratio. Moreover, the relation of the gapwidth to gap center ratios to the refractive index ratio for the CFOG is close that for the maximum omnidirectional gap of the FSL.

Based on the major gaps of the FSL, high reflection for an arbitrary incident direction, called an omnidirectional gap or omni-gap [21–23], can be determined by the overlap of the bandgap for the cases of normal incidence, called NI state and grazing incidence for TE and TM polarizations, as referred to as TE1, and TM1 states, as shown in Fig. 3(a). We find that the ranges of the major holes of TM1 are within the TE1 [11].

In summary, analogous to one Bragg condition for the traditional bilayers, the broad bandgap and high reflectance for the FSL is found by the midline of the major gaps and expressed by two concise critical conditions, called the effective Bragg conditions. Moreover, the highest performance of the reflector for the FSL is determined by the centers of the effective Bragg conditions, which are analogous to quarter wave condition for the traditional bilayers. Similar to the highly omnidirectional reflection properties of the quarter wave stack in bilayers [20–23], the existence of omnidirectional reflection in the FSL is found by the center of the omni-gap based on the effective Bragg conditions. From this concept, we have potential to deal with novel quasicrystals for design and application in various fields and potential to reach higher performance than traditional periodical crystals.

## Acknowledgments

The authors acknowledge the support in part by the National Science Council of Taiwan under grant numbers NSC 100-2221-E-002-021 and NSC 101-2221-E-002-030.

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