Bragg-like interference in one-dimensional double-period quasicrystals

Three quasi-Bragg conditions (QBCs) in one-dimensional double-period quasicrystals based on high reflectance from interference are proposed. Analytical formulas for these QBCs are derived using band-edge equations. All of these QBCs have conditions that are different from that of the traditional Bragg condition. The QBC at quarter-wave thickness in double-period quasicrystals is also discontinuous in different regions of the gap map. In contrast, the traditional Bragg condition for periodic cases, which lies at quarter-wave thickness, is continuous in different regions of the gap map. It is found that there are three thickness conditions with the maximum reflectance occurring in the midpoints of the QBCs in double-periodic quasicrystals, which is analogous to the quarter-wave thickness in traditional periodic crystals. These QBCs in double-period cases are different not only from the traditional Bragg condition in periodic cases, but also from those in Fibonacci and Thue-Morse quasicrystals.

Figure 1

Transmission spectra for sixth-order DPSLs with (a) $n_b$ = $n_0$, (b) $n_b$ = $n_A$, (c) $n_b$ = $n_B$ at normal incidence, and (d) their eigenfunction $cos\left(KL\right)$. The normalized frequency Ω is defined by Ω = ω$D$/2π$c$. The parameters of the system are $n_A$ = 1.5, $n_B$ = 2.5, $N$ = 1, $D$ = 1.0 μm, and $F$ = 0.3. The gray areas in (a)–(d) correspond to the region of the major gaps that are sufficiently large.

Figure 2

A gap map for DPSLs with $v$ = 6. All of the parameters are the same as those for Fig. 1. The thickness filling factor $F$ is defined by $F$ = dA/(dA + dB). The gray areas are major gaps. The red line marks the QBC for the major gaps. The QBCs of the up, middle, and the down major gaps for DPSLs are denoted as UBC, MBC, and DBC, respectively. The cyan (light gray) and blue (dark gray) dotted lines denote the half-wave lines $B_{A,p}$ and $B_{B,q}$, respectively. The character $x$ represents the midpoint of the QBC in region (0,0). The squares and triangular symbols are the left and right ends in region (0,0).

Figure 3

Normalized frequency and transmission spectra for DPSLs of generation orders: (a) $v$ = 4, (b) $v$ = 5, and (c) $v$ = 6, at normal incidence. The parameters of the system are $n_A$ = 1.5, $n_B$ = 2.5, $n_b$ = $n_0$, $N$ = 1, $D$ = 1.0 μm, and $F$ = 0.3. The gray areas in (a)–(c) correspond to the region of the major gaps that are sufficiently large.

Figure 4

(a) The generation order $v$ and the minimum transmittance for DPSLs at normal incidence. (b) The total number of layers $2{}^{\mathrm{u}}$ and the minimum transmittance for BPSLs at normal incidence. The parameters of the system are the same as those for Fig. 3. The blue (solid) lines denote the frequency of the minimum transmittance. The green (dotted) lines and the red (dashed) lines each denote band edges and their corresponding band centers.

Figure 5

(a) The thickness filling factor and (b) the frequency of the center of the major gaps in region (0,0) for DPSLs with $v$ = 6 and BPSLs with the same number of layers as the DPSLs. The parameters of the system are the same as those for Fig. 2. The dashed lines correspond to the midpoint of the QBC of the up, middle, and down major gaps of DPSLs and the quarter-wave thickness of BPSLs, denoted as CU, CM, CD, and QW(BPSL). The solid lines correspond to the maximum gap centers of the up, middle, and down major gaps of a DPSL and the major gap of a BPSL, denoted as XU, XM, XD, and XG(BPSL).

Figure 6

The gap width at the center of the major gaps in region (0,0) for structures that are the same as those in Fig. 5. The gap width is normalized in the same way as Ω. The parameters of the system are the same as those for Fig. 5. The dashed lines correspond to the midpoint of the QBC of the up, middle, and down major gaps of DPSLs and the quarter-wave thickness of BPSLs, denoted as CU, CM, CD, and QW(BPSL). The solid lines correspond to the maximum gap centers of the up, middle, and down major gaps of a DPSL and the major gap of a BPSL, denoted as XU, XM, XD, and XG(BPSL).

Figure 7

The generation order $v$ and the minimum transmittance for DPSLs at (a) θ = 45° with TE polarization and (b) θ = 45° with TM polarization. The total number of layers $2{}^{\mathrm{u}}$ and the minimum transmittance for BPSLs at (c) θ = 45° with TE polarization and (d) θ = 45° with TM polarization. The parameters of the system are the same as those for Fig. 4. The blue (solid) lines denote the frequency of the minimum transmittance. The green (dotted) lines and red (dashed) lines each denote band edges and their corresponding band centers.

Figure 8

The ${\left|E\right|}^2$ distribution at (a) CU and (b) CM for the QBCs in DPSLs with $v$ = 6. (c) The ${\left|E\right|}^2$ distribution at the quarter-wave thickness of BPSLs that have the same number of layers as those in Fig. 8. Here, ${\left|E\right|}^2$ is the normalized squared electric, and $x$ is the normalized length. The parameters of the system are the same as those for Fig. 3. The (blue) dashed lines are the boundaries of the different layers. The character $A$ (in yellow areas) and the dots (in green areas) represent materials $A$ and $B$, respectively. White areas denote the bonding media where $n_b$ = $n_0$.