Localized mode in dielectric photonic crystals has received intensive interest in both fundamental physics and for device applications [1–4]. This property is widely used in telecommunications, fibers and lasers. A dielectric periodic bilayer with defect (PBD) is one of the fundamental structures, which consists of two parallel structures with bilayers that reflect light back and forth successively in the cavity to generate a cavity localized field [5, 6]. Moreover, the localized modes in the PBD appear in the frequency region of the band gap of the traditional periodic bilayer in the transmission spectra. As the incident field conforms to the resonant conditions in the PBD, the electric field is highly localized in the defect layer [7, 8]. A variety of optical characteristics result when the arrangements of the structure and the optical thickness of the defect layer are changed. For the PBD structure, there are three important properties that allow diverse designs for optical devices [9]. Firstly, localized mode appears in defect layers of the same thickness as the periodic number increases. Secondly, the electric fields are symmetrically distributed within the structure. Finally, the positions of localized modes in the PBD, which appear around the odd quarter-wave thicknesses, are the same as those in the distributed Bragg reflector (DBR) with different periodic numbers [10, 11].

Recently, considerable effort has been placed on quasperiodic system since the discovery of quasicrystals [12–14]. Due to the particular arrangement of quasicrystals, the properties of photon propagation in the quasicrystal are different from those in periodic and disordered structures [15–17]. Thus, the quasicrystals have received a lot of interest in the designing of optical materials and devices. Since devices with a quasicrystal system in one dimension are easily manufactured, this study examines a 1-D quasicrystal structure [18, 19]. For quasiperiodic systems, one of the well-known structures is the Thue-Morse sequence [20, 21]. The existence of strongly localized modes has been proposed by inserting a defect layer into the supperlattice structures [22]. This paper proposes a new structure: a one-dimensional symmetrical Thue-Morse supperlattice (STMS). The STMS is formed by inserting a space layer between two parallel structures that are constructed with the symmetry of a Thue-Morse sequence. Moreover, we propose the novel method to find the localized modes in the STMS. Therefore, it is necessary to determine whether there is strongly localized mode in a STMS. If there is localized mode in a STMS, whether the optical properties of STMS’s and PBD’s are different? Firstly, do the localized modes also exist around the quarter-wave thickness in a STMS that are similar to a PBD? Secondly, does the frequency of the resonant mode for a fixed thickness of the space layer in the transmission spectra of the STMS depend on the generation orders? Finally, if the optical thickness of the space layer is changed for a fixed frequency, do resonant modes appear periodically, which are similar to that in a PBD?

For the sake of simplicity, a one-dimensional Thue-Morse structure, which consists of two different materials, A and B, according to the iteration rule: A→AB and B→BA, is considered. The structure of the system with a lower order is given by ν≥2 with S₂ = AB, S₃ = ABBA, S₄ = ABBABAAB, etc. The STMS is constructed by inserting a space layer (d_s) into two parallel and symmetrical Thue-Morse dielectric supperlattice; for instance, S´₂d_sS₂, S´₃d_sS₃, S´₄d_sS₄, etc. The arrangement of S´ is opposite to the arrangement of S. However, a PBD can be denoted as P´_md_DP_m, where P_m = (AB)^m and P´_m = (BA)^m and m is the number of periods.

The relationship between the resonant peak in traditional major gaps in the transmission spectrum for the STMS with different generation orders and a PBD is considered firstly. The traditional gap regions are found in gap map of fundamental Thue-Morse structure [23]. The transmission spectra for the generation orders 5 and 6 in the STMS are shown in Fig. 1(a) and 1(b). The gray areas are the regions of the major gap in the band structure of the fundamental Thue-Morse supperlattice. The squared electric field of the light propagation corresponding to the signs A and B in Fig. 1(a) and 1(b) are shown in Fig. 1(c) and 1(d). We find that there is strong localization near the space layer. From Fig. 1, it is seen that there are two properties that are similar to the PBD. The first property is that the resonant modes appear on a conformed thickness of space layer and the corresponding frequencies appear in the traditional major gaps. The other property is that the minimum value for the transmittance decreases as the generation order increases.

 

Fig. 1 The transmission spectra in the STMS’s with generation orders (a) ν = 5, and (b) ν = 6, for normal incidence from air. The gray areas correspond to the region of the major gap in the band structure of the Thue-Morse superlattice. The |E|² distribution in the ν₅ STMS for (c) Ω = 0.2308 and (d) Ω = 0.5192, respectively, which correspond to the signs, A and B, in Fig. 1(a). The parameters of the system are n_A = 1, n_B = 2, n_D = 2, d_A = 2/3µm, d_B = 1/3µm, and d_S = 4/3µm. The normalized frequency, Ω, is defined by Ω = ωD/2πc, where D = d_A + d_B.

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However, there are two differences between these two systems. Firstly, by comparing to the fundamental region of the fixed gap in the PBD, it can be found that there are oscillations for the fundamental region of the major gap in the STMS. The fundamental regions of the major gaps approach to a fixed region as the generation order increases. Secondly, the positions of resonant peaks maintain a fixed value in the gap of a PBD with different periodic numbers. However, the positions of resonant peaks appear arbitrarily in the major gap for the STMS with different generation orders.

The gap map diagram for a ν = 6 STMS versus the thickness of the space layer, d_S, is shown in Fig. 2(a).The gray areas are the regions of the major gaps in the band structure of the fundamental Thue-Morse supperlattice. There is an important physical relationship between the resonant wavelength and the thickness of the space layer. For shorter resonant wavelength, the appearance of the resonance is in a shorter period as the space layer increases its thickness while a longer period for longer resonant wavelength. In Fig. 2(a), the resonant lines corresponding to different modes shift from a small value of d_S to a large value of d_S periodically in major gaps. There is periodical variation between adjacent resonant lines in the major gaps. The period function of resonant lines for different values of frequency is written as

 

Fig. 2 (a)The plot of the change of the normalized frequency for resonant lines versus the thickness of the space layer for normal incidence in a ν = 6 STMS, for n_A = 1, n_B = 2. The gray areas correspond to the major gaps of the Thue-Morse superlattice. The signs, A, B, C and D, show the intersection points of the red midline in the major gaps and the black resonant lines. (b) Φ_S,U/π in the up major gap and Φ_S,D/π in the down major gap, versus the generation orders. The blue solid line and the green dashed line correspond to Φ_S,U/π and Φ_S,D/π, respectively

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In order to understand further the relationship between the upper and lower resonant lines in higher generation orders, the intersection of the resonant lines and the middle lines in the major gap are used. The A and B signs are the first and second intersections, respectively, of the resonant lines and the middle lines in the down major gap. Similarly, the C and D signs are the first and second intersections, respectively, of the resonant lines and the middle lines in the up major gap. According to the relationship between d_S and P_S, the phase Φ_S is written as

It is shown that difference of Φ_S between points A and B is the multiple of π in Eq. (2). This is also true for points C and D. Similarly, difference of Φ_D for two adjacent resonant lines with a fixed value of Ω is also π in the PBD. The phase Φ_D in the PBD can be written as Φ_D = (d_D/P_S). Moreover, the values of Φ_S oscillate as the generation order increases in the STMS, as shown in Fig. 2(b). They approach a fixed value as the generation order increases. The sum of Φ_S,U/π around the three semi-quarter-wave thickness and Φ_S,D/π around the semi-quarter-wave thickness is unity for any generation order. The semi-quarter-wave is defined as an intersection of k_Ad_A = π/4 and k_Bd_B = π/4. k_A and k_B are the wavevectors in layers A and B defined by k_A = n_Aωcos(θ_A)/c and k_B = n_Bωcos(θ_B)/c, respectively. θ_A and θ_B are the angles of the wavevectors in layers A and B, respectively. However, the positions of the resonant lines are constant for all periodic numbers in gap map in the PBD. Thus, value of Φ_D /π does not oscillate, and the sum of Φ_D/π around the three quarter-wave thickness and Φ_D /π around the quarter-wave thickness is not unity.

Next, we consider the transmission for the STMS in terms of Φ_S, as shown in Fig. 3.It is seen that the positions of the resonant peak are arbitrary for the up and down major gaps, in Figs. 3(a) and 3(b), respectively. Although the Φ_S of the resonant peak are arbitrary, the period of the Φ_S to appear the resonant peak is π for all generation orders. For the PBD, the Φ_D of the resonant peak appear on mπ, where m is an integer, in Fig. 3(c). Moreover, the peak becomes sharper for ν→∞ in these two systems.

 

Fig. 3 The transmission spectra in the STMS versus Φ_S/π at the middle positions are in (a) the up major gap and (b) the down major gap, respectively. (c) The transmission spectra versus Φ_D/π at the middle position in the fundamental major gap for the PBD. The dashed red, the dotted green and the solid blue lines correspond to the generation orders ν = 3 to 5 in the STMS and to the numbers of period, m = 2, 4 and 8, in the PBD, respectively. (d) The FWHM and the group delay, dΦ_S/dω, on the first whole resonant line in major gap versus the generation orders of the STMS. The black solid line and the blue dotted line correspond to the up and down major gaps for the FWHM. The red solid line and the green dotted line correspond to the up and down major gaps for the group delay

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The effect on the full width at half maximum (FWHM) and the group delay, dΦ_S/dω, for different generation orders in the STMS is of interest. The minimum value for the FWHM and the maximum value for the group delay on the first whole resonant line in major gap are chosen as a reference. The relationship between the FWHM and the group delay for different generation orders is shown in Fig. 3(d). It is seen that the value of the FWHM decreases as the generation order increases. In other words, the sharpness of the resonant peaks increases as the generation order increases. However, the value for the group delay increases as the generation order increases. It is seen that the values of the FWHM and values for the group delay around the semi-quarter-wave thickness and the three semi-quarter-wave thickness are very close. Similarly, there are the same situations for the FWHM and group delay in the PBD. The value of the FWHM also decreases as the periodic number increases. Moreover, the value of group delay also increases as the periodic number increases.

In order to further understand the properties of localization in the one and three semi-quarter-wave thicknesses for the STMS for different generation orders, the squared electric field of the light propagation is considered, as shown in Fig. 4.It is shown that the squared electric fields are almost invariably symmetrical in the one and three semi-quarter-wave thickness for the ν₅ STMS shown in Figs. 4(a) and 4(b), and the ν₆ STMS shown in Figs. 4(c) and 4(d). In particular, there is stronger light localization simultaneously in the space layer. There is greater localization enhancement as the generation order increases. Similarly, this is also true for the quarter-wave thickness in the PBD with the same thicknesses d_A and d_B. The space mode can also be changed by changing the thickness of the space layer d_S. In other words, one resonant line represents only one mode style. For example, there is one period of wavelength for the squared electric field in the space layer on the sign, C, in Fig. 2(a), but there are two periods of wavelength for the squared electric field in the space layer on the sign, D, in Fig. 2(a). However, the first resonant line is an incomplete line at the down major gap that has a space layer of insufficient thickness, as shown in Fig. 2(a). Thus, the squared electric field in the space layer has an imperfect period of wavelength. Furthermore, the squared electric field adds one period when d_S increases from one resonant line to the next. This is also true for the PBD. Moreover, these two structures differ in that the peak values of squared electric field are the same in the one and three quarter-wave thickness in the PBD. However, the peak values of squared electric field are different in the one and three semi-quarter-wave thickness in the STMS.

 

Fig. 4 The |E|² distribution in the ν₅ STMS, for (a) d_S = 0.37µm and Ω = 0.519 in the three semi-quarter-wave thickness and (b) d_S = 0.245µm and Ω = 0.2315 in the semi-quarter-wave thickness, respectively, and in the ν₆ STMS, for (c) d_S = 0.43µm and Ω = 0.5 in the three semi-quarter-wave thickness and (d) d_S = 0.135µm and Ω = 0.25 in the semi-quarter-wave thickness, respectively, which correspond to the signs, C and A, in Fig. 2. The white areas symbolize the bonding media. The yellow, green and light blue areas symbolize materials A and B and the space layer, respectively.

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In this paper, we proposed a novel method to determine the occurrence of the localized mode in one-dimensional symmetric Thue-Morse quasicrystals based on the plot of the change of the normalized frequency for resonant lines. We find that there are three properties of the localized mode in the STMS. Firstly, the localized modes exist around odd semi-quarter-wave thickness in the STMS rather than the ones around quarter-wave thickness in a traditional PBD. Secondly, the frequency of localized mode in the transmission spectra of the STMS changes for different generation orders. Lastly, even if the thickness of the space layer in the first localized mode appears randomly for different generation orders, the localized modes appear periodically. The last situation is similar to that in the PBD when the thickness of the space layer is increased for a fixed frequency. We also find that the value of FWHM decreases and the value of group delay increases as the generation order increases. The sharpness of the resonant peaks becomes sharp as the generation order increases. These results show that the optical properties of STMS’s are similar to those for traditional PBD. Thus, they are suitable for use in advanced optical devices. Moreover, the propose method may be extended to investigate the localization problems for other quasiperiodic systems.