Ultra-broadband polarization-independent omnidirectional reflectors via dielectric-reversal quasicrystal heterostructure multilayer films

Since quasicrystals provide additional freedom to expand photonic bandgap, dielectric quasiperiodic sequences of broadbands become crucial for implementing low-loss omnidirectional reflectors or filters. The current major challenge is to find an elegant scheme to construct angle-insensitive multilayer structures with fewer layer numbers. In this work, we create an appealing quasiperiodic ordering of the dielectric-reversal Fibonacci sequence to optimize the properties of omnidirectional reflectors. We have concatenated it to a periodic sequence to construct the omnidirectional reflector of hybrid multilayers with thin thickness and reflectivity above 99% over a visible scope from 436 to 559 nm. An ultrabroad polarization-independent omnidirectional bandgap of the relative bandwidth of 39.8% has been achieved by cascading two dielectric-reversal ordering sequences. The self-similarity of Fibonacci multilayer structures along with this characteristic quasiperiodic order guarantees the existence of perfect omnidirectional reflections at any scale. This ordering strategy of dielectric-reverse essentially differs from previous conjugate and recursion methods, which will significantly enrich the variety of quasiperiodic orders in theory and applied science domains.


Introduction
Omnidirectional photonic band gap (OPBG) [1] has attracted researchers' great attention because of strong reflection at any incident angle with a fixed bandwidth, which can be utilized as omnidirectional reflectors (ODRs) or filters. ODRs have potential applications, such as microcavities, coaxial waveguides, antenna substrates. One-dimensional (1D) common periodic photonic crystal (PC) [2] is sensitive to incident angle, so unfit for ODRs. Quasicrystal multilayers [3] are in particular 1D PCs constructed according to certain quasiperiodic sequences with perfect long-range order and lack of translational symmetry. Self-similarity arrangement and periodicity deviation lead to more complex transmittance properties and broadening of photonic bandgaps (PBGs). Different materials of semiconductors [4], superconductors [5], plasmas [6], graphene [7], etc. have been induced to broaden omnidirectional reflection bandgaps (OBGs) with fewer layers, but the issues of high cost, large absorption loss, and low threshold of laser-induced damage limit their potential applications in some specific domains. Dielectric multilayer reflectors show extremely low loss in the optical range. Various quasiperiodic sequences, such as Fibonacci [8], Octonacci [9], Thue-Morse [10], maximum length [11], and Rudin-Shapiro (RS) [12] have been studied for high reflectivity in omni-direction. Conjugation operation [12] and heterostructures [13] are usually performed to optimize the properties of omnidirectional reflection. However, hundreds of constitutive layers and too many defects for large incident angles hinder realistic fabrication procedures and practical applications. Novel approaches in the quest for low-loss ODRs of better performances are urgently needed for certain specific optical applications.

Theoretical principle and method
Since Fibonacci quasicrystals show significant forbidden frequency bands with self-similar energy spectrum [14], Fibonacci sequence (FS) [15] has been investigated intensively. FS is defined as a generation of two kinds of materials with lower L and higher H refractive indices, following the iteration rule of S 0 = L, S 1 = H, S 2 = HL, S 3 = HLH, S 4 = HLHHL, S 5 = HLHHLHLH,……, S m = S m−1 S m−2 , m ≥ 2, where m represents the FS order. Generally, TM-polarized mode is more sensitive to incident angle than TE mode due to the Brewster angle; omnidirectional bandgaps are always defined by the PBGs of TM mode [12]. Thereby, we only consider the TM polarization in the following discussions. The periodic multilayer and the 4 th and 5 th FS multilayers are investigated to demonstrate the effect of sequence configuration on dispersion properties of PBGs. Considering the factors of low loss and mature coating technology [16], we select TiO 2 and SiO 2 with the refractive indices of n H (TiO 2 ) = 2.627 and n L (SiO 2 ) = 1.44 as the constituent materials of the multilayer structures, with each layer thickness satisfying the quarter-wavelength condition. Different from the PBGs of the periodic multilayer in Fig. 1a, the 4 th (Fig. 1b) and 5 th (Fig. 1c) FSs exhibit wider and more abundant characteristics of PBGs.
Various combinations of reflection windows in FSs provide more design ideas to optimize the PBG properties. Figure 2a depicts the structure diagrams of the 5 th FS(HLHHLHLH), dielectric-reversed FS′(LHLLHLHL) by reversing the high-and low-refractive index constitutive materials, and the combination of FS|FS′ (HLHHLHLH|LHLLHLHL) by concatenating the FS to FS′, respectively, where H denotes TiO 2 and L denotes SiO 2 . The Fourier spectra of FS with the long-range quasiperiodic order consist of isolated Bragg peaks in reciprocal space [17] which correspond to possible PBGs in the frequency domain, so we calculated their Fourier spectra by the algorithm of discrete Fourier transform to analyze the recursive properties. These FS sequences can be converted to binary sequences by defining H = 1 and L = 0; the Fourier spectra were got by solving the following discrete Fourier formula: where X k is the discrete Fourier value and x n is the binary quasiperiodic sequence. For better clarity, supposing the layer numbers of all sequences are set 520, Fig. 2b-d shows the calculated discrete Fourier spectra for the FS, FS′, and FS|FS′ of the 4 th to 6 th orders. As evident from these results, the FS and FS′ exhibit the same Fourier spectra due to the same order of binary sequences, while the FS|FS′ exhibits completely different Fourier spectra (blue dash lines) for different orders. Neglecting the DC term of the first peaks, the most noticeable peaks are separated with different distances, and the minimum distance is obtained in the 5 th FS|FS′ (Fig. 2c), implicating that the interesting optical phenomenon of close PBGs can be exploited in this dielectricreverse heterostructure.
We numerically calculated the reflection spectra of the 5 th (FS) 4 , (FS′) 4 , and (FS|FS′) 2 separately by using the CST Microwave Studio to verify the reflection properties. Each sequence contains 32 layers. As can be seen from Fig. 2e, f, the FS and FS′ structures really exhibit similar PBG distribution, as expected, with a lot of passband defects interspersed between the main reflection bands. While, for the case of the FS|FS′ in Fig. 2g, there is just one narrow passband defect appearing between two hyperreflective bands. These reflection spectra are consistent with the predictions of the Fourier spectra in Fig. 2c. Abundant PBG distributions in quasiperiodic sequences originate from the defect layers of "HH" or "LL" disrupting the sequence periodicity. However, more layer defects induce more passband defects. Compared with the excess defect layers initiated at the interfaces of the repeated 5 th FS or FS′, the FS|FS′ multilayers conquer this drawback, with a narrow transmission gap substituting the complex passband defects in Fig. 2g. This structural

Heterostructure of FS|FS′ and PC
According to the heterostructures method [18], two or more periodic or quasiperiodic multilayer structures can be stacked to construct hybrid sequences to expand PBGs [4]. Hence, we took advantage of consecutive PBGs of 1D PC (HL) 16 16 always covers the defect gap of the (FS|FS′) 2 , which predicts that an ultrawide reflection band could be realized in the tandem structure of (HL) 16 and (FS|FS′) 2 . It is found that the order of constituent sequences causes significant effects on the defect feature of heterostructures at large incident angles. Based on the numerical simulation, we built a stacked multilayer model consisting of (HL) 16 and (FS|FS′) 2 of the same central wavelength of 550 nm on an infinite BK7 glass substrate (n sub = 1.52), described as Sub|(HL) 16 (FS|FS′) 2 with the total thickness of 4.73 μm. Figure 3d-f shows the measured reflection spectra as the incident angle changes from 0 to 85°. For the normal incidence, a wide PBG of total reflection almost covering the whole visible scope has been achieved from 442 to 727 nm. With the incident angle increasing, the PBG blueshifts gradually. Due to the sufficiently broad bandwidth, even at 85° the PBG still partly overlaps with that at 0°, as indicated by the blue shadow area. Ultimately, overcoming the effect of blueshift, a polarizationindependent omnidirectional reflection of high reflectivity R > 99% is obtained in the wavelength range from 436 to 559 nm with a relative bandwidth of 24.7%, which is much larger than the periodic PC (HL) 32 in the same conditions.
In Table 1, we illustrate the achieved OBG parameters of the periodic PC and different hybrid heterostructure multilayers with the same 64 layers. Compared to the 1D PC of (HL) 32 , the OBG of the heterostructure composed of (HL) 16 and (FS|FS′) 2 is enlarged 2.55 times. This is the first time such a perfect dielectric polarization-independent ODR is realized with so wide relative bandwidth and so thin thickness < 5 μm by the periodic and quasiperiodic cascading fashion manner. The more layers of the periodic sequence, the higher reflectivity with fewer defects. The design of 64 layers with the thickness < 5 μm is entirely feasible for the existing technology in the realistic preparation procedure.

Heterostructure of FS|FS′ structures
If two (FS|FS′) 2 sequences of different central wavelengths λ c exhibit complementary reflection spectra at any incident angle, it is possible to superimpose their PBGs to expand the global reflection band. In this work, two (FS|FS′) 2 constituent sequences of λ c1 = 513 nm and λ c2 = 587 nm were selected, whose cascaded structure diagram of Sub|(FS 1 |F S 1 ′) 2 |(FS 2 |FS 2 ′) 2 is shown in Fig. 4d. Figure 4a-c exhibits their separate reflectivity spectra indicated with the yellow and red shadow areas for different incidence angles of 0°, 45°, and 85°, where the orange shadows denote the superimposed areas. All the colored reflection bands exactly overlap each other, concatenating head-to-tail to merge into an ultrawide reflection band. The reflectivity spectra at the specific incident angles of 0° and 85° are exhibited in Fig. 4e, respectively. A wide omnidirectional PBG of the relative bandwidth 39.8% is achieved in the wavelength scope from 403 to 603 nm, which is much wider than other reported similar dielectric multilayers. To demonstrate the OBG properties are determined by the TM-polarized mode, Fig. 4f presents in the TM-and TE-polarized full reflectivity spectra as a function of incidence angle and wavelength, where two white lines indicate the borderlines for polarization-independent OBG. It demonstrates that the TE-polarized PBGs are always wider than the TM-polarized, thereby the bandwidths of OBG are exactly determined by the TM mode. Compared with the cascaded periodic PCs of Sub| (H 1 L 1 ) 16 (H 2 L 2 ) 16 in the same conditions, as shown in Table 1, the 5 th FS|FS′ sequence is a good candidate for constituting broadband ODR. Unfortunately, two obvious defects appear in the consecutive reflection bands for 85° in Fig. 4e, making the OBG less than perfect. Thus, we need to find an effective solution to eliminate small defects interfering.
In the above-mentioned study on the heterostructure of FS|FS′ and PC, we found the order of constituent sequences may cause significant effects on the defect feature of heterostructures for large incident angles. Therefore, we exchanged the order of (FS 1 |FS 1 ′) 2 and (FS 2 |FS 2 ′) 2 . Figure 5a presents the reflectivity spectra of the Sub|(FS 2 |FS 2 ′) 2 |(FS 1 |FS 1 ′) 2 at 0° and 85°. Compared to the former of Sub|(FS 1 |FS 1 ′) 2 |(F S 2 |FS 2 ′) 2 , the later exchanged heterostructure exhibits the similar OBG range, but for the large incident angle, such as 85°, the defects become quite distinct from the former. Both two defects shift toward short wavelength, broadening the perfect OBG range denoted by the shaded area, achieving an optimized OBG of the relative bandwidth of 28% for the high reflectivity of R > 98%. To further clarify the advantage of this dielectric-reverse heterostructure, in Fig. 5b we magnify the reflection spectra at different incident angles of 0°, 60°, and 85°. It is found that strong reflection can be achieved readily in a wide wavelength scope with the relative bandwidth > 46% for incident angles from 0° to 60°. For the larger incident angle such as 85°, although certain defects appear, the perfect OBG from 448 to 594 nm is broad enough for practical applications of ODR, which can be obtained at any scale due to the self-similarity of Fibonacci multilayer structures. These results turn out the appropriate arrangement of constituent sequences order is effective to improve the defect state in the omnidirectional reflection region. Moreover, random or gradual small offsets of each layer's thicknesses are likely to provide ample opportunities to further optimize the properties of reflection broadbands, which are going to be discussed in the future works.

Conclusion
In this paper, we proposed an effective scheme of dielectricreversal quasiperiodic ordering for perfect ODRs of fewer dielectric layers by enlarging OBG and eliminating defects. It is proved that the 5 th FS|FS′ is an excellent candidate as a component sequence to challenge the task of broadband omnidirectional reflection in dielectric multilayer structures. The iconic feature of close reflection broadbands makes it easier to superimpose other periodic and quasiperiodic sequences to bring about angle-insensitive total reflections. The appropriate arrangement of constituent sequences order and central wavelength selection can optimize the ODR properties. The characteristics of wide OBG, fewer layer numbers, and high reflectivity enable them applicable for applications such as multidirectional reflectors of the laser cavity, laser protection, laser damage resistance, etc.
Author contribution HW carried on simulations and wrote the partial manuscript text. XG prepared figures and text typesetting. XZ carried out the partial digital calculation. GD proposed the idea and reviewed the manuscript.