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Abstract
We propose an innovative design of highly reflective omnidirectional mirror based on chirped-type aperiodic and hybrid-order aperiodic-periodic multilayered structures in the spectral range from 400 nm to 3000 nm that contains a larger part of solar radiation. The aperiodic sequences considered are Fibonacci (FIB) and Thue-Morse (ThMo), while a power law function has been adopted to tune the thickness of layers. The chirping functions parameters were optimized to obtain maximum averaged reflectivity over the entire angular range. The numerical calculations were performed by a recursive formula via the scattering matrix method. For all the structures, the chirping effect improved the omnidirectional properties of multilayers in terms of average reflectivity and quasi omnidirectional spectral width (q-ODW). In detail, it was revealed that the ThMo based structures provide the best performance in terms of both averaged reflectivity and q-ODW. For ThMo chirped multilayer the q-ODW reached the value of 1605 nm with incident angular span 0-60°. On the other hand, the hybrid-order structure based on ThMo shows an q-ODW up to 2500 nm. Finally, for all the investigated structures the average reflectivity reaches a value larger than 0.93 using nearly 100 layers.
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1. Introduction
Thanks to their enticing characteristics, photonic crystal structures have long been in the spotlight of optics-based research. Such structures offer the ability of light manipulation. An interaction between an electromagnetic wave going into a photonic crystal with its periodically various dielectric functions, eventually, can result in photonic bands. If the propagation of light is disallowed, in similarity to the electronic bands, energy will lie within specific region namely, photonic band gap [1]. This behavior can tailor such structures for synthetization of omnidirectional mirrors (OMs). These kinds of mirrors are able to totally reflect the radiation in certain range of wavelengths for all angles of incidence in both transverse electric (TE) and transverse magnetic (TM) polarizations [2,3].
For enlargement of omnidirectional reflectivity range, a plethora of theoretical and experimental works have been done so far [4,5]. From theoretical point of view, S. Kumar Singh et al. proposed the incorporation of FIB quasiperiodic and a periodic structure by means of transfer matrix method [6,7]. They found the perfect reflection could be obtained if the bandgap of two substructures, SiO2 and Si, were closed to each other for all incident angles. Also, using the same method and for enhancement of omnidirectional reflectivity range, a structure with an exponentially varying refractive index was suggested by S. Sharma and colleagues [8]. They showed a structure composed by a periodic array of two alternate layers of dielectric/semiconductor with different values of refractive indices could be considered as a perfect reflector in a specific wavelength of 1500 nm. Furthermore, J. O. Estevez et al. [9] could make an increase of 2.3 times, in the complete angular range of 0° to 90°, compared with their own previous work, the omnidirectional width (ODW) for porous Si dielectric mirrors through a Gaussian profile refractive index. In another work, V. Kumar and coworkers [10] numerically tried to improve range of omnidirectional reflection in one-dimensional Si-based photonic crystal through making a defect in the multilayered structure, and proved that in three different angles of 0°, 45°, and 89° degrees, there is 100% of reflectivity. Moreover, Xifre-Perez et al. [11] and Bruyant and coworkers [12] increased the ODW up to 319 and 340 nm, respectively. Moreover, it has been shown that nonlinearity can affect the reflection properties of one-dimensional photonic crystal. O. Habli et al. [13] numerically designed 1D photonic crystal composed of nonlinear organic material (poly-9BCMU), and found that photonic band gap could be controlled by intensity of electric field at normal incident angle.
In the last years several optimization algorithms have been used to design omnidirectional mirrors. Estrasa-Wiese et al. proposed a procedure to design a broadband photonic crystal in a desired wavelength range using a stochastic optimization [14]. Eric Oritz-Vasquez and colleagues also managed, via genetic algorithm, to design and optimize one-dimensional photonic crystals made by porous Si. Their structure was verified with different functions to reach maximum amount of reflectivity, more than 90%, in visible spectral range [15]. Finally, H. Kim and coworkers, theoretically proposed a Bragg reflector based on SiO2/TiO2 by means of the optimization of refractive index of porous SiO2 to attain broad-band as well as broad-angled high solar reflectivity from visible to near-infrared frequencies [16].
Experimentalists have also attempted to widen the omnidirectional range via different techniques using sol-gel deposition [17], via molecular beam epitaxy [18], or through thermal evaporation and spin-casting [19]. Jena et al. with the aim of magnetron sputtering methods have obtained different values for ODW [20]. In other attempts, working on refractive index, J. O. Estevez and coworkers [21], and A. David Ariza-Flores and colleagues [22,23] through in respective Gaussian profile refractive index and Bragg type chirped layers with a growing thickness fabricated the complete visible range OM with porous Si. The latter group reached 95% for reflectivity in two different incident angles: 8°, and 68°. Nevertheless, each procedure has its own specific defect and weakness.
One of the challenging methods to enlarge the omnidirectional bandwidth of mirrors with broadband feedback and wide frequency is the chirped-type structures [12]. In these types of structures, the layer thickness changes gradually as a function of their depths. Furthermore, it has been confirmed that a hybrid-order-based structure, that is the juxtaposition of an aperiodic structure with a periodic one, can increase the ODW [24]. So, in this work, for the first time, we have designed a chirped multilayered structure based on aperiodic and hybrid-order, aperiodic-periodic, sequences to maximize the quasi-ODW (q-ODW) in the spectral ranges Rs, from 400 to 3000 nm, that is a crucial region for the solar radiation [25].
At first, let us define the investigated multilayer structures. Such structures are made of two types of dielectric materials, namely A and B, with respective refractive indexes, nA = 1.5, nB = 3.5. If the substitutional rule of a dielectric structure indicated by two symbols of A and B is g(A)=AB and g(B)=A, the well-known quasicrystal FIB multilayer is built. Therefore, the first generations of such multilayer will be A, AB, ABA, ABAAB, and so forth. Such kind of structures have long-range order but lack translational symmetry [26].
Moreover, such symbols can also have another substitutional rule: g(A)=AB and g(B)=BA with the generations of A, AB, ABBA, ABBABAAB, ABBABAABBAABABBA, and so on which will produce ThMo multilayer; an aperiodic structure with a singular continuous frequency spectrum [27]. The thickness of j-th layer is written in terms of λ/4 stack varying the wavelength gradually: dj = λc(j)/(4nj) with j = 0 … Nl–1 where Nl is the total number of layers and nj is the refractive index of j-th layer. The wavelength, λc(j) increases according to the depth of structures following the relation:
where ${\lambda _{min}}$ and ${\lambda _{max}}\textrm{}$ are the extreme wavelengths of proposed spectral range, 400 and 3000 nm respectively, and ξ=j/(Nl-1). Furthermore, $f(\xi )$ is a smooth function varying between 0 and 1 once its arguments changes from $\xi $=0 to $\xi $=1. In particular, we take into account the power law chirping function defined as [28]:In the following the parameter of α has been adjusted to maximize the average reflectivity, Rav, which is given be following formula:
In the current study we have also investigated a multilayered structure constituted by the combined sequence of an aperiodic sequence, as FIB or ThMo, with the periodic ones to increase the average reflectivity in the interested spectral range and to enlarge the ODW. In Fig. 1 it has been reported the reflectivities at normal incidence of multilayers without chirping effect, dj = λ0/(4nj), where λ0 = ( λmin + λmax)/2 is the central wavelength of our investigated range. We observe that the periodic multilayer has a high reflectivity spectral region whereas FIB and ThMo multilayers show an oscillatory behavior and hence they are unable to act as a perfect mirror. So, this motivated us to combine a periodic structure with the aperiodic one (FIB or ThMo) to construct hybrid-order structure FIB-PER and ThMo-PER multilayers to cover those frequency regions which cannot be reflected in ThMo and FIB multilayers.
Fig. 1. Reflectivity of a) periodic multilayer, Nl = 32; b) FIB multilayer, Nl = 55; c) ThMo multilayer, Nl = 64; in the Rs spectral range wavelength.
2. Electromagnetic calculation method
The numerical simulations about electromagnetic propagation were performed through scattering matrix method [28], in which each layer is characterized by a 2 × 2 matrix. We consider a multilayer system with N layers and N + 1 interfaces surrounded by the air (n0 = nN + 1 = 1), as reported in Fig. 2.
The electromagnetic wave with amplitude equal to 1 impinges on the interface labeled with 0, with the incident angle of θ0 and it is refracted with an angle θ1 ruled by the Snell’s law. In general, on the m-th interface we can write:
Therefore, taking into account the backward propagation from (m + 1)-th interface into m-th interface, the transmission and reflection coefficients, tm and rm, respectively, will be given from the following recursive relations [30]:
3. Results and discussion
We investigate the contribution of chirping function on the q-ODW and on the Rav for aperiodic and hybrid-order aperiodic periodic multilayers. While the possibility to use aperiodic structure for omnidirectional mirror has been investigated in Refs. [31–34]. Until now, to the best of our knowledge, no work has been published about the chirping effect in aperiodic-based multilayers.
3.1 Chirped aperiodic multilayer
We have investigated the effect of chirping on the omnidirectional reflection properties of ThMo and FIB multilayers as a function of the number of layers. For each structure, the α parameter of chirping function has been chosen to maximize the Rav therefore the reflectivity spectra have been calculated in the angular range between 0° and 70°.
3.1.1. Fibonacci chirped multilayers
The omnidirectional mirror properties of FIB multilayer have been investigated starting from the 10th order FIB multilayer, composed of 55 layers, until the 13th order with 233 layers. In Fig. 3 we report the reflectivity spectra for two polarizations of no-chirped FIB (the left column, corresponding to 3a), 3c), 3e), 3 g)) compared with the optimized reflectivity, in terms of parameter α, of chirped FIB (the right column, corresponding to 3b), 3d), 3f), and 3 h)). Here, it emerges clearly the effect of chirping on the omnidirectional properties of systems. In fact, for no-chirped FIB structure the shape of reflectivity spectra does not change significantly by increase in the number of layers; the band gaps regions are spaced out from strong transmission resonances, and it is possible to have a high reflectivity region only in a limited angular range.
Fig. 3. Reflectivity for TE and TM polarizations as function of incident angle and wavelength of FIB multilayer with no-chirped (left column) and chirped (right column) for different length systems: 55, 89, 144, 233 layers.
On the other hand, the reflectivity of chirped FIB shows a strong dependence on the number of layers with large spectral region with high reflectivity value. Due to narrow and isolated transmission resonances (that are deep holes in reflectivity spectra) it is useful to evaluate the omnidirectional high reflectivity properties by reporting in a 2D graph the reflectivity spectra for two polarizations and for several incident angles.
In Fig. 4 we have shown the reflectivity spectra of chirped FIB for both polarizations in an angular range 0°-60° with step of 5°. A valuable q-ODW appears in a FIB multilayer with 89 layers, and it increases at 1080 nm for FIB with 233 layers. However, beyond the omnidirectional spectral region we observe a larger region of high reflectivity with the presence of some weak transmission peaks that can be classified as q-ODW [28]. For example, for Nl = 144, it can be observed that the quasi-omnidirectional region starts from λ=1300 nm until to about λ=2500 nm.
Fig. 4. Reflectivity spectra for TE and TM polarization for chirped FIB multilayers for different incident angles in a range between 0° and 60° with step of 5°. Nl is the number of layers.
Finally, in Fig. 5, we report the average reflectivity calculated over the entire investigated range (400-3000 nm), Rav (see Eq. (3)) as function of the number of layers for no-chirped and chirped FIB multilayers. An improvement of Rav of about 14% is obtained by using chirped structures. Furthermore, just for chirped FIB with 83 layers we have Rav > 0.94.
Fig. 5. Average reflectivity of chirped and no-chirped FIB multilayers as function of the number of layers.
3.1.2. Thue-Morse chirped multilayers
To investigate the chirping effect on ThMo multilayer, we have adopted ThMo multilayers from the 5th to the 8th order that are composed of 32 until to 256 layers. In Fig. 6 we illustrated the reflectivity spectra for both polarizations in relation to an angular range of 0-60°. By increasing the number of layers, the transmission peaks reduce until to obtain for Nl = 128 a q-ODW, in the 0-60° angular range, of 730 nm. For the larger system under investigation the q-ODW is larger than 1600 nm and it covers the entire spectral region under investigation from λ=1.39 µm. The behavior of Rav for no-chirped and chirped structures is reported in Fig. 7. Also, in this case we observe an improvement of about 14% of Rav respect to the no-chirped structures, the Rav is about 0.96 for chirped ThMo with 128 layers.
Fig. 6. Reflectivity spectra for TE and TM polarization at different incident angles in a range of 0°-60° with step of 5° of chirped ThMo multilayer for different number of layers, Nl.
Fig. 7. Average reflectivity for chirped and no-chirped ThMo multilayers as function of the number of layers.
3.2 Chirped hybrid-order aperiodic-periodic multilayers
As shown in Fig. 1, the juxtaposition of periodic multilayer with the aperiodic one makes it possible to enlarge the ODW. The chirping function reported in Eq. (2) has been applied using the following chirping function:
In the following it has been reported the optimized reflectivity respect to two parameters, α and β for FIB-PER structures of 87, 121, 176 and 265 layers, corresponding to 10th, 11th 12th and 13th orders following from a 32-layered periodic arrangement that are enough to guarantee the theoretical reflection bandwidth obtainable with the refractive index contrast, nA/nB. In Fig. 9 we report the reflectivity for different incident angles in the range between 0° and 60°. By comparing omnidirectional properties of chirped FIB-PER with the chirped FIB (Fig. 4), we observe that the adding periodic part gives an improvement of high reflectivity spectral region for the FIB multilayer of 10th and 11th order, in fact the q-ODW becomes 580 and 810 nm respect to 0 and 670 nm, respectively. For larger FIB multilayer (12th and 13th order), the juxtaposition of periodic part generates some strong transmission peaks in the spectra, however the q-ODW value passes to be 1190 nm respect to the value of 1100 nm for FIB.
Fig. 9. Reflectivity spectra for TE and TM polarizations at different incident angles in a range of 0°-60° of chirped FIB-PER multilayer for different number of layers: 87, 121, 176, 265
On the other hand, by analyzing the hybrid-order ThMo-PER in Fig. 10, and by comparing it with the ThMo structure (Fig. 6) we observe an enlargement of quasi-ODW for all the ThMo multilayers. The transmission peaks are increasingly rarer as the number of layers increases.
Fig. 10. Reflectivity spectra for TE and TM polarizations at different incident angles in a range of 0°-60° of chirped ThMo-PER multilayer for different number of layers: 64, 96, 160, 288.
Observing the Fig. 11, where is reported a resume of Rav for the investigated structures, it is interesting to note that Rav does not drastically change after 100 layers while, ThMo, FIB and FIB-PER structures have the value of about 0.93 for the average reflectivity within about 60 layers. However, for small number of layers, less than 50, ThMo and ThMo-PER multilayers have higher Rav than FIB ones.
Fig. 11. Rav of investigated systems as function of the number of layers. The Rav for FIB and FIB-PER for 100, 150, 200, and 250 has been obtained by truncating the 14th FIB order (Nl = 377)
Eventually, in Fig. 12 it has been reported the q-ODW as a function of number of layers for incident angle range between 0° and 60°. The ThMo structures show a large sensitivity of q-ODW respect to the number of layers. In other words, the q-ODW value for hybrid-order structure of ThMo-PER and ThMo reached approximately 2500 and 1700nm, respectively. While the FIB based systems have a sort of plateau after about 150 layers. However, comparing the q-ODW for FIB multilayer with other structures once they are composed of fewer than 150 layers, one can simply find that FIB structure has the highest amount of q-ODW value, nearly 1200 nm. For better comparison in q-ODW values, we have reported, in Table 1, a summary of the results of our work with the others structures previously published.
Fig. 12. The q-ODW as function of the number of layers for ThMo, FIB, ThMo-PER and FIB-PER structures for 0°≤θ≤60°.
Table 1. Development of omnidirectional mirrors for previously published works with different multilayer strategies and materials
4. Conclusions
We have proposed new structures to design the broadband omnidirectional mirror based on the chirping of layers thicknesses in aperiodic and hybrid aperiodic-periodic structures. The calculations have been exploited by means the scattering matrix method. The results show, potentially, a strong improvement of q-ODW and of the average reflectivity respect to the other devices proposed in literature.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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