The engineering of electromagnetic modes at optical frequencies in artificial dielectric structures, with periodic, quasiperiodic or random variation in the refractive index, was first proposed independently by Yablonovitch, Kohmoto and John in 1987 [1-3]. Such kinds of complex dielectrics are often considered as photonic crystals or quasicrystals, and can be much easily obtained in a onedimensional multilayered configuration. Their attributes have generated worldwide research and development of sub-µm and µm size active and passive photonic devices such as a single-mode and non-classical sources, guided wave devices, resonant cavity detection, and components for optical communication.

Quasicrystals are non-periodic structures constructed following a simple deterministic generation. Among the different structural sequences used for quasicrystal fabrication one finds Cantor-like, Fibonacci (FN), generalized Fibonacci, Thue-Morse (TM), generalized ThueMorse, Period Doubling and Rudin-Shapiro. Although the FN system has been the subject of most study in this field. Electronic properties of FN multilayer structures are quite widely investigated (see for instance [4-7], and references therein). There are also studies on vibrational frequencies and atom displacements [8], as well as the propagation and localization of elastic waves [5,9-11]. Although less investigated than FN ones, the TM multilayered systems have also deserved significant attention. Their electronic properties have been studied, for instance, in [4,6,12,13]. In addition, some properties of elastic waves in quasiregular TM structures are reported in [14,15].

From the optical point of view, previously done work has focused on the study of the localization of light waves within FN quasiperiodic multilayer quasicrystals which leads to the appearance of photonic bandgaps, in a similar way to those existing in periodic structures. The photonic localization in the dielectric microstructure bears an analogy to the electronic localization in crystals [3]. Later on the concept of localization was recognized as applicable to any type of wave, such as acoustical wave [16]. Kohmoto et al. introduced the first system based on optical FN multilayers capable of localizing photons [2]. In 1994 Gellermann et al. showed experimental proof of the existence of bandgaps in the spectrum of FN dielectric multilayers [17]. There are also reports in the literature about the obtention of omnidirectional bandgaps using FN quasiperiodic structures [18,19]. A study on the features of the optical propagation in porous silicon-based FN polytype multilayers was put forward by Agarwal and Mora-Ramos [20]. Optical properties of non-Fibonacci 1D dielectric quasiregular systems have also been studied by some authors [20-41].

On the other hand, in a couple of works on quasiperiodic dielectric multilayers, E. Maciá outlined the possibility of designing hybrid dielectric heterostructures [42, 43]. Soon after, Wen et al. put forward the realization of an omnidirectional reflector made of on the basis of a Fibonacci-Bragg(periodic) hybrid system [44]. Then, for some years, the study of hybrid dielectric systems was practically out of literature. It is worth mentioning that during such period A. Montalbán and his collaborators lead a research on the behavior of oscillation modes in hybrid 1D periodic/quasiregular chains of point masses [45-47]. One of the main results of these works is the appearance of a selective spatial confinement of the vibration amplitudes, whether it is within the multilayered periodic part or whether it takes place within the quasiregular region of the structure.

Motivated by those reports on mechanical oscillations, in 2009 we presented some preliminary theoretical reports on the optical propagation in porous silicon-based multilayers designed as hybrid combinations of FN and Bragg mirror (BM) dielectric sections [48-50]. In these works we obtained an enhancement of in-gap microcavity mode localization as well as of the omnidirectionality of the reflectance properties of the structures proposed. Moreover, a significant selective spatial localization was detected for some propagating modes. That is, there are wavelengths for which the intensity of the electric field confines almost totally within the FN section or within a BM one. Some refinements of these results, but still preliminary ones were published in a more recent article [51]. It is worth mentioning that, later than the publication of our 2009 reports there have been some communications in relation with optical properties of hybrid periodic/ quasiregular and quasiregular/quasiregular hybrid dielectric heterostructures [52-54]. There is also a very recent experimental and theoretical report on tunable resonant transmission in a kind of porous silicon-based heterostructures [55]. The main interest in these works is focused on their use as reflectors and filters, and nothing is said there about the confinement or localization of the optical waves.

In this article we are aimed at developing a more detailed study of the selective spatial localization of the electric field of light waves propagating along hybrid FN-BM heterostructures. The discussion will consider both FN-BM-FN and BM-FN-BM multilayer designs together with the effects of the angle incidence and the variation of the dielectric contrast by changing the refractive index of the two basic layers involved. The work is organized with a section that follows on, presenting the model and the calculation scheme. Then there is a section containing the analysis and discussion of the obtained results. Finally, the main conclusions of the work shall be given.

A hybrid structure is defined as a combination of two independent structures with different properties, in this case a periodic and an aperiodic structure, to form a structure with superior features. As we previously mentioned, for the periodic part we choose the classical BM structure, while for the aperiodic we chose the FN one. These structures are produced with the use of the following substitution rules: A_j → A_{j – 1}B_{j – 1}, B_j → A_{j – 1}B_{j – 1} and A_j → A_{j – 1}B_{j – 1}, B_j → A_{j – 1} for the periodic and quasiperiodic structure, respectively. Both constituents have A₀ = A and B₀ = B as a seed. So, for instance, the third generation period structure is given by BM(3) = ABABABAB while the fourth generation Fibonacci structure is given by FN(4) = ABAABABA. Once they are constructed, the hybrid assembly design implies putting them together in a sandwich configuration, for which we are considering two different ways: BM-FN-BM and FN-BM-FN. In our particular case, we use the pair of refractive indices n_A = 1.8 and n_B = 1.2 which are commonly used in the fabrication of porous silicon heterostructures for photonic applications [20,34,36,39]. The choice for the layer widths is made according the quarter-λ rule: d_A,B = λ₀/4n_A,B, with λ₀ = 800 nm. The Figure 1 schematically shows the BM-FN-BM design, with the indication of the incident, reflected and transmitted waves. Viewed in this way, the proposed structure can be considered as a system of two or more defects (depending upon the particular order of the FN generation) included in the otherwise periodic configuration. A bulky defect would be obtained if the central (FN) part were made of layers with values of the refractive index distinct to those used in the BM part. But this possibility will not be discussed here, rather it could be the subject of another study.

The calculation of the electric field of s and p polarizations in the hybrid dielectric multilayer uses the well known transfer matrix method. Its details can be found elsewhere [56-58]. The approach makes provision for the variation of the angle of incidence as well as for the discontinuity in the dielectric properties when passing from a layer A to a layer B and viceversa. Within this framework, we are able to calculate the reflection and transmission coefficients, but our main interest is the obtention of the relative electric field intensity (defined as the quotient |E(z,ω)/E_i|², with E_i being the amplitude of the incident signal) as a function of the position z in the structure, measured from the point of light incidence. The expression that allows us to evaluate the electric field amplitude in terms of the transfer matrix is

where w_αβ are the elements of the matrix that connects the field at the extreme left-hand interface with the layer in which the coordinate z lies. The quantity r(ω) is the complex reflectance of the whole structure, and the factor γ₀ appears as a result of the imposition of suitable boundary conditions on the field components at the interface of light incidence (see Figure 1). Then, the field intensity is simply the absolute square of the electric field, |E|², which is normalized to the amplitude of the incident wave.

The Figure 2 contains the calculated optical properties of hybrid Bragg-Fibonacci heterostructures in different configurations. The color density graphics correspond to the relative electric field intensity associated to the optical mode with energy hω (vertical axis), at a given position z in the multilayer structure. On the side insets, one can see the corresponding normal-incidence reflectance, R = |r(ω)|², in each case.

In the graphics 2a and 2b we are representing the results for hybrid BM3-FN4-BM3, and aBM3-aFN4-aBM3 dielectric heterolayers. The prefix “a” indicates that the structure considered has the refractive indices of its layers

exchanged; that is, n_A ↔ n_B. Both BM3 and FN4 parts consist of eight monolayers. On the other hand, the FN4 contains a total of five “A” layers whilst there are only four in the BM3. This makes the physical thickness of each part to have a different value. According to the λ/4 design, the BM3 and aBM3 parts have the same width: 1111 nm. However, the total thicknesses of the FN4 and aFN4 substructures are different: 1056 nm and 1167 nm, respectively. Color bars located above each graphics serve as intensity-scale indicators: from zero (deep blue) at the left to maximum (dark red) intensities at the right. In each case, the maximum value of the relative intensity attained within the structures appears at the top right of the figure. The arrows are pointing at the positions of such maximum intensity peaks within the structures.

From the analysis of Figures 2(a) and 2(b) we can see that the relative field intensity of some particular modes show a selective spatial confinement, whereas the intensity of the electric field of the remaining propagating optical modes in the energy interval considered distributes rather homogeneously throughout the structure, with the normal occurrence of maxima and minima resulting from the multiple interference. Of course, the far lefthand part of the system shows the presence of illuminated zones even in the energy region within the photonic band gap. This is due precisely to its proximity to the surface of light incidence. The maximum relative intensity of the BM3-FN4-BM3 corresponds to a mode with energy immediately above the gap. The most of the spatial location of the associated electric field squared amplitude is approximately at the middle point of the left Bragg mirror. One readily sees that when z > 1000 nm, the relative intensity decreases significantly. It is possible to observe also the spatial localization of rather high field intensity in the left-hand Bragg multilayer. The energy of such a mode is now immediately below the gap. However the values of |E|² in this case are smaller. For these two optical modes, the confinement of a greater part of the electric field amplitude at the left of the system can be explained as follows: The remaining parts of the hybrid heterostructure are acting as mirrors that reflect the light back to the left (notice that the region containing the right-hand Bragg part is colored in darker blue which means that almost no light is propagating throughout it).

There are two signals of low, but nonzero relative intensity with spatial field localization mainly in the central FN4 substructure. These correspond to the pair of microcavity modes within the photonic band gap. Such modes can be related with the situation of a “double A-defect”, introduced by the FN substructure. The same double microcavity is present in the aBM3-aFN4-aBM3 configuration Figure 2(b). They are at the same energy position of those of the BM3-FN4-BM3 structure; but from the reflection spectrum one may see that they are sharply defined, reaching almost the unity transmission. With regard to their spatial field dependence, we can say that the change in the dielectric contrast (with n_A now having the lower of the two values) has led to the change in the relative intensity localization. In this case the microcavity modes and not the bandgap-edge ones exhibit the strongest confinement, which takes place within the aFN4 region this time.

Figures 2(c) and 2(d) show the calculated relative intensity and total reflectance for hybrid FN4-BM3-FN4 and aFN4-aBM3-aFN4 multiple heterolayers. This is not the typical Bragg-microcavity optical complex design. Now the photonic band gap appears split into two parts separated by a region containing two main propagating modes, whose relative intensities confined mostly within the left-hand FN4 or aFN4 substructures. Interestingly, the process of multiple internal reflections in the hybrid system makes these two optical signals to travel along the structure in such a way that, close to its right-hand end, they seem to merge into a wave with a an intermediate frequency. This frequency is the one at which there is a local minimum of the reflectance in the separated transmission gap structure.

However, the highest relative field intensity in these structures corresponds to a microcavity-like mode located close to the high-frequency edge of the photonic band gap. In this case, the main spatial confinement of the field occurs in the central BM part of the hybrid multilayer, although non-vanishing field intensities can be observed in the two adjacent FN4 substructures as well. In all, the highest values of the relative intensities attained at the maxima indicated by the arrows, are smaller than those occurring in the BM3-FM4-BM3 structures of Figures 2(a) and 2(b).

The physical reason behind the behavior of the light propagation in the hybrid dielectric heterostructures is the process of multiple internal reflections associated to the interfaces that separate regions of different dielectric properties. The discussion made above assumes that the light incidence to be normal to the layers. However, the electromagnetic usually reaches the structure forming an angle with the normal to the layer’s plane. Then, there will be multiple Snell-type refraction processes at the interfaces. Taking this into account, the Figure 3 presents the spatial distribution of the calculated relative field intensity for particular optical modes as a function of the angle of incidence.

Figure 3(a) contains the relative field intensity of both the s and p polarizations of an optical wave signal with a frequency mode of 1.8955 eV propagating in the FN4- BM3-FN4 hybrid dielectric heterostructure considered above. In the case of the p polarization, the field intensity distributes rather homogeneously over all the spatial