The analogs between electron transport and propagation of the optical waves in dielectric structures opened the possibility of the implementation of Bloch oscillations for electromagnetic waves in photonic crystals (PC) [
1,
2]. The photonic analog of the abovementioned effect appears when a PC is subjected to a slowly varying refractive index or a geometric parameter modulation, resulting in a linear tilting of the band structure. Such ‘chirped’ PCs give rise to a set of equidistant frequency levels [
3], i.e., the optical counterpart of the WannierStark ladders (WSLs) in semiconductor superlattices. Recently, different methods have been adopted to tilt the photonic band for the observation of WSLs and photon Bloch oscillations (PBOs) [
4
7]. In confined Bragg mirrors, the band structure modifications are due to the gradual change of the lateral confinement [
4]. On the other hand, in geometrically chirped PCs, the band structure modulation arises from a gradual increase in the thicknesses of the layers [
5,
6]. Furthermore, in gradedindex optical superlattices, the index gradient comes from a linear modification in the refractive indices of the layers [
7,
8], and so on. The use of abovementioned PCs with gradient in optical thickness has evoked special interests in many applications due to their novel properties [
9]. Such multilayer structures originate a new type of FabryPerot cavity where the reflectors are replaced by nonpropagating regions associated with the local periodicity of the structure. If the linear gradient is considered, the distance between the band edges where the PBOs occur can be maintained constant. Hence, the period of PBOs remains constant with the change in the frequency of the incident wave. In case of nonlinear gradient, the distance between the band edges and, therefore, the period of the resulting PBOs can be tuned by changing the frequency of the incident light. For example in 2005 Lousse and Fan [
8] reported the tunable terahertz Bloch oscillations in the chirped photonic crystals, with the potential applications in several fields, like biomedical sensing. Such useful photonic structures can be fabricated with different materials. Recently, onedimensional photonic superlattices made of porous silicon (PSi) have allowed the demonstration of optical analogs [
10] of electronic phenomena [
11,
12], such as PBOs, Zener tunneling, and Anderson localization [
6,
7,
13,
14].
Porous silicon provides good flexibility in the design of optical devices due to its easy fabrication technique [
15
17] and tunable optical properties. PSi can be obtained by electrochemical etching of doped silicon wafers, which allows the fabrication of several types of onedimensional (1D) porous silicon photonic bandgap structures, such as distributed Bragg reflectors [
18], omnidirectional mirrors [
19
22], FabryPerot optical microcavities [
23,
24], waveguides [
25], rugate filters [
26], and optical biosensors [
27
31].
In the present work, we demonstrate the theoretical and experimental evidence of WSLs, using dualperiodical multilayer structures (shown schematically in Figure ), with a linear gradient in refractive indices, based on porous silicon. Theoretical evidence of the presence of PBOs in such structures is also presented.
Dualperiodic structures
The optical properties of dualperiodical (DP) structures have been theoretically and experimentally reported by several groups [
32,
33]. Recently, Pérez et al. [
34] reported DP structures from PSi multilayers. Dualperiodic structure (Figure ) is composed of two substructures,
A and
B, repeating alternatively in the sequence
. The
A_{
n
} and
B_{
m
}are in turn composed of two different periodic units,
a and
b, respectively, where subscripts,
n and
m, are the number of periods for
a and
b in the
A and
B substructures, respectively. Both
a and
b consist of a pair of layers with high and low refractive indices. The thickness of the double layer
a is
d_{
a
} =
d_{1} +
d_{2}d_{1} and
d_{2} being the thicknesses for the layers with the high (
n_{1}) and low (
n_{2}) refractive indices, respectively. Similarly, the double layer
b has thickness
d_{
b
} =
d_{3} +
d_{4};
d_{3} and
d_{4} being the thicknesses for layers with the high and low refractive index as well. In particular, the following sequence was used:
A_{2}B_{4}A_{2}B_{4}A_{2}B_{4}A_{2}B_{4}A_{2}B_{4}A_{2}B_{4}=
for the infrared region. If the substructure
B is considered as a defective layer, the frequency intervals where the resonances of the transmission peaks appear can be reduced by increasing the number of periods
a in the substructure
A. On the other hand, if substructure
A is a defective layer, the frequency intervals of the resonances can be increased by reducing the number of periods
b in the substructure
B. When identical
A_{
n
}B_{
m
}structures are coupled, a degenerate mode repulsion arises. Each degenerate optical resonance splits up and a miniphotonic band forms [
32
34]. Due to the periodicity of the structure, the miniphotonic bands are separated by photonic band gaps in which propagation is prohibited. Moreover, when
N>1, there are
N−1 defect layers; therefore,
N−1 resonance modes and
N−1 transmission peaks will appear in the spectra. By adjusting the structural parameters, it is possible to tune the number, frequency, and full width at halfmaximum (FWHM) of the resonance modes, opening the possibility to fabricate optical filters based on porous silicon multilayers. Such DP photonic structures are very promising in the field of optoelectronics, optical communications, and optical biosensors [
34].
Furthermore, 1D translational symmetry of the system should be broken by introducing a small gradient in the refractive indices along the depth of the DP structure to obtain PBOs in periodic 1D photonic crystals. The gradient in the refractive indices results in a spatial tilting of the miniband and photonic band gaps in which the resonances, due to defects in DP structure, change slightly while preserving the mode coupling. In this way, the extended photonic states are turned into a discrete sequence of energy levels with level spacing
E, which is an optical equivalent of a WSL in frequency domain. The refractive index gradient in layers is given by
n = (

)/
, where the subscripts
z_{1}and
z_{
m
} are the first and the mth layer along the depth within the sample. This gradient is the optical counterpart of the external electric field used in electronic superlattices.