We have investigated the photonic Wannier-Stark ladder in the system of coupled electromagnetic cavities, which consists of a stack of metallic plates structured with subwavelength apertures and where the tilted potential effect is mimicked by imposing the gradient variation of refractive index. Making an analogy to its quantum counterpart and assuming the translational property of its solutions, we have shown the photonic ladder has the eigenenergies, that is, frequencies, in a geometrical series. Within the approximation of small gradient, the ladder states manifest the equidistant frequency spacing in the spectrum. By both analytical derivation and numerical simulation, we have illustrated the geometrically progressed energies of the photonic Wannier-Stark ladder.

A lot of similar physical properties have been revealed between electrons in a solid and photons in a periodic structure [

Recently, the two concepts advance to electromagnetic (EM) waves in an analogy way, driven by the physical similarity of wave systems and the phenomenological robustness of classical waves [

In this work, we have investigated the photonic WSL in the system of coupled EM cavities, where the biased potential effect is mimicked by imposing the gradient variation of refractive index. A numerical model is constructed, using a stack of metallic plates patterned with subwavelength apertures. By calculating the transmission, we have observed that the transmission peaks of the ladder states have the equidistant frequency spacing in the spectrum, that is, their frequencies (eigenenergies) showing an arithmetical series, in the case of small gradient. Specifically, the frequencies of the peaks evolve into a geometrical series in the case of large gradient, which points out the generalized photonic WSL with geometrically progressed energies.

We start with the quantum WSL problem. Consider a particle of mass

(a) The Kronig-Penney potential (coupled quantum wells) titled by a static electric field, where

The photonic WSL has been proved in one-dimensional coupled EM cavities [

It can be proved that a geometric series,

In order to illustrate the photonic WSL states in a specific system, we employ a metallic plate patterned with narrow H-fractal slits as EM cavity. The resonant state supported by the exotic fractal pattern has been found to be responsible for transmission enhancement of EM wave through such metallic plates [

The coupled cavity system is 8 stacked metallic plates on which a periodic array of fractal slits was cut with lattice constant smaller than the relevant incident wavelength to avoid the grating effect, illustrated schematically in Figure

(a) The 8 stacked metallic plates on which a periodic array of fractal slits was generated. In the schematic picture, the plate has 2 × 2 unit cells, each of which is a five-level H-fractal slit. (b) The 8 plates are embedded, respectively, inside 8 dielectrics, which have the refractive index,

To impose a tilted potential effect, we embed the plates in the dielectrics, which have the geometrically increasing refractive index,

When

After a small variation,

While a large variation,

(a) The peak frequencies in the three cases, periodic, small

We have labeled the frequency values of all identified peaks from Figure

From the experimental perspective, the critical requirement for observing the proposed WSL is to fabricate the dielectrics with designed refractive index

In conclusion, by making an analogy to the quantum counterpart and assuming the translational properties of the solution

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant nos. 11104198 and 11474212), the Natural Science Foundation of Jiangsu Province (Grant no. BK20141191), and a Project Funded by the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. The authors thank Professors Gang Wang and Zhi Hong Hang for the beneficial discussions.