One-dimensional photonic crystals (1DPCs) are periodic arrays of dielectric materials that prevent electromagnetic (EM) waves with frequencies within the photonic bandgap (PBG) from propagating. With careful design, certain 1DPCs can exhibit an omnidirectional bandgap that blocks EM waves from all incident angles, for both transverse electric (TE) and transverse magnetic (TM) polarized waves. These types of 1DPCs are known as omnidirectional reflectors (ORs). To increase the bandgap width of OR, a cascaded 1DPC structure can be used. However, such structures become bulky when many 1DPC layers are combined. In this paper, we propose an optical transformed cascaded 1DPC structure that achieves a wide omnidirectional bandgap while minimizing the overall thickness.

First, three single-layer ORs based on individual 1DPC structures, illuminated by TM wave, are simulated, and their omnidirectional reflection bands are calculated using the transfer matrix method. The results show reflection bands of 433?453, 466?488, and 499?524 nm, respectively. Next, these three 1DPC structures are cascaded to form a wider-band OR. Finally, the thickness of the cascaded structure is reduced using transformation optics principles. The permittivity and permeability of the compressed structure are recalculated, and the magnetic permeability is adjusted to be non-magnetic for ease of fabrication. The omnidirectional reflectance of the transformed cascaded structure is computed using the full-wave simulator HFSS.

The omnidirectional reflection bandwidth of the cascaded 1DPC structure ranges from 446 to 494 nm, with a total bandwidth of 48 nm. The reflection bandwidths at different incident angles for the TM wave are calculated using the transfer matrix method, as shown in Fig. 3. The original cascaded 1DPC structure is then compressed to different sizes using transformation coefficients of 0.8, 0.5, and 0.1. The dielectric materials required for this transformation are calculated based on the principle of transformation optics. Subsequently, the dielectric materials are adjusted to be non-magnetic for easier fabrication in the optical regime. Using HFSS, the omnidirectional bandgaps are calculated for the optical transformed cascaded 1DPC structure with the ideal permittivity and permeability derived from transformation optics. The omnidirectional reflection bandwidths for transformation coefficients of 0.8, 0.5, and 0.1 are found to remain within the 446?494 nm range, as shown in Figs. 5(b)?(d). It is observed that the omnidirectional reflection bandwidths remain unchanged compared to the original cascaded 1DPC structure. Furthermore, for the optical transformed cascaded 1DPC structure with non-magnetic dielectric materials, the omnidirectional reflection bandwidth is also consistent with that of the original cascaded structure, ranging from 446 to 494 nm, as shown in Figs. 5(b)?(d). Deviations outside the reflection bandwidth are attributed to the approximation of refractive indices in non-magnetic materials, which do not affect practical applications.

This proposed dielectric omnidirectional reflector, based on an optical transformed cascaded 1DPC structure, features a wide omnidirectional reflection band while maintaining minimal thickness. The thickness and dielectric properties are determined by the transformation coefficient used in the spatial transformation. In addition, the structure can be made non-magnetic for simpler fabrication, without compromising its omnidirectional reflection properties. This reflector has promising applications in integrated optical circuits where broad omnidirectional reflection bandgaps are essential.