# Design and Experiment of Wideband Filters Based on Double-Layered Square-Loop Arrays in the F-Band
## 1 Introduction  
In recent years, electromagnetic (EM) metamaterials have gained significant attention due to their unique physical properties, which are not commonly observed in natural materials. The unusual physical characteristics of metamaterials (MMs) lead to a variety of special electromagnetic effects, such as negative refractive index [1; 2] and applications in invisible cloaks [3; 4]. Other applications include filters [5; 6], absorbers [7; 8], modulators [9; 10], and a number of others. As a basic wave control device, filters are widely used in imaging [11; 12], sensing [13; 14], and other fields [15]. Compared to traditional filters, a metamaterial filter has advantages of being ultra-thin, low cost, and light weight. Moreover, the operating frequencies of MM filters have been expanded from the microwave frequency range [16; 17] to the THz range [18; 19] and even to the optical range [20; 21].  
EM metamaterials typically consist of periodically patterned sub-wavelength metallic elements embedded within or on top of a dielectric substrate. The transmittance of subwavelength elements in a metal film is enhanced when the film is placed in a periodic array. Many kinds of MMs in single andmultiple configurations have been proposed and studied in recent years. For example, transmittance and reflectance response of single layer consisted with a split-ring resonator (SRR) pair is studied at optical frequencies [22]. In one study [23], transmission spectra of terahertz pulses through periodic array of subwavelength holes with various hole widths are presented. However, the bandwidth and maximum transmission of the MMs with a single layer is restricted. [PERSON] et al. [24] proposed a THz filter integrated with a subwavelength structured antireflection coating to reduce surface reflection loss and increase the bandwidth at frequencies ranging from 0.2 THz to 1.4 THz. [PERSON] et al. [25] proposed a thin, low-cost bandpass filter composed of double-layer tin foil with a hexagonal lattice array of circular holes and a 3-dB bandwidth is 0.20 THz from 0.81 to 1.01 THz. Bandpass filters were reported based on double-stacked metamaterial layers separated by an air gap with a 3-dB bandwidth of 78 GHz and 0.5 THz center frequency [26].  
In this paper, we report a theoretical and experimental study on the transmission characteristics of continuous short millimeter waves through a metal-dielectric-metal (MDM) structure sandwiched between subwavelength periodic air square-loops. The normalized transmittance spectra through the two different structures were analyzed to explain the formation of two resonant peaks, \(f_{1}\) and \(f_{2}\). The surface current and electric field distributions of these resonant peaks were calculated to further understand the transmission mechanism. In addition, transmission of oblique incident waves through the structure and periodic misalignment were studied. A prototype of the proposed MDM structure was fabricated and the transmission spectra can be obtained with a vector network analyzer. The results from both simulations and experiments show that the proposed structure has a potential application as a wideband passband filter.