# Frequency Selective Surfaces: A Review
## 3 Frequency Selective Surfaces (FSSs)
### Operating Theory of FSS  
FSS structures (capacitive and inductive), also called spatial filters, are analogous to the microwave filters according to the circuit theory. Filtering characteristics of FSS can be categorized into the four kinds, including low pass, high pass, stop band, and passband. Low pass FSS filters permit a lower range of frequencies to pass through the structure, while bypass higher range of frequencies. High pass FSS filter operation is a counterpart of low pass filter function by applying Babinet principle. Similarly, stopband FSS filter blocks undesired frequencies while passband FSS filter allows only specific frequency range. For a desired resonant operation, FSSs are designed by periodic arrays of metal patches and/or slots etched on a dielectric material. The appropriate selection of FSSs array elements, shape, dimension, and substrate material is the most important part of the design process.  
The operational theory of FSS based structures has been explained by [PERSON] in detail [2]. Figure 2 shows the functionality of FSS achieved by a complementary self-resonating network. Simply, when EM waves are incident on FSS structure, they incite electric currents into the array elements. The level of coupling energy defines the amplitude of the produced currents. However, these generated currents also work as EM sources and they create additional scattered fields. Incident EM fields combined with these scattered fields make up the resultant field in the surrounding of FSS. Consequently, the required currents and field characteristics can be obtained by properly designed elements and create the filter response.  
The patch elements (dipole FSSs) are designed to act as stopband filters for incoming plane waves and work as completely reflecting surfaces in a narrow band of frequencies. Their counterparts, the aperture elements (slot FSSs), demonstrate passband characteristics, means that they behave as transparent surfaces to incident EM waves within the working band of frequencies. Nevertheless, to accomplish the functional necessities for a variety of EM applications, conventional FSSs come across a limited use due to their inadequate filter response, low angular stability, and narrow bandwidth.  
Figure 3a shows two basic types of element, equivalent circuits, and filters responses for low and high pass FSSs [2]. FSS patches create resistance (R) and inductance (L), while gaps among the FSS elements generate capacitance (C). Simple electrostatic principle applies to manipulate the physical significance of these passive values for different FSS elements, e.g., L of two parallel wires and C generated by a parallel plate capacitor. So, a required filter response is constructed by the combination of these capacitive and inductive elements. However, any change in the FSS dimensional parameters leads to an equivalent variation in the L and C values. Physically, when a unit cell of an FSS is illuminated by the EM wave, it can be converted to an equivalent resonance circuit. The resonance frequency can be found by (3), in which L and C represent the equivalent inductance and the capacitance of an FSS unit cell, respectively.  
\[\mathrm{f_{r}}=\frac{1}{2\pi\mathrm{\sqrt{LC}}} \tag{3}\]  
Choosing an appropriate array element is very crucial in designing FSS. While various unit cell geometries have been implemented, out of which some are easily controllable and so more famous in  
Figure 2: The functionality of an FSS.  
FSS community. A classification of frequently used FSS elements types is summarized in [4] based on their resonant properties. For example, non-resonant element (patch, wire grid) can be modeled by a capacitance, whereas single resonant element (like loop, cross, dipole) can be represented by a series combination of capacitor and inductor. It is to note that the number of resonances has a direct relationship with the count of lumped elements.