Frequency selective surface

This technique uses the method of moments (MoM) in combination with a Bloch wave expansion of the electromagnetic field to yield a matrix eigenvalue equation for the propagation bands.

However, it can also be readily expanded to dielectric structures, using the well-known interior and exterior equivalent problems commonly employed in ordinary spatial domain method of moments formulations.

[3] For PEC structures, the electric field E is related to the vector magnetic potential A via the well-known relation: and the vector magnetic potential is in turn related to the source currents via: where To solve equations (1.1.1) and (1.1.2) within the infinite periodic volume, we may assume a Bloch wave expansion for all currents, fields and potentials: where for simplicity, we assume an orthogonal lattice in which α only depends on m, β only depends on n and γ only depends on p. With this assumption, and, where lx, ly, lz are the unit cell dimensions in the x,y,z directions respectively, λ is the effective wavelength in the crystal and θ0, φ0 are the directions of propagation in spherical coordinates.

Equation (1.3.3) is not strictly correct however, since it is only the tangential components of electric field that are actually zero on the surface of the PEC scatterer.

This inexactness will be resolved presently, when we test this equation with the electric current basis functions - defined as residing on the surface of the scatterer.

This matrix equation is easy to implement and requires only that the 3D Fourier transform (FT) of the basis functions be computed, preferably in closed form.

It's evident from (1.4.2) that the EFIE could become singular whenever the free space wavenumber is exactly equal to one of the wave numbers in any of the 3 periodic coordinate directions.

Historically, the first approach to solving for fields reflected and transmitted by FSS was the spectral domain method (SDM), and it is still a valuable tool even today [Scott(1989)].

The dimension of the matrix is determined by the number of current basis functions on each individual scatterer and can be as small as 1×1 for a dipole at or below resonance.

The spectral domain method is based on Floquet's principle, which implies that when an infinite, planar, periodic structure is illuminated by an infinite plane wave, then every unit cell in the periodic plane must contain exactly the same currents and fields, except for a phase shift, corresponding to the incident field phase.

Substituting equations (2.2.1) into (2.1.1) and (2.1.2) yields the spectral domain Greens function relating the radiated electric field to its source currents (Scott [1989]), where we now consider only those components of the field vectors lying in the plane of the FSS, the x-y plane: where, One notices the branch point singularity in the equation above (the inverse square root singularity), which is no problem thanks to the discrete spectrum, as long as the wavelength never equals the cell spacing.

Equation (2.3.3) is not strictly correct, since only the tangential components of electric field are actually zero on the surface of the PEC scatterers.

This inexactness will be resolved presently when (2.3.3) is tested with the current basis functions, defined as residing on the surface of the scatterer.

All of these matrix equations are very simple to implement and require only that the 2D Fourier transform (FT) of the basis functions be computed, preferably in closed form.

The RWG (Rao–Wilton–Glisson) basis functions (Rao, Wilton and Glisson [1982]) are a very versatile choice for many purposes and have a transform that is readily computed using area coordinates.

Equations (2.4.2) and (2.3.1) have been used to solve for the electric current J and then the scattered fields E to compute reflection and transmission from various types of FSS (Scott[1989]).

The main thing is that both free space and transmission lines admit traveling wave solutions with a z-dependence of the form:

The magnetic field jump condition for the FSS mirrors the Kirchhoff current division law for the equivalent circuit.

For all but the most tightly packed dipole arrays (the brickwork-like "gangbuster" low-pass filters), a first order understanding of FSS operation can be achieved by simply considering the scattering properties of a single periodic element in free space.

Bandpass filters may be constructed using apertures in conducting planes, which are modeled as a shunt element consisting of a parallel connection of an inductor and a capacitor.

The real FSS has a reflection null at 18.7 GHz (the frequency at which the wavelength equals the unit cell dimension of .630"), which is not accounted for in the equivalent circuit model.

The null is known as a Wood's anomaly and is caused by the inverse square root singularity in the spectral domain Green's function (3.1) going to infinity.

As an example of how to use FSS equivalent circuits for quick and efficient design of a practical filter, we can sketch out the process that would be followed in designing a 5-stage Butterworth filter (Hunter [2001], Matthaei [1964]) using a stack of 5 frequency selective surfaces, with 4 air spacers in between the FSS sheets.

The cutoff frequency will be scaled to 7 GHz and the filter will be matched to 377 Ohms (the impedance of free space) on the input and output sides.

At this point in the development, the series inductors in the prototype L,C ladder network will now be replaced by sub-half-wavelength air spacers (which we will model as transmission lines) between the FSS layers.

3.1.3-1, in which we compare the ABCD matrix of a series inductor with the ABCD matrix of a short transmission line (Ramo [1994]), in order to obtain the proper length of transmission line between the shunt capacitors (sub-resonant FSS layers) to produce a Butterworth filter response.

It is well known that a series inductor represents an approximate lumped circuit model of a short transmission line, and we'll exploit this equivalence to determine the required thickness of the air spacers.

Typically FSSs are fabricated by chemically etching a copper-clad dielectric sheet, which may consist of Teflon (ε=2.1), Kapton, (ε=3.1), fiberglass (ε-4.5) or various forms of duroid (ε=6.0, 10.2).

Applications of FSS range from the mundane (microwave ovens) to the forefront of contemporary technology involving active and reconfigurable structures such as smart skins.

Radomes EM absorbers For multi-spectral camouflage, like the Saab Barracuda, FSS can be used to allow certain frequencies to penetrate, so communication and GPS isn't blocked.