Frequency selective surface

See also: Photonic crystal and Metamaterial

A Frequency Selective Surface (FSS) is a planar, periodic structure designed to reflect, transmit or absorb electromagnetic fields based on their frequency. In this sense, an FSS is a type of Optical filter in which the filtering is accomplished by virtue of the regular, periodic arrangement of planar scatterers on the surface of the FSS. Frequency selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as microwave ovens, Antenna Radomes, Metamaterials, and Stealth technology. Sometimes frequency selective surfaces are referred to as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as Photonic crystals.

Many factors are involved in understanding the operation and application of frequency selective surfaces. These include analysis techniques, design principles, manufacturing techniques and methods for integration into space, ground and airborne platforms.

FSS Analysis - First Principles

The Spectral Domain Method (Overview)

Background

Historically, the first approach to solving for fields reflected and transmitted by FSS was the spectral domain method (SDM), and it's still a valuable tool even today. The spectral domain method is known at Ohio State University as the periodic method of moments (PMM). The SDM starts out with an assumed Floquet/Fourier series solution for all fields, currents and potentials whereas the PMM starts out with a single scatterer, then adds in all of the scatterers in the infinite plane (in the spatial domain), then uses a transformation to yield the spectral domain representation of the fields. Both approaches are effectively the same approach, in the sense that they both assume an infinite planar structure which gives rise to a discrete Fourier series representation for the fields.

Advantages/Disadvantages

The spectral domain method has one very important advantage over other - strictly numerical - solutions to Maxwell's equations for FSS. And that is that it yields a matrix equation of very small dimensionality, so it is amenable to solution on virtually any type of computer. The dimension of the matrix is determined by the number of current basis functions on each individual scatterer and can be as small as 1x1 for a dipole at or below resonance. The matrix elements however take longer to compute than with volumetric approaches such as FEM. Volumetric approaches require that a volume surrounding the unit cell be gridded accurately, and can require many thousands of elements for an accurate solution, though the matrices are usually sparse.

Motivation

The spectral domain method is based on Floquet's principle, which says that if an infinite, planar, periodic structure is illuminated by an infinite plane wave, then every unit cell in the periodic plane will contain exactly the same currents and fields, except for a phase shift, corresponding to the incident field phase. This principle allows all currents, fields and potentials to be written in terms of a modified Fourier series, which consists of an ordinary Fourier series multiplied by the incident field phase. If the periodic plane occupies the x-y plane, then the Fourier series is a 2-dimensional Fourier series in x,y.

Field equations for 2D PEC frequency selective surfaces

Perfectly electrically conducting (PEC) periodic surfaces are not only the most common but also the easiest to understand mathematically, as they admit only electric current sources J. This section presents the spectral domain method for analyzing a free-standing (no substrate) PEC FSS. The electric field E is related to the vector magnetic potential A via the well-known relation:

${\mathbf {E}}~=~-jk\eta \left[{\mathbf {A}}+{\frac {1}{k^{2}}}\nabla (\nabla \bullet {\mathbf {A}})\right]~~~~~~~~~~~~~~(1.1)$

and the vector magnetic potential is in turn related to the source currents via:

$\nabla ^{2}{\mathbf {A}}+k^{2}{\mathbf {A}}~=~-{\mathbf {J}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.2)$

where

$k^{2}~=~\omega ^{2}(\mu \epsilon )=2\pi /\lambda ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1.3)$

Plane wave expansion of the fields

Frequency selective surfaces are frequently stratified in the direction normal to the plane of the surface. That is, all dielectrics are stratified and all metallic conductors are considered stratified as well, and they will be regarded as perfectly planar. As a result, we are excluding metallic vias (wires perpendicular to the plane of the FSS) which could potentially connect currents from different strata of the FSS structure. With this type of a stratified structure in mind, we can then use a plane wave expansion for the fields in and around the FSS.

To solve equations (1.1) and (1.2) for the doubly-periodic surface, we consider an infinite 2D periodic surface occupying the entire x-y plane, and assume a discrete plane wave expansion for all currents, fields and potentials (see also Fourier Optics):

${\mathbf {J} }(x,y,z)~=~\sum _{mn}~{\mathbf {J} }(\alpha _{m},\beta _{n})~e^{j(\alpha _{m}x+\beta _{n}y\pm \gamma _{mn}z)}~~~~~~(2.1a)$

${\mathbf {E} }(x,y,z)~=~\sum _{mn}~{\mathbf {E} }(\alpha _{m},\beta _{n})~e^{j(\alpha _{m}x+\beta _{n}y\pm \gamma _{mn}z)}~~~~~(2.1b)$

${\mathbf {A} }(x,y,z)~=~\sum _{mn}~{\mathbf {A} }(\alpha _{m},\beta _{n})~e^{j(\alpha _{m}x+\beta _{n}y\pm \gamma _{mn}z)}~~~~(2.1c)$

where for mathematical simplicity, we assume a rectangular lattice in which α only depends on m and β only depends on n. In the equations above,

$\alpha _{m}~=~k~\sin \theta _{0}~\cos \phi _{0}~+~{\frac {2m\pi }{l_{x}}}~~~~~~~~~~~~(2.2a)$

$\beta _{n}~=~k~\sin \theta _{0}~\sin \phi _{0}~+~{\frac {2n\pi }{l_{y}}}~~~~~~~~~~~~~~(2.2b)$

$\gamma _{mn}~=~{\sqrt {~k^{2}~-~\alpha _{m}^{2}~-~\beta _{n}^{2}}}~~~~~~~~~~~~~~~~~(2.2c)$

and,

$k~=~2\pi /\lambda ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.3)$

where l_x, l_y are the dimensions of the unit cell in the x,y directions respectively, λ is the free space wavelength and θ₀, φ₀ are the directions of an assumed incident plane wave, with the FSS regarded as lying in the x-y plane. In (2.2c), the root is taken which has a positive real part and non-positive imaginary part).

Integral equation for free-standing PEC FSS

Substituting equations (2.1) into (1.1) and (1.2) yields the spectral domain Greens function relating the radiated electric field to its source currents (where we now consider only those components of the field vectors lying in the plane of the FSS, the x-y plane):

${\mathbf {E}}(\alpha _{m},\beta _{n})~=~{\frac {jk\eta }{\sqrt {k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}}}}~{\mathbf {G}}_{mn}~{\mathbf {J}}(\alpha _{m},\beta _{n})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3.1)$

where,

${\mathbf {G}}_{mn}~=~\left({\begin{matrix}1-{\frac {\alpha _{m}^{2}}{k^{2}}}&-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}\\-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}&1-{\frac {\beta _{n}^{2}}{k^{2}}}\end{matrix}}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3.2)$

We notice 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. With this, the electric field boundary condition on the surface of PEC material within a unit cell becomes:

$~\sum _{mn}~{\frac {1}{\sqrt {k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}}}}~{\mathbf {G} }_{mnp}~{\mathbf {J} }(\alpha _{m},\beta _{n})~e^{j(\alpha _{m}x+\beta _{n}y)}~=~-{\mathbf {E} }^{inc}(x,y)~~~~~~~~~~~~~~~~(3.3)$

where again, we are restricting our attention to the x,y components of currents and fields, which lie in the plane of the scatterer.

Equation (3.3) is not strictly correct, since the tangential electric field is zero only on the surface of the PEC scatterers. This inexactness will be resolved presently when we test with the current basis functions, defined as residing on the surface of the scatterer.

In this type of problem, the incident field is considered a plane wave expressed as

$~{\mathbf {E} }^{inc}(x,y)=~{\mathbf {E} }^{inc}(\alpha _{0},\beta _{0})~e^{j(\alpha _{0}x+\beta _{0}y)}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3.4)$

in the x-y plane.

Method of Moments (MoM) solution

As is usual in the method of moments, we assume an expansion for the source currents over some known set of basis functions with unknown weighting coefficients J_j:

$~{\mathbf {J} }(x,y,z)~=~\sum _{j}~J_{j}~{\mathbf {J} }_{j}(x,y,z)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4.1)$

Substituting (4.1) into (3.3) and then testing the resulting equation with the i-th current basis function (i.e., dotting from the left and integrating over the domain of the i-th current basis function, thereby completing the quadratic form) produces the i-th row of the matrix equation as:

$~\sum _{j}~J_{j}~\left[~\sum _{mn}~{\frac {{\mathbf {J} }_{i}(-\alpha _{m},-\beta _{n})~{\mathbf {G} }_{mn}~{\mathbf {J} }_{j}(\alpha _{m},\beta _{n})}{\sqrt {k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}}}}\right]~=~-{\mathbf {J} }_{i}(-\alpha _{0},-\beta _{0})~\bullet ~{\mathbf {E} }^{inc}(\alpha _{0},\beta _{0})~~~~~~~~~~~~~(4.2)$

This matrix equation is very simple to implement and requires only that the 2D FT of the basis functions be computed, preferably in closed form. It's interesting to note the similarity of this equation to the Bloch wave - MoM method equation for computing ω-β diagrams for periodic volumes. Equation (4.2) may be readily modified for analyzing FSS with surrounding dielectrics, and even complex multi-layer FSS structures (Scott [1989]).

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

Computing reflection and transmission coefficients

Equation (4.2) has been used to compute reflection and transmission from various types of FSS (Scott[1989]). The reflected field is due to the currents on the FSS (the field radiated by the FSS) and the transmitted field is equal to the radiated field plus the incident field, and differs from the reflected field only for the n=0, m=0 order.

FSS Analysis - Equivalent Circuits

Transmission line equivalent circuits

We know both from an intuitive understanding of FSS operation and from first-principles analysis, that FSS can be understood qualitatively and quantitatively over certain bands through the use of equivalent transmission line circuits. The free space between FSS sheets (as well as in front and in back) may be represented by a transmission line and the FSS itself may be represented in terms of lumped RLC networks placed in parallel across the transmission line.

Resonant circuits to represent resonant scatterers

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. A dipole or patch in free space will strongly reflect energy for wavelengths comparable in size to the object itself, for example when the dipole is 1/2 wavelength in length. For frequencies below this first resonance (or frequencies between the first and second resonance), the object will reflect little energy. So, this resonance phenomenon observed with dipoles and patches leads naturally to the notion of modeling them as a resonant circuit connected in parallel across a transmission line - in this case the element is a series connection of a capacitor and inductor, which produces a reflective short circuit at resonance. This type of structure would be known as a band-reject or band-stop filter. 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.

FSS Design

FSS are typically resonance region structures (wavelength comparable to element size and spacing). FSS can be classified either by their form or by their function. Morphologically, Ben Munk classified FSS elements into 2 broad categories: those that are "wire-like" (one-dimensional) and those that are "patch-like" (two-dimensional) in appearance. His lifelong preference was for the one-dimensional wire-like FSS structures, and they do seem to have advantages for many applications. Frequency selective surfaces, as any type of filter, may also be classified according to their function, and these usually fall into 3 categories: low-pass, high-pass and bandpass, in addition to some specialty classes like band-reject and notch filters. FSS may be made to be absorptive as well, and absorption is usually over some frequency band.

FSS Fabrication

Typically FSS 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). The sheet may range in thickness from a few thousandths of an inch to as much as 20-40 thousandths.

FSS Applications

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

See also

• Fourier optics
• Bloch wave - MoM method
• Photonic crystal
• Metamaterial
• Bragg diffraction

References

• Munk, Benedikt (2000), Frequency Selective Surfaces: Theory and Design, John Wiley, ISBN 978-0-471-37047-5
• Vardaxoglou, John (1997), Frequency Selective Surfaces, John Wiley, ISBN 978-0863801969
• Wu, Te-Kao (1995), Frequency Selective Surfaces, John Wiley, ISBN 978-0-471-31189-8
• Scott, Craig (1989), The Spectral Domain Method in Electromagnetics, Artech House, ISBN 0-89006-349-4

External links

• Everything You Wanted to Know About FSS

Category:Photonics Category:Optics Category:Metamaterials Category:Physical optics Category:Fourier analysis

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