Frequency selective surface

See also: Photonic crystal and Metamaterial

A Frequency Selective Surface (FSS) is a doubly-periodic, planar structure designed to pass or reflect electromagnetic fields based on the frequency of the field. 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.

A variety of factors is involved in understanding the operation and application of frequency selective surfaces. These include analysis techniques (ranging from the spectral domain method, and the closely-related periodic method of moments developed in the 1970's, to the modern finite element methods), design principles, manufacturing techniques and methods for integration into space, ground or airborne platforms.

FSS Analysis

Spectral Domain Method

Field equations for 2D PEC frequency selective surfaces

Perfectly electrically conducting (PEC) periodic surfaces are not only the most common but 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 confined to the plane of the surface. As a result, we are excluding metallic vias 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 may assume a discrete plane wave expansion for all currents, fields and potentials:

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

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

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

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. Note that k in equations (1.1) and (1.2) comes from the time derivative in Maxwell's equations and is the free space propagation constant, proportional to frequency as we see in equation (1.3). On the other hand, k₀ in the equations above comes from our assumed Bloch wave solution given by equations (2.1) & (2.2). As a result, it represents the propagation constant in the periodic medium. These two k's, i.e. the free space propagation constant and the propagation constant of the Bloch wave, are in general different thereby allowing for dispersion in our solution.

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:

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

where,

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

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

$~\sum _{mn}~{\frac {1}{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}~{\mathbf {G}}_{mnp}~{\mathbf {J}}(\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}~=~{\mathbf {0}}~~~~~~~~~~~~~~~~(3.3)$

Since we are seeking characteristic modes (eigenmodes) of the structure, there is no impressed E-field on the RHS of this electric field integral equation (EFIE). Equation (3.3) is not strictly correct, since only the tangential electric field is zero on the surface of the PEC scatterer. This inexactness will be resolved presently when we test with the current basis functions, defined as residing on the surface of the scatterer.

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:

$~{\mathbf {J}}(x,y,z)~=~\sum _{j}~J_{j}~{\mathbf {J}}_{j}(x,y,z)~~~~~~~~~~~~~~~~~(3.4)$

Substituting (3.4) 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 eigenvalue problem as:

$~\sum _{j}~J_{j}~\left[~\sum _{mnp}~{\frac {{\mathbf {J}}_{i}(-\alpha _{m},-\beta _{n},-\gamma _{p})~{\mathbf {G}}_{mnp}~{\mathbf {J}}_{j}(\alpha _{m},\beta _{n},\gamma _{p})}{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}\right]~=~{\mathbf {0}}~~~(3.5)$

This matrix equation is very simple to implement and requires only that the 3D FT of the basis functions be computed, preferably in closed form. With this method, computing bands of a 3D photonic crystal is as easy as computing reflection and transmission from a 2D periodic surface. In fact, equation (3.5) is identical to the basic EFIE for PEC FSS (Scott [1989]), the only difference being the stronger singularity in 3D which accelerates convergence of the triple sums.

Computing bands

To compute bands of the crystal (i.e. k-k₀ diagrams), we may assume values for (k₀, θ₀, φ₀) and then search for those values of k which drive the determinant of the impedance matrix to zero. Equation (3.5) has been used to compute bands in various types of doped and undoped photonic crystals (Scott[1998], Scott [2002]).

FSS Design

FSS are typically resonance region structures (wavelength comparable to element spacing)

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.

See also

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

References

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

External links

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