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<title>Frequency selective surface</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Frequency selective surface</span></span>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks em-sidebar"><tbody><tr><th class="sidebar-title"><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Electricity" title="Electricity">Electricity</a></li>

<li><a href="Magnetism" title="Magnetism">Magnetism</a></li>

<li><a href="Optics" title="Optics">Optics</a></li>

<li><a href="History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li>

<li><a href="Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li>

<li><a href="List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li>

<li><a href="List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrostatics" title="Electrostatics">Electrostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Charge_density" title="Charge density">Charge density</a></li>

<li><a href="Electrical_conductor" title="Electrical conductor">Conductor</a></li>

<li><a href="Coulomb's_law" title="Coulomb's law">Coulomb law</a></li>

<li><a href="Electret" title="Electret">Electret</a></li>

<li><a href="Electric_charge" title="Electric charge">Electric charge</a></li>

<li><a href="Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li>

<li><a href="Electric_field" title="Electric field">Electric field</a></li>

<li><a href="Electric_flux" title="Electric flux">Electric flux</a></li>

<li><a href="Electric_potential" title="Electric potential">Electric potential</a></li>

<li><a href="Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li>

<li><a href="Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li>

<li><a href="Gauss's_law" title="Gauss's law">Gauss's law</a></li>

<li><a href="Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li>

<li><a href="Permittivity" title="Permittivity">Permittivity</a></li>

<li><a href="Polarization_density" title="Polarization density">Polarization</a></li>

<li><a href="Electric_potential_energy" title="Electric potential energy">Potential energy</a></li>

<li><a href="Static_electricity" title="Static electricity">Static electricity</a></li>

<li><a href="Triboelectric_effect" title="Triboelectric effect">Triboelectricity</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetostatics" title="Magnetostatics">Magnetostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Amp%C3%A8re's_circuital_law" title="Ampère's circuital law">Ampère's law</a></li>

<li><a href="Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li>

<li><a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></li>

<li><a href="Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li>

<li><a href="Magnetic_field" title="Magnetic field">Magnetic field</a></li>

<li><a href="Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li>

<li><a href="Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li>

<li><a href="Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li>

<li><a href="Magnetization" title="Magnetization">Magnetization</a></li>

<li><a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li>

<li><a href="Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li>

<li><a href="Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li>

<li><a href="Displacement_current" title="Displacement current">Displacement current</a></li>

<li><a href="Eddy_current" title="Eddy current">Eddy current</a></li>

<li><a href="Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li>

<li><a href="Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li>

<li><a href="Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li>

<li><a href="Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li>

<li><a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a></li>

<li><a href="Jefimenko's_equations" title="Jefimenko's equations">Jefimenko equations</a></li>

<li><a href="Larmor_formula" title="Larmor formula">Larmor formula</a></li>

<li><a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li>

<li><a href="London_equations" title="London equations">London equations</a></li>

<li><a href="Lorentz_force" title="Lorentz force">Lorentz force</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li>

<li><a href="Poynting_vector" title="Poynting vector">Poynting vector</a></li>

<li><a href="Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrical_network" title="Electrical network">Electrical network</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Alternating_current" title="Alternating current">Alternating current</a></li>

<li><a href="Capacitance" title="Capacitance">Capacitance</a></li>

<li><a href="Current_density" title="Current density">Current density</a></li>

<li><a href="Direct_current" title="Direct current">Direct current</a></li>

<li><a href="Electric_current" title="Electric current">Electric current</a></li>

<li><a href="Electric_power" title="Electric power">Electric power</a></li>

<li><a href="Electrolysis" title="Electrolysis">Electrolysis</a></li>

<li><a href="Electromotive_force" title="Electromotive force">Electromotive force</a></li>

<li><a href="Electrical_impedance" title="Electrical impedance">Impedance</a></li>

<li><a href="Inductance" title="Inductance">Inductance</a></li>

<li><a href="Joule_heating" title="Joule heating">Joule heating</a></li>

<li><a href="Kirchhoff's_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's laws</a></li>

<li><a href="Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li>

<li><a href="Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li>

<li><a href="Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li>

<li><a href="Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li>

<li><a href="Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li>

<li><a href="Voltage" title="Voltage">Voltage</a></li>

<li><a href="Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="AC_motor" title="AC motor">AC motor</a></li>

<li><a href="DC_motor" title="DC motor">DC motor</a></li>

<li><a href="Electric_machine" title="Electric machine">Electric machine</a></li>

<li><a href="Electric_motor" title="Electric motor">Electric motor</a></li>

<li><a href="Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li>

<li><a href="Induction_motor" title="Induction motor">Induction motor</a></li>

<li><a href="Linear_motor" title="Linear motor">Linear motor</a></li>

<li><a href="Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li>

<li><a href="Permeance" title="Permeance">Permeance</a></li>

<li><a href="Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li>

<li><a href="Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li>

<li><a href="Rotor_(electric)" title="Rotor (electric)">Rotor</a></li>

<li><a href="Stator" title="Stator">Stator</a></li>

<li><a href="Transformer" title="Transformer">Transformer</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li>

<li><a href="Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li>

<li><a href="Four-current" title="Four-current">Four-current</a></li>

<li><a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li>

<li><a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li>

<li><a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li>

<li><a href="Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li>

<li><a href="Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li>

<li><a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Michael_Faraday" title="Michael Faraday">Faraday</a></li>

<li><a href="Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>

<li><a href="Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li>

<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>

<li><a href="Joseph_Henry" title="Joseph Henry">Henry</a></li>

<li><a href="Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li>

<li><a href="John_Hopkinson" title="John Hopkinson">Hopkinson</a></li>

<li><a href="Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li>

<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>

<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>

<li><a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li>

<li><a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a></li>

<li><a href="Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>

<li><a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li>

<li><a href="Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li>

<li><a href="William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li>

<li><a href="F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li>

<li><a href="George_Singer" title="George Singer">Singer</a></li>

<li><a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li>

<li><a href="Nikola_Tesla" title="Nikola Tesla">Tesla</a></li>

<li><a href="J._J._Thomson" title="J. J. Thomson">Thomson</a></li>

<li><a href="Alessandro_Volta" title="Alessandro Volta">Volta</a></li>

<li><a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li>

<li><a href="Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td>

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<p>A <b>frequency-selective surface</b> (<b>FSS</b>) is a thin, repetitive surface (such as the screen on a microwave oven) designed to reflect, transmit or absorb <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a> based on the frequency of the field. In this sense, an FSS is a type of <a href="Optical_filter" title="Optical filter">optical filter</a> or <a href="Metal-mesh_optical_filter" title="Metal-mesh optical filter">metal-mesh optical filters</a> in which the filtering is accomplished by virtue of the regular, periodic (usually metallic, but sometimes dielectric) pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSSs also have properties which vary with incidence angle and <a href="Polarization_(waves)" title="Polarization (waves)">polarization</a> as well; these are unavoidable consequences of the way in which FSSs are constructed. Frequency-selective surfaces have been most commonly used in the radio signals of the <a href="Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a> and find use in applications as diverse as the aforementioned <a href="Microwave_oven" title="Microwave oven">microwave oven</a>, <a href="Antenna_(radio)" title="Antenna (radio)">antenna</a> <a href="Radome" title="Radome">radomes</a> and modern <a href="Metamaterial" title="Metamaterial">metamaterials</a>. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as <a href="Photonic_crystals" class="mw-redirect" title="Photonic crystals">photonic crystals</a>.

</p><p>Many factors are involved in understanding the operation and application of frequency selective surfaces. These include analysis techniques, operating principles, design principles, manufacturing techniques and methods for joining these structures into space, ground and airborne platforms.

</p>

<div class="mw-heading mw-heading2"><h2 id="Bloch_Wave_MOM_method">Bloch Wave MOM method</h2></div>

<p><b>Bloch wave – MoM</b> is a <a href="First_principle" title="First principle">first principles</a> technique for determining the photonic <a href="Electronic_band_structure" title="Electronic band structure">band structure</a> of triply-periodic electromagnetic media such as <a href="Photonic_crystal" title="Photonic crystal">photonic crystals</a>. It is based on the 3-dimensional spectral domain method,<sup id="cite_ref-FOOTNOTEKastner1987_1-0" class="reference"><a href="#cite_note-FOOTNOTEKastner1987-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> specialized to triply-periodic media. This technique uses the <a href="Boundary_element_method" title="Boundary element method">method of moments</a> (MoM) in combination with a <a href="Bloch_wave" class="mw-redirect" title="Bloch wave">Bloch wave</a> expansion of the electromagnetic field to yield a matrix eigenvalue equation for the propagation bands. The eigenvalue is the frequency (for a given propagation constant) and the eigenvector is the set of current amplitudes on the surface of the scatterers. Bloch wave - MoM is similar in principle to the <a href="Plane_wave_expansion_method" title="Plane wave expansion method">plane wave expansion method</a>, but since it additionally employs the method of moments to produce a surface integral equation, it is significantly more efficient both in terms of the number of unknowns and the number of <a href="Plane_wave" title="Plane wave">plane waves</a> needed for good convergence.

</p><p>Bloch wave - MoM is the extension to 3 dimensions of the <a class="mw-selflink-fragment" href="#Spectral_domain_method_(overview_and_mathematical_introduction)">spectral domain MoM method</a> commonly used for analyzing 2D periodic structures such as frequency selective surfaces (FSS). In both cases, the field is expanded as a set of eigenfunction modes (either a Bloch wave in 3D or a discrete plane wave - aka <a href="Floquet_theory" title="Floquet theory">Floquet mode</a> - spectrum in 2D), and an integral equation is enforced on the surface of the scatterers in each unit cell. In the FSS case, the unit cell is 2-dimensional and in the photonic crystal case, the unit cell is 3-dimensional.

</p>

<div class="mw-heading mw-heading3"><h3 id="Field_equations_for_3D_PEC_photonic_crystal_structures">Field equations for 3D PEC photonic crystal structures</h3></div>

<p>The Bloch wave - MoM approach will be illustrated here for the case of perfectly electrically conducting (PEC) structures admitting only electric current sources, <b>J</b>. 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.<sup id="cite_ref-FOOTNOTEHarrington1961_2-0" class="reference"><a href="#cite_note-FOOTNOTEHarrington1961-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In dielectric problems, there are twice as many unknowns - <b>J</b> & <b>M</b> - and also twice as many equations to enforce - continuity of tangential <b>E</b> & <b>H</b> - at the dielectric interfaces.<sup id="cite_ref-FOOTNOTEScott1998_3-0" class="reference"><a href="#cite_note-FOOTNOTEScott1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>For PEC structures, the electric field <b>E</b> is related to the vector magnetic potential <b>A</b> via the well-known relation:

</p>

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ~=~-jk\eta \left[\mathbf {A} +{\frac {1}{k^{2}}}\nabla (\nabla \bullet \mathbf {A} )\right]}">

</mrow>

<mrow>

<mrow>

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<mfrac>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ~=~-jk\eta \left[\mathbf {A} +{\frac {1}{k^{2}}}\nabla (\nabla \bullet \mathbf {A} )\right]}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.1.1" class="reference nourlexpansion" style="font-weight:bold;">1.1.1</span></td></tr></tbody></table>

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

</p>

<msup>

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</msup>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {A} +k^{2}\mathbf {A} ~=~-\mathbf {J} }</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.1.2" class="reference nourlexpansion" style="font-weight:bold;">1.1.2</span></td></tr></tbody></table>

<p>where

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{2}~=~\omega ^{2}(\mu \varepsilon )}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.1.3" class="reference nourlexpansion" style="font-weight:bold;">1.1.3</span></td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Bloch_wave_expansion_of_the_fields">Bloch wave expansion of the fields</h3></div>

<p>To solve equations (<b><a href="#math_1.1.1">1.1.1</a></b>) and (<b><a href="#math_1.1.2">1.1.2</a></b>) within the infinite periodic volume, we may assume a <a href="Bloch_wave" class="mw-redirect" title="Bloch wave">Bloch wave</a> expansion for all currents, fields and potentials:

</p>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} (x,y,z)~=~\sum _{mnp}~\mathbf {J} (\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.1a" class="reference nourlexpansion" style="font-weight:bold;">1.2.1a</span></td></tr></tbody></table>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (x,y,z)~=~\sum _{mnp}~\mathbf {E} (\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.1b" class="reference nourlexpansion" style="font-weight:bold;">1.2.1b</span></td></tr></tbody></table>

</mrow>

</mrow>

</munder>

</mrow>

<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (x,y,z)~=~\sum _{mnp}~\mathbf {A} (\alpha _{m},\beta _{n},\gamma _{p})~e^{j(\alpha _{m}x+\beta _{n}y+\gamma _{p}z)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.1c" class="reference nourlexpansion" style="font-weight:bold;">1.2.1c</span></td></tr></tbody></table>

<p>where for simplicity, we assume an orthogonal lattice in which α only depends on <i>m</i>, β only depends on <i>n</i> and γ only depends on <i>p</i>. With this assumption,

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{m}~=~k_{0}~\sin \theta _{0}~\cos \phi _{0}~+~{\frac {2m\pi }{l_{x}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.2a" class="reference nourlexpansion" style="font-weight:bold;">1.2.2a</span></td></tr></tbody></table>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{n}~=~k_{0}~\sin \theta _{0}~\sin \phi _{0}~+~{\frac {2n\pi }{l_{y}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.2b" class="reference nourlexpansion" style="font-weight:bold;">1.2.2b</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{p}~=~k_{0}~\cos \theta _{0}~+~{\frac {2p\pi }{l_{z}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.2c" class="reference nourlexpansion" style="font-weight:bold;">1.2.2c</span></td></tr></tbody></table>

<p>and,

</p>

<msub>

</mrow>

</msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle k_{0}~=~2\pi /\lambda }</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.2.3" class="reference nourlexpansion" style="font-weight:bold;">1.2.3</span></td></tr></tbody></table>

<p>where <i>l</i><sub><i>x</i></sub>, <i>l</i><sub><i>y</i></sub>, <i>l</i><sub><i>z</i></sub> are the unit cell dimensions in the <i>x</i>,<i>y</i>,<i>z</i> directions respectively, λ is the effective wavelength in the crystal and θ<sub>0</sub>, φ<sub>0</sub> are the directions of propagation in <a href="Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinates</a>.

</p><p>The quantity <i>k</i> in equations (<b><a href="#math_1.1.1">1.1.1</a></b>) and (<b><a href="#math_1.1.2">1.1.2</a></b>) comes originally from the time derivative in <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> and is the <i>free space</i> propagation constant (actually, the propagation constant of whatever dielectric medium the metallic scatterers are embedded in), proportional to frequency as in equation (<b><a href="#math_1.1.3">1.1.3</a></b>). On the other hand, <i>k</i><sub>0</sub> in the equations above comes from the <i>assumed Bloch wave solution</i> given by equations (<b><a href="#math_1.2.1">1.2.1</a></b>) & (<b><a href="#math_1.2.2">1.2.2</a></b>). As a result, it represents the propagation constant inside the periodic medium, inversely proportional to wavelength. These two <i>k's</i>, i.e. the free space propagation constant (proportional to frequency) and the propagation constant of the Bloch wave (inversely proportional to wavelength), are in general different thereby allowing for dispersion in the solution. The band diagram is essentially a plot of <i>k</i> as a function of <i>k</i><sub>0</sub>.

</p><p>The Bloch wave expansions in equations (<b><a href="#math_1.2.1">1.2.1</a></b>) are nothing more than exponential <a href="Fourier_series" title="Fourier series">Fourier series</a> multiplied by the cell-to-cell propagation factor: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{j(\alpha _{0}x+\beta _{0}y+\gamma _{0}z)}.}">

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{j(\alpha _{0}x+\beta _{0}y+\gamma _{0}z)}.}</annotation>

</semantics>

</math></span><img src="./f5587dd705b572216e07846f2530a1f04d79038e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.332ex; height:2.843ex;" alt="{\displaystyle e^{j(\alpha _{0}x+\beta _{0}y+\gamma _{0}z)}.}" loading="lazy"></span> The Bloch wave expansions are chosen since any field solution within an infinite periodic volume must have the same periodicity as the medium itself, or stated another way, the fields in neighboring cells must be identical up to a (real or complex) propagation factor. In passbands the propagation factor is an exponential function with purely imaginary argument and in the stop bands (or band gaps) it is a decaying exponential function whose argument has a real component.

</p><p>The wave numbers α<sub>0</sub>, β<sub>0</sub> and γ<sub>0</sub> satisfy the relations: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{0}|~\leq ~{\frac {\pi }{l_{x}}}~,~~~~|\beta _{0}|~\leq ~{\frac {\pi }{l_{y}}}~,~~~~|\gamma _{0}|~\leq ~{\frac {\pi }{l_{z}}}~~~~}">

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle |\alpha _{0}|~\leq ~{\frac {\pi }{l_{x}}}~,~~~~|\beta _{0}|~\leq ~{\frac {\pi }{l_{y}}}~,~~~~|\gamma _{0}|~\leq ~{\frac {\pi }{l_{z}}}~~~~}</annotation>

</semantics>

</math></span><img src="./95f8efc16c92ef49c7c36a4f5341cfdb8beba85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.839ex; height:5.509ex;" alt="{\displaystyle |\alpha _{0}|~\leq ~{\frac {\pi }{l_{x}}}~,~~~~|\beta _{0}|~\leq ~{\frac {\pi }{l_{y}}}~,~~~~|\gamma _{0}|~\leq ~{\frac {\pi }{l_{z}}}~~~~}" loading="lazy"></span> and outside of these ranges, the bands are periodic.

</p><p>The Bloch waves are periodic functions of space, with periods <i>l</i><sub><i>x</i></sub>, <i>l</i><sub><i>y</i></sub>, <i>l</i><sub><i>z</i></sub> and the bands are periodic functions of wavenumber, with periods: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{l_{x}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{l_{x}}}}</annotation>

</semantics>

</math></span><img src="./daa9126f0d764789b37e33da0bf7f78198b58509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.331ex; height:5.676ex;" alt="{\displaystyle {\frac {2\pi }{l_{x}}}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{l_{y}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{l_{y}}}}</annotation>

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</math></span><img src="./c28af84ba1c5bd6eb1428c8afa9a26f28e3ff20f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:3.331ex; height:6.009ex;" alt="{\displaystyle {\frac {2\pi }{l_{y}}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{l_{z}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{l_{z}}}}</annotation>

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</math></span><img src="./6696a1846b490cfc316a10720999235a75998d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.331ex; height:5.676ex;" alt="{\displaystyle {\frac {2\pi }{l_{z}}}}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Integral_equation_for_PEC_media">Integral equation for PEC media</h3></div>

<p>Substituting equations (<b><a href="#math_1.2.1">1.2.1</a></b>) into (<b><a href="#math_1.1.1">1.1.1</a></b>) and (<b><a href="#math_1.1.2">1.1.2</a></b>) yields the spectral domain Greens function relating the radiated electric field to its source currents:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\alpha _{m},\beta _{n},\gamma _{p})~=~{\frac {jk\eta }{k^{2}-\alpha _{m}^{2}-\beta _{n}^{2}-\gamma _{p}^{2}}}~\mathbf {G} _{mnp}~\mathbf {J} (\alpha _{m},\beta _{n},\gamma _{p})}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.3.1" class="reference nourlexpansion" style="font-weight:bold;">1.3.1</span></td></tr></tbody></table>

<p>where,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{mnp}~=~{\begin{pmatrix}1-{\frac {\alpha _{m}^{2}}{k^{2}}}&-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}&-{\frac {\alpha _{m}\gamma _{p}}{k^{2}}}\\-{\frac {\alpha _{m}\beta _{n}}{k^{2}}}&1-{\frac {\beta _{n}^{2}}{k^{2}}}&-{\frac {\beta _{n}\gamma _{p}}{k^{2}}}\\-{\frac {\alpha _{m}\gamma _{p}}{k^{2}}}&-{\frac {\beta _{n}\gamma _{p}}{k^{2}}}&1-{\frac {\gamma _{p}^{2}}{k^{2}}}\end{pmatrix}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.3.2" class="reference nourlexpansion" style="font-weight:bold;">1.3.2</span></td></tr></tbody></table>

<p>is the tensor Green's function in the spectral domain. Note that the spatial domain convolution has been transformed into a simple multiplication in the spectral domain, consistent with the convolution theorem for Fourier transforms.

</p><p>With this equation for the electric field, the electric field boundary condition (requiring that the total tangential electric field be zero on the surface of PEC scatterer) becomes:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{mnp}~{\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} }</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.3.3" class="reference nourlexpansion" style="font-weight:bold;">1.3.3</span></td></tr></tbody></table>

<p>Since we are seeking <a href="Normal_mode" title="Normal mode">characteristic modes (eigenmodes)</a> of the structure, there is no impressed E-field on the RHS of this electric field integral equation (EFIE). Equation (<b><a href="#math_1.3.3">1.3.3</a></b>) 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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Method_of_moments_(MoM)_solution">Method of moments (MoM) solution</h3></div>

<p>As is usual in the method of moments, the source currents are now expanded in terms of a sum over some known set of basis functions with unknown weighting coefficients <i>J</i><sub><i>j</i></sub>:

</p>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} (x,y,z)~=~\sum _{j}~J_{j}~\mathbf {J} _{j}(x,y,z)}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.4.1" class="reference nourlexpansion" style="font-weight:bold;">1.4.1</span></td></tr></tbody></table>

<p>Different structures will have different sets of basis functions for representing the currents on the elements and as in the ordinary spatial domain method of moments, the solution (in this case, the band diagram) <i>is a function of the set of basis functions used</i>.

</p><p>Substituting (<b><a href="#math_1.4.1">1.4.1</a></b>) into (<b><a href="#math_1.3.3">1.3.3</a></b>) and then testing the resulting equation with the <i>i</i>-th current basis function (i.e., dotting from the left and integrating over the domain of the <i>i</i>-th current basis function, thereby completing the quadratic form) produces the <i>i</i>-th row of the matrix eigenvalue equation for a 3-dimensional array of PEC scatterers as:

</p>

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<annotation encoding="application/x-tex">{\displaystyle \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} }</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1.4.2" class="reference nourlexpansion" style="font-weight:bold;">1.4.2</span></td></tr></tbody></table>

<p>As in all MoM formulations, the reaction concept in electromagnetics<sup id="cite_ref-FOOTNOTEHarrington1961_2-1" class="reference"><a href="#cite_note-FOOTNOTEHarrington1961-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERumsey1954_4-0" class="reference"><a href="#cite_note-FOOTNOTERumsey1954-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> was used in obtaining this equation. The electric field boundary/continuity conditions are "tested" (or enforced) by being integrated against electric current basis functions (for dielectric structures, the magnetic field continuity conditions are additionally tested by being integrated against magnetic current basis functions), and this is how the electric (and magnetic) field boundary conditions are converted into a matrix equation via the method of moments. This process is wholly analogous to that used to decompose a periodic function into its Fourier sine & cosine components, the only difference being that in this case the basis functions are not necessarily orthogonal, merely linearly independent.

</p><p>This matrix equation is easy to implement and requires only that the 3D <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> (FT) of the basis functions be computed, preferably in closed form.<sup id="cite_ref-FOOTNOTEScott1998_3-1" class="reference"><a href="#cite_note-FOOTNOTEScott1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In fact, computing bands of a 3D photonic crystal with this method is no more difficult than computing reflection and transmission from a 2D <a href="Frequency_selective_surfaces" class="mw-redirect" title="Frequency selective surfaces">periodic surface</a> using the <a class="mw-selflink-fragment" href="#Spectral_domain_method_(overview_and_mathematical_introduction)">spectral domain method</a>. This is because equation (<b><a href="#math_1.4.2">1.4.2</a></b>) is identical to the basic EFIE for a freestanding PEC FSS (see <a class="mw-selflink-fragment" href="#Method_of_moments_(MoM)_solution">Frequency selective surface</a> eq. <b><a href="#math_1.4.2">1.4.2</a></b>),<sup id="cite_ref-FOOTNOTEScott1989_5-0" class="reference"><a href="#cite_note-FOOTNOTEScott1989-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the only difference being the stronger singularity in 3D which significantly accelerates convergence of the triple sums, and of course the fact that the vectors are now 3-dimensional. As a result, an ordinary PC is sufficient to compute bands of many types of photonic crystals.

</p><p>It's evident from (<b><a href="#math_1.4.2">1.4.2</a></b>) 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. This can happen for example when the free space wavelength exactly equals the lattice spacing. This is a statistically rare occurrence in computational practice and corresponds to a propagation anomaly similar to a Wood's reflection anomaly for gratings.

</p>

<div class="mw-heading mw-heading3"><h3 id="Computing_bands">Computing bands</h3></div>

<p>To compute bands of the crystal (i.e. <i>k</i>-<i>k</i><sub>0</sub> diagrams), successive values of frequency (<i>k</i>) are tried - in conjunction with pre-selected values of propagation constant (<i>k</i><sub>0</sub>) and propagation direction (<i>θ</i><sub>0</sub> & <i>φ</i><sub>0</sub>) - until a combination is found which drives the determinant of the matrix to zero. Equation (<b><a href="#math_1.4.2">1.4.2</a></b>) has been used to compute bands in various types of doped and undoped <a href="Photonic_crystal" title="Photonic crystal">photonic crystals</a>.<sup id="cite_ref-FOOTNOTEScott1998_3-2" class="reference"><a href="#cite_note-FOOTNOTEScott1998-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEScott2002_6-0" class="reference"><a href="#cite_note-FOOTNOTEScott2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Not surprisingly, doping photonic crystals with defects provides a means for designing photonic passbands, in just the same way that doping semiconductors with chemical impurities provides a means for designing electronic passbands.

</p><p>For many subsectional basis functions, such as those having a half-sine or triangular shape along a round wire, the FT of the basis function for negative wave numbers -α, -β, -γ is the complex conjugate of the basis function FT for positive wave numbers. As a result, the matrix in eqn. (<b><a href="#math_1.4.2">1.4.2</a></b>) is <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian</a>. And as a result of that, only half the matrix needs to be computed. And a second result is that the determinant is a purely real function of the real-valued wavenumber <i>k</i>. Zeroes generally occur at zero-crossings (inflection points, where curvature is zero), so a simple root-finding algorithm such as <a href="Newton's_method" title="Newton's method">Newton's method</a> is usually sufficient to find the roots to a very high degree of accuracy. If may still be useful however to plot the determinant as a function of <i>k</i>, to observe its behavior near the zeros.

</p><p>As a matter of computational convenience, whenever the matrix is larger than 2×2 it's vastly more efficient to compute the determinant either by reducing the matrix to <a href="Triangular_matrix" title="Triangular matrix">upper triangular</a> form using <a href="QR_decomposition" title="QR decomposition">QR decomposition</a> or to <a href="Gaussian_elimination#Computing_determinants" title="Gaussian elimination">compute the determinant</a> by reducing to <a href="Gaussian_elimination#Echelon_form" title="Gaussian elimination">echelon form</a> using <a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, rather than trying to actually compute the determinant of the matrix directly.

</p>

<div class="mw-heading mw-heading2"><h2 id="Analysis_–_first_principles_approaches">Analysis – first principles approaches</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Spectral_domain_method_of_moments_(overview_and_mathematical_introduction)">Spectral domain method of moments (overview and mathematical introduction)</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Background">Background</h4></div>

<div class="mw-heading mw-heading5"><h5 id="History">History</h5></div>

<p>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 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 <i>spatial</i> 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.

</p>

<div class="mw-heading mw-heading5"><h5 id="Advantages_and_disadvantages">Advantages and disadvantages</h5></div>

<p>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 1×1 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.

</p>

<div class="mw-heading mw-heading5"><h5 id="Floquet's_principle">Floquet's principle</h5></div>

<p>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. 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 <i>x</i>-<i>y</i> plane, then the Fourier series is a 2-dimensional Fourier series in <i>x</i>, <i>y</i>.

</p>

<div class="mw-heading mw-heading5"><h5 id="Plane_wave_spectrum">Plane wave spectrum</h5></div>

<p>As in <a href="Fourier_optics" title="Fourier optics">Fourier optics</a>, the Floquet–Fourier series expansion of fields and currents <i>in the plane of the FSS</i> leads immediately to the discrete plane wave spectrum representation of the fields on either side of the FSS.

</p>

<div class="mw-heading mw-heading4"><h4 id="Field_equations_for_2D_PEC_frequency_selective_surfaces">Field equations for 2D PEC frequency selective surfaces</h4></div>

<p>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 <b>J</b>. This section presents the spectral domain method for analyzing a free-standing (no substrate) PEC FSS. The electric field <b>E</b> is related to the vector magnetic potential <b>A</b> via the well-known relation (Harrington [2001], Scott [1989], Scott [1997]):

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ~=~-jk\eta \left[\mathbf {A} +{\frac {1}{k^{2}}}\nabla (\nabla \bullet \mathbf {A} )\right]}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.1.1" class="reference nourlexpansion" style="font-weight:bold;">2.1.1</span></td></tr></tbody></table>

<p>and the vector magnetic potential is in turn related to the source currents via (Harrington [2001], Scott [1997]):

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {A} +k^{2}\mathbf {A} ~=~-\mathbf {J} }</annotation>

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<p>where

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<annotation encoding="application/x-tex">{\displaystyle k^{2}~=~\omega ^{2}(\mu \varepsilon )=(2\pi /\lambda )^{2}}</annotation>

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</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.1.3" class="reference nourlexpansion" style="font-weight:bold;">2.1.3</span></td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Plane_wave_expansion_of_the_fields_in_source-free_media">Plane wave expansion of the fields in source-free media</h4></div>

<p>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, since plane waves are the eigenfunction solution to the vector wave equations in <i>source-free media</i>.

</p><p>To solve equations (<b><a href="#math_2.1.1">2.1.1</a></b>) and (<b><a href="#math_2.1.2">2.1.2</a></b>) for a free-standing, 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 (Tsao [1982], Scott [1989], <a href="Fourier_optics" title="Fourier optics">Fourier optics</a>):

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<annotation encoding="application/x-tex">{\displaystyle \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)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.1a" class="reference nourlexpansion" style="font-weight:bold;">2.2.1a</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle \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)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.1b" class="reference nourlexpansion" style="font-weight:bold;">2.2.1b</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle \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)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.1c" class="reference nourlexpansion" style="font-weight:bold;">2.2.1c</span></td></tr></tbody></table>

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

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

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<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{m}~=~k~\sin \theta _{0}~\cos \phi _{0}~+~{\frac {2m\pi }{l_{x}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.2a" class="reference nourlexpansion" style="font-weight:bold;">2.2.2a</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{n}~=~k~\sin \theta _{0}~\sin \phi _{0}~+~{\frac {2n\pi }{l_{y}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.2b" class="reference nourlexpansion" style="font-weight:bold;">2.2.2b</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msqrt>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{mn}~=~{\sqrt {~k^{2}~-~\alpha _{m}^{2}~-~\beta _{n}^{2}}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.2c" class="reference nourlexpansion" style="font-weight:bold;">2.2.2c</span></td></tr></tbody></table>

<p>and,

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k~=~2\pi /\lambda }</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.2.3" class="reference nourlexpansion" style="font-weight:bold;">2.2.3</span></td></tr></tbody></table>

<p>where <i>l</i><sub><i>x</i></sub>, <i>l</i><sub><i>y</i></sub> are the dimensions of the unit cell in the <i>x</i>,<i>y</i> directions respectively, λ is the free space wavelength and θ<sub>0</sub>, φ<sub>0</sub> are the directions of an assumed incident plane wave, with the FSS regarded as lying in the <i>x</i>-<i>y</i> plane. In (<b><a href="#math_2.2.2c">2.2.2c</a></b>), the root is taken which has a positive real part and non-positive (i.e., either negative or zero) imaginary part).

</p>

<div class="mw-heading mw-heading4"><h4 id="Integral_equation_for_free-standing_PEC_FSS">Integral equation for free-standing PEC FSS</h4></div>

<p>Substituting equations (<b><a href="#math_2.2.1">2.2.1</a></b>) into (<b><a href="#math_2.1.1">2.1.1</a></b>) and (<b><a href="#math_2.1.2">2.1.2</a></b>) 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:

</p>

</mrow>

<msub>

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</msub>

<msub>

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</msub>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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})}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.3.1" class="reference nourlexpansion" style="font-weight:bold;">2.3.1</span></td></tr></tbody></table>

<p>where,

</p>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

</mrow>

</mrow>

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<msup>

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</msup>

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<mtd>

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<msub>

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</msub>

<msub>

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</msub>

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<msup>

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<msub>

</mrow>

</msub>

<msub>

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</msub>

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<msup>

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</msup>

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</mrow>

</mtd>

<mtd>

<mfrac>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {G} _{mn}~=~{\begin{pmatrix}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{pmatrix}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.3.2" class="reference nourlexpansion" style="font-weight:bold;">2.3.2</span></td></tr></tbody></table>

<p>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. With this, the electric field boundary condition on the surface of PEC material within a unit cell becomes (Scott [1989]):

</p>

</mrow>

</munder>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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)}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.3.3" class="reference nourlexpansion" style="font-weight:bold;">2.3.3</span></td></tr></tbody></table>

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

</p><p>Equation (<b><a href="#math_2.3.3">2.3.3</a></b>) 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 (<b><a href="#math_2.3.3">2.3.3</a></b>) is tested with the current basis functions, defined as residing on the surface of the scatterer.

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

</p>

<msup>

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</mrow>

</msup>

<msup>

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</mrow>

</msup>

<msub>

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</msub>

<msub>

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</msub>

<msup>

<msub>

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</msub>

<msub>

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</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ^{inc}(x,y)=~\mathbf {E} ^{inc}(\alpha _{0},\beta _{0})~e^{j(\alpha _{0}x+\beta _{0}y)}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.3.4" class="reference nourlexpansion" style="font-weight:bold;">2.3.4</span></td></tr></tbody></table>

<p>in the x-y plane.

</p>

<div class="mw-heading mw-heading4"><h4 id="Method_of_moments_(MoM)_solution_2">Method of moments (MoM) solution</h4></div>

<p>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 <i>J</i><sub><i>j</i></sub> (Scott [1989]):

</p>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} (x,y,z)~=~\sum _{j}~J_{j}~\mathbf {J} _{j}(x,y,z)}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.4.1" class="reference nourlexpansion" style="font-weight:bold;">2.4.1</span></td></tr></tbody></table>

<p>Substituting (<b><a href="#math_2.4.1">2.4.1</a></b>) into (<b><a href="#math_2.3.3">2.3.3</a></b>) and then testing the resulting equation with the <i>i</i>-th current basis function (i.e., dotting from the left and integrating over the domain of the <i>i</i>-th current basis function, thereby completing the quadratic form) produces the <i>i</i>-th row of the matrix equation as (Scott [1989]):

</p>

</mrow>

</munder>

<msub>

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</msub>

<mrow>

<mrow>

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<mfrac>

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<msub>

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<msub>

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<msqrt>

<msup>

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</msup>

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</msubsup>

</msqrt>

</mfrac>

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</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

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</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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})}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2.4.2" class="reference nourlexpansion" style="font-weight:bold;">2.4.2</span></td></tr></tbody></table>

<p>This is the <i>i</i>-th row of the electric field integral equation (EFIE) for a free-standing metallic FSS. Equation (<b><a href="#math_2.4.2">2.4.2</a></b>) may be readily modified for analyzing FSS with surrounding dielectric sheets (substrates and/or superstrates), and even complex multi-layer FSS structures (Scott [1989]). 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. There is a striking similarity between eqn. (<b><a href="#math_2.4.2">2.4.2</a></b>) above, and the <a href="Bloch_wave_%E2%80%93_MoM_method" class="mw-redirect" title="Bloch wave – MoM method">Bloch wave - MoM method</a> eqn. (<b><a href="#math_2.4.2">2.4.2</a></b>) for computing ω–β diagrams for triply-periodic electromagnetic media such as <a href="Photonic_crystals" class="mw-redirect" title="Photonic crystals">photonic crystals</a> (Scott [1998], Scott [2002], available on researchgate.net). Given this similarity, eqn. (<b><a href="#math_2.4.2">2.4.2</a></b>) and its numerous variants in dielectric-layered FSS structures (Scott [1989]) could also be used (with the RHS set to zero) to find surface waves in complex FSS structures.

</p><p>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 <a href="Barycentric_coordinate_system#Barycentric_coordinates_on_triangles" title="Barycentric coordinate system">area coordinates</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Computing_reflection_and_transmission_coefficients">Computing reflection and transmission coefficients</h4></div>

<p>Equations (<b><a href="#math_2.4.2">2.4.2</a></b>) and (<b><a href="#math_2.3.1">2.3.1</a></b>) have been used to solve for the electric current <b>J</b> and then the scattered fields <b>E</b> 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 <span class="texhtml"><i>m</i> = 0</span>, <span class="texhtml"><i>n</i> = 0</span> order (the zero order).

</p><p>Or else the numerical method with periodical boundary conditions can serve as a powerful tool to compute the coefficients of FSS.

</p>

<div class="mw-heading mw-heading2"><h2 id="Equivalent_circuits_–_introduction">Equivalent circuits – introduction</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Background_2">Background</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Overview">Overview</h4></div>

<p>For wavelengths greater than the FSS lattice dimensions, only one – out of the infinitude of Floquet modes – actually propagates. All of the others are (exponentially decaying in the z-direction, normal to the plane of the FSS, since the quantity under the root in (<b><a href="#math_2.2c">2.2c</a></b>) is negative. And for FSS spacings greater than roughly a tenth of a wavelength or so, these evanescent wave fields have negligible effect on FSS stack performance. So, for practical purposes, in the frequency bands for which we'll be likely to use the FSS, a single propagating wave will be sufficient to capture the significant properties of a multi-layer FSS stack. This single propagating wave can be modeled in terms of an equivalent transmission line.

</p><p>The FSS sheet may be represented in terms of lumped RLC networks placed in parallel across the transmission line. The shunt admittance FSS model is exact only for an infinitesimally thin FSS, for which the tangential electric field is continuous across the FSS; for finite thickness FSS, a tee or pi network can be used as a better approximation.

</p>

<div class="mw-heading mw-heading4"><h4 id="Free_space_as_a_transmission_line">Free space as a transmission line</h4></div>

<p>Both free space and transmission lines admit TEM traveling wave solutions, and even the TE/TM plane waves in free space can be modeled using equivalent transmission line models. The main thing is that both free space and transmission lines admit traveling wave solutions with a z-dependence of the form:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\pm jkz}}</annotation>

</semantics>

</math></span></span>

</p><p>One can construct equivalent transmission lines as follows:

</p><p>For TEM waves,

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{0}~=~{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}}</annotation>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle k_{0}~=~\omega {\sqrt {\mu _{0}\varepsilon _{0}}}}</annotation>

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</math></span></span>

</p><p>For TE waves,

<msub>

</mrow>

</msub>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle Z_{0}~=~{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}~/~\cos \theta }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle k_{0}~=~\cos \theta ~\omega {\sqrt {\mu _{0}\varepsilon _{0}}}}</annotation>

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</math></span></span>

</p><p>For TM waves,

<msub>

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</msub>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle Z_{0}~=~{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}~\cos \theta }</annotation>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle k_{0}~=~\cos \theta ~\omega {\sqrt {\mu _{0}\varepsilon _{0}}}}</annotation>

</semantics>

</math></span></span>

where <i>θ</i> is the angle off-normal that the incident wave makes with respect to the FSS. <i>Z</i><sub>0</sub> for <a href="Impedance_of_free_space" title="Impedance of free space">free space</a> is 377 Ohms.

</p>

<div class="mw-heading mw-heading4"><h4 id="Shunt_circuit_resonators_and_FSS">Shunt circuit resonators and FSS</h4></div>

<p>Circuit elements placed in parallel across an equivalent transmission line have some factors in common with thin FSS. The continuity of tangential electric field condition for thin FSS mirrors the voltage continuity condition on either side of the shunt circuit elements. The magnetic field jump condition for the FSS mirrors the Kirchhoff current division law for the equivalent circuit. For sufficiently thick FSS sheets, a more general pi or tee model will likely be required for good approximation to the real FSS.

</p>

<div class="mw-heading mw-heading4"><h4 id="Resonant_circuits_can_approximately_model_resonant_scatterers.">Resonant circuits can approximately model resonant scatterers.</h4></div>

<p>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 (and for 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.

</p><p>One-dimensional line gratings can be modeled as shunt inductors (for polarization parallel to the lines) or shunt capacitors (for polarization perpendicular to the lines). Tightly packed "gangbuster" dipole arrays are lowpass structures that can be modeled using shunt capacitors.

</p>

<div class="mw-heading mw-heading4"><h4 id="Resonant_circuit_R,L,C_values_must_be_determined_from_first_principles_analysis">Resonant circuit R,L,C values must be determined from first principles analysis</h4></div>

<p>The exact circuit topology and element values of an equivalent circuit for a FSS sheet have to be determined using first-principles codes. A bandpass mesh-type FSS sheet is a parallel connection of L,C and bandstop patch-type FSS sheet is a series connection of L,C and in both cases, the L,C values are determined from the center frequency and bandwidth of the filter.

</p>

<div class="mw-heading mw-heading3"><h3 id="Reflection_and_transmission_properties_of_bandpass_and_bandstop_FSS_and_equivalent_circuits_–_introduction">Reflection and transmission properties of bandpass and bandstop FSS and equivalent circuits – introduction</h3></div>

<p>The equivalent transmission line circuit models for FSS came into being from the observation that FSS yield reflection and transmission properties that are very similar to the reflection and transmission properties of inductors and capacitors placed in parallel across a transmission line.

</p>

<div class="mw-heading mw-heading4"><h4 id="Bandstop_FSS_filter_equivalent_circuit_and_reflection_response">Bandstop FSS filter equivalent circuit and reflection response</h4></div>

<p>The two fundamental types of FSS are shown in Fig. 2.4.1-1 to the right - the bandpass mesh-type FSS and the bandstop patch-type FSS (<a href="Metal-mesh_optical_filters" class="mw-redirect" title="Metal-mesh optical filters">Metal-mesh optical filters</a>). The equivalent circuit for a patch-type bandstop FSS is shown in Fig. 2.4.1-2. The impedance of the series connection of the inductor and the capacitor is (Desoer, Kuh [1984]):

<msub>

<mtext>shunt</mtext>

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<annotation encoding="application/x-tex">{\displaystyle Z_{\text{shunt}}~=~j\omega L~+~{\frac {1}{j\omega C}}}</annotation>

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</math></span></span>

or,

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{\text{shunt}}~=~{\frac {1-\omega ^{2}LC}{j\omega C}}}</annotation>

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</math></span></span>

and this series connection of an inductor and capacitor produces a zero impedance (short circuit) condition when

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<annotation encoding="application/x-tex">{\displaystyle \omega _{0}^{2}~=~{\frac {1}{LC}}}</annotation>

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</math></span></span>

</p><p>At the short circuit condition, all incident energy is reflected, and so this is the equivalent circuit of a resonant patch bandstop filter.

</p><p>The magnitude of the reflection coefficient is:

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</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |R|={\frac {\omega CZ_{0}}{\sqrt {\omega ^{2}C^{2}Z_{0}^{2}~+~4~(1~-~\omega ^{2}LC)^{2}}}}}</annotation>

</semantics>

</math></span></span>

where <i>Z</i><sub>0</sub> is the characteristic impedance of the transmission line.

</p><p>The frequencies for the upper and lower 3 dB points are given as the solution to the equation:

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1,2}^{2}~=~{\frac {-b\pm {\sqrt {b^{2}~-~4ac}}}{2a}}}</annotation>

</semantics>

</math></span></span>

where,

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a~=~(LC)^{2}}</annotation>

</semantics>

</math></span></span>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b~=~-(C^{2}Z_{0}^{2}~/~4~+~2LC)}</annotation>

</semantics>

</math></span></span>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c~=~1}</annotation>

</semantics>

</math></span></span>

</p><p>So, if the center frequency and the width of the resonance are determined from first principles codes, the L,C of the equivalent circuit may be readily obtained by fitting the reflection response of the equivalent resonant circuit to the reflection response of the actual FSS, and in this way, the circuit parameters L,C are readily extracted. Once that is done, then we can use the equivalent circuit model for multi-layer FSS design. Any nearby dielectrics should be included in the equivalent circuit.

</p><p>For small values of ω, the impedance of the inductor, jωL, is smaller than the impedance of the capacitor, 1/jωC, therefore the capacitor dominates the shunt impedance and so the patch-type bandstop FSS is capacitive below resonance. We'll use this fact in section 2.3.1 to design a lowpass FSS filter using equivalent circuits.

</p>

<div class="mw-heading mw-heading4"><h4 id="Bandpass_FSS_filter_equivalent_circuit_and_transmission_response">Bandpass FSS filter equivalent circuit and transmission response</h4></div>

<p>The equivalent circuit for a mesh-type bandpass FSS is shown in Fg. 2.4.2-1. The admittance of the parallel connection of inductor and capacitor is (Desoer, Kuh [1984]):

<msub>

<mtext>shunt</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{\text{shunt}}~=~{\frac {1-\omega ^{2}LC}{j\omega L}}}</annotation>

</semantics>

</math></span></span>

and this admittance is zero (open-circuit condition) when

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{0}^{2}~=~{\frac {1}{LC}}}</annotation>

</semantics>

</math></span></span>

</p><p>When the parallel combination of inductor and capacitor produces an open circuit, all energy is transmitted.

</p><p>In the same way, the magnitude of the transmission coefficient of the bandpass filter is:

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |T|={\frac {2\omega L/Z_{0}}{\sqrt {4\omega ^{2}L^{2}/Z_{0}^{2}~+~(1~-~\omega ^{2}LC)^{2}}}}}</annotation>

</semantics>

</math></span></span>

</p><p>Below resonance, the admittance of the inductor, 1/jωL is greater than the admittance of the capacitor jωC, therefore the mesh-type bandpass FSS is inductive below resonance.

</p>

<div class="mw-heading mw-heading4"><h4 id="Comparison_of_equivalent_circuit_response_and_actual_FSS_response">Comparison of equivalent circuit response and actual FSS response</h4></div>

<p>Fig. 2.4.3-1 shows the comparison in reflection between a single-layer crossed dipole FSS and its fitted equivalent circuit. The equivalent circuit is a series connection of a capacitor and inductor placed in parallel across the transmission line, as in Fig. 2.4.1-2. This resonator produces a short circuit condition at resonance. The fit is very good below the resonance though not nearly as good above.

</p><p>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 (<b><a href="#math_3.1">3.1</a></b>) going to infinity. Physically, this represents a uniform plane wave propagating in the plane of the FSS. In the spatial domain, the coherent summation of all of the spatial domain Green's function's becomes infinite, so that any finite current produces an infinite field on the surface of the FSS. As a result, all currents must be zero under this condition.

</p><p>This example illustrates the usefulness and shortcomings of the simple equivalent circuit model. The equivalent circuit only includes features related to the individual scattering element, not features related to the periodic array, such as interactions between the scatterers.

</p>

<div class="mw-heading mw-heading4"><h4 id="FSS_duality_versus_circuit_duality">FSS duality versus circuit duality</h4></div>

<div class="mw-heading mw-heading5"><h5 id="FSS_duality">FSS duality</h5></div>

<p>If a mesh type FSS is created from a patch type FSS in such a way that the metal portions or the former are replaced by aperture portions of the latter, then the two FSS are said to be duals of one another. Duality only strictly applies when no dielectric substrates are present, therefore it is only approximately satisfied in practice, though even when dielectric substrates are present, duality can be useful in FSS design. As a side note, Pathological FSS patterns such as a checkerboard FSS may be treated as the limit of the patch and mesh as the patch (and aperture) size approaches the unit cell size, with electrical connections of the mesh retained in the limit. For dual FSS, the reflection coefficient of the patch will be equal to the transmission coefficient of the mesh.

</p><p>Circuit duality

</p><p>The dual circuit of the bandstop filter can be obtained simply equating the reflection coefficient of the bandstop FSS to the transmission coefficient of the bandpass FSS to obtain (if we use L<sub>1</sub>, C<sub>1</sub> for the bandstop patch FSS and L<sub>2</sub>, C<sub>2</sub> for the bandpass mesh FSS):

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}~=~C_{1}~{\frac {Z_{0}^{2}}{4}}}</annotation>

</semantics>

</math></span></span>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{2}~=~L_{1}~{\frac {4}{Z_{0}^{2}}}}</annotation>

</semantics>

</math></span></span>

</p><p>This produces a bandpass circuit (with parameters L<sub>2</sub>, C<sub>2</sub>) which is the dual of the bandstop circuit (with parameters L<sub>1</sub>, C<sub>1</sub>).

</p>

<div class="mw-heading mw-heading2"><h2 id="FSS_equivalent_circuits_-_applications_to_FSS_design">FSS equivalent circuits - applications to FSS design</h2></div>

<p>Once the transmission line equivalent circuit has been determined, multi-layer FSS design becomes much simpler and more intuitive - like ordinary filter analysis and design. Now while it is certainly possible to design multi-layer FSS structures using first principles codes and generalized scattering matrices (GSM), it is far easier, quicker and more intuitive to use equivalent circuit models for FSS design, since it is possible to leverage decades' worth of research performed on electrical filter analysis and design and bring it to bear on FSS structures. And, FSS filters are even easier to design than waveguide filters since the incidence angle does not vary with frequency.

</p>

<div class="mw-heading mw-heading3"><h3 id="Butterworth_lowpass_filter_design_using_FSS_equivalent_circuits">Butterworth lowpass filter design using FSS equivalent circuits</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Starting_point:_prototype_lumped_L,_C_Butterworth_filter">Starting point: prototype lumped <i>L</i>, <i>C</i> Butterworth filter</h4></div>

<p>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 <a href="Butterworth_filter" title="Butterworth filter">Butterworth filter</a> (Hunter [2001], Matthaei [1964]) using a stack of 5 frequency selective surfaces, with 4 air spacers in between the FSS sheets.

</p><p>The lowpass prototype L,C ladder network is shown in Fig. 3.1.1-1 (Hunter [2001]). 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. The idea we'll follow is that the shunt capacitors will eventually be replaced by sub-resonant (capacitive) patch-type FSS sheets and the series inductors will be replaced by air spacers between the 5 FSS layers. Short transmission lines are approximately equivalent to series inductors.

</p>

<div class="mw-heading mw-heading4"><h4 id="Transmission_response_of_prototype_lumped_L,_C_filter">Transmission response of prototype lumped <i>L</i>, <i>C</i> filter</h4></div>

<p>The transmission magnitude and phase response of the scaled Butterworth L,C filter is shown in Fig. 3.1.2-1. Transmission magnitude is flat in the passband (below the 7 GHz cutoff frequency) and has a monotonically decreasing skirt on the high frequency side of the passband. The phase through the filter is linear throughout the 7 GHz passband, making this filter an ideal choice for a linear phase filter application, for example in the design of an ultra-wideband filter that approximates a true time delay transmission line. This is the baseline lumped L,C filter that will be the starting point for our 5-layer FSS Butterworth filter design.

</p><p>Now we begin the process of transforming the prototype Butterworth lumped L,C filter into an equivalent FSS Butterworth filter. Two modifications of the baseline lumped L,C filter will be necessary, in order to obtain the corresponding FSS filter. First, the series inductors will be replaced by their equivalent transmission line sections, and then the shunt capacitors will be replaced by capacitive frequency selective surfaces.

</p>

<div class="mw-heading mw-heading4"><h4 id="First_transformation:_replace_series_inductors_with_transmission_line_spacers">First transformation: replace series inductors with transmission line spacers</h4></div>

<p>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. The thickness of the air spacers may be determined as shown in Fig. 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.

</p><p>With the thickness of the air spacers between sheets now determined, the equivalent circuit now takes on the form shown in Fig. 3.1.4-1:

</p>

<div class="mw-heading mw-heading4"><h4 id="Second_transformation:_Replace_shunt_capacitors_with_capacitive_patch_FSS_below_resonance">Second transformation: Replace shunt capacitors with capacitive patch FSS below resonance</h4></div>

<p>Now the only thing left to do is to find the lowpass FSS that yields the shunt capacitance values called out in Fig. 2.3.1-4. This is usually done through trial and error. Fitting a shunt capacitor to a real FSS is done by repeated running of a first principles code to match the reflection response of the shunt capacitor with the reflection from a capacitive FSS. Patch-type FSS below resonance will produce a capacitive shunt admittance equivalent circuit, with closer packing of elements in the FSS sheet yielding higher shunt capacitance values in the equivalent circuit.

</p>

<div class="mw-heading mw-heading2"><h2 id="Fabrication">Fabrication</h2></div>

<p>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). The sheet may range in thickness from a few thousandths of an inch to as much as 20–40 thousand.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

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

</p>

<div class="mw-heading mw-heading3"><h3 id="Microwave_ovens">Microwave ovens</h3></div>

<div class="mw-heading mw-heading3"><h3 id="Antennas">Antennas</h3></div>

<p><a href="Radome" title="Radome">Radomes</a>

</p><p>EM absorbers

</p>

<div class="mw-heading mw-heading3"><h3 id="Camouflage">Camouflage</h3></div>

<p>For <a href="Multi-spectral_camouflage" title="Multi-spectral camouflage">multi-spectral camouflage</a>, like the <a href="Saab_Barracuda" class="mw-redirect" title="Saab Barracuda">Saab Barracuda</a>, FSS can be used to allow certain frequencies to penetrate, so communication and GPS isn't blocked.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Fourier_optics" title="Fourier optics">Fourier optics</a></li>

<li><a href="Photonic_crystal" title="Photonic crystal">Photonic crystal</a></li>

<li><a href="Metamaterial" title="Metamaterial">Metamaterial</a></li>

<li><a href="Bragg_diffraction" class="mw-redirect" title="Bragg diffraction">Bragg diffraction</a></li>

<li><a href="Diffraction_grating" title="Diffraction grating">Diffraction grating</a></li>

<li><a href="Bloch_wave" class="mw-redirect" title="Bloch wave">Bloch wave</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 20em;">

<li id="cite_note-FOOTNOTEKastner1987-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKastner1987_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKastner1987">Kastner 1987</a>.</span>

</li>

<li id="cite_note-FOOTNOTEHarrington1961-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEHarrington1961_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHarrington1961_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHarrington1961">Harrington 1961</a>.</span>

</li>

<li id="cite_note-FOOTNOTEScott1998-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEScott1998_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEScott1998_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEScott1998_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFScott1998">Scott 1998</a>.</span>

</li>

<li id="cite_note-FOOTNOTERumsey1954-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERumsey1954_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRumsey1954">Rumsey 1954</a>.</span>

</li>

<li id="cite_note-FOOTNOTEScott1989-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEScott1989_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFScott1989">Scott 1989</a>.</span>

</li>

<li id="cite_note-FOOTNOTEScott2002-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEScott2002_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFScott2002">Scott 2002</a>.</span>

</li>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.shephardmedia.com/news/landwarfareintl/dsei-saab-barracuda-reveals-new-camouflage-and-shows-jltv-with-mobile-camouflage-system/">"DSEi 2023: Saab Barracuda reveals new camouflage system"</a>. <i>shephardmedia.com</i>. 2023-09-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-05</span></span>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFHarrington2001" class="citation cs2">Harrington, Roger (2001), <i>Time-Harmonic Electromagnetic Fields</i>, John Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-20806-8</bdi></cite></li>

<li>Hunter, Ian, <i>Theory and Design of Microwave Filters</i> (IEE: 2001).</li>

<li>Matthaei, George L.; Young, Leo and Jones, E. M. T., <i>Microwave Filters, Impedance-Matching Networks, and Coupling Structures</i>, McGraw-Hill, 1964}.</li>

<li><cite id="CITEREFMunk2000" class="citation cs2">Munk, Benedikt (2000), <i>Frequency Selective Surfaces: Theory and Design</i>, John Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-37047-5</bdi></cite></li>

<li>Ramo, S.; Whinnery, J. R. and Van Duzer T., <i>Fields and Waves in Communication Electronics</i>, Wiley, 1994 978-0471585510}.</li>

<li><cite id="CITEREFRaoWiltonGlisson1982" class="citation cs2">Rao, S.M.; Wilton, Donald; Glisson, Allen (1982), "Electromagnetic scattering by surfaces of arbitrary shape", <i>IEEE Transactions on Antennas and Propagation</i>, <b>30</b> (3), IEEE Trans. Antennas Propagat.: 409, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982ITAP...30..409R">1982ITAP...30..409R</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAP.1982.1142818">10.1109/TAP.1982.1142818</a></cite></li>

<li>W. Mai <i>et al</i>., <i>Prism-Based DGTD With a Simplified Periodic Boundary Condition to Analyze FSS With D2n Symmetry in a Rectangular Array Under Normal Incidence</i>, in <i>IEEE Antennas and Wireless Propagation Letters</i>. <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/8656552">doi: 10.1109/LAWP.2019.2902340</a></li>

<li><cite id="CITEREFScott1989" class="citation cs2">Scott, Craig (1989), <i>The Spectral Domain Method in Electromagnetics</i>, Artech House, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989ah...bookQ....S">1989ah...bookQ....S</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89006-349-4</bdi></cite></li>

<li><cite id="CITEREFScott1997" class="citation cs2">Scott, Craig (1997), <i>Introduction to Optics and Optical Imaging</i>, IEEE Press, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998iooi.book.....S">1998iooi.book.....S</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0780334403</bdi></cite></li>

<li><cite id="CITEREFScott1998" class="citation cs2">Scott, Craig (1998), <i>Analysis, Design and Testing of Integrated Structural Radomes Built Using Photonic Bandgap Structures</i></cite></li>

<li><cite id="CITEREFScott2002" class="citation cs2">Scott, Craig (2002), <i>Spectral Domain Analysis of Doped Electromagnetic Crystal Radomes Using the Method of Moments</i></cite></li>

<li><cite id="CITEREFTsaoMittra1982" class="citation cs2">Tsao, Chich-Hsing; Mittra, Raj (1982), "A Spectral Iteration Approach for Analyzing Scattering from Frequency Selective Surfaces", <i>IEEE Transactions on Antennas and Propagation</i>, <b>30</b> (2), IEEE Trans. Antennas Propagat. Vol. AP-30, No. 2, March 1982: <span class="nowrap">303–</span>308, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982ITAP...30..303T">1982ITAP...30..303T</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAP.1982.1142779">10.1109/TAP.1982.1142779</a></cite></li>

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