Doubly Periodic - BLB

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<ul><li><p>8/10/2019 Doubly Periodic - BLB</p><p> 1/7</p><p>1712</p><p>J.</p><p>Opt.</p><p>Soc.</p><p>Am.</p><p>A/Vol.</p><p>7,</p><p>No.</p><p>9/September</p><p>1990</p><p>Boag</p><p>t</p><p>al.</p><p>Analysis</p><p>of</p><p>diffraction</p><p>from</p><p>doubly</p><p>periodic</p><p>arrays</p><p>of</p><p>perfectly</p><p>conducting</p><p>bodies</p><p>by</p><p>using</p><p>a</p><p>patch-current</p><p>model</p><p>Amir</p><p>Boag,</p><p>Yehuda</p><p>Leviatan,</p><p>and</p><p>Alona</p><p>Boag</p><p>Department</p><p>of</p><p>Electrical</p><p>Engineering,</p><p>Technion-Israel</p><p>Institute</p><p>of Technology,</p><p>Haifa</p><p>32000,</p><p>srael</p><p>Received</p><p>November</p><p>29,</p><p>1989;</p><p>accepted</p><p>May</p><p>10, 1990</p><p>A</p><p>novel</p><p>solution</p><p>is</p><p>presented</p><p>for</p><p>the</p><p>problem</p><p>of</p><p>three-dimensional</p><p>electromagnetic</p><p>scattering</p><p>of</p><p>aplane</p><p>wave</p><p>from</p><p>a</p><p>doubly</p><p>periodic</p><p>infinite</p><p>array</p><p>of</p><p>perfectly</p><p>conducting</p><p>bodies.</p><p>A set of</p><p>fictitious</p><p>spatially</p><p>periodic</p><p>and</p><p>properly</p><p>modulated</p><p>patches</p><p>of magnetic</p><p>current</p><p>is</p><p>used</p><p>to</p><p>simulate</p><p>the</p><p>scattered</p><p>field.</p><p>These</p><p>patch</p><p>currents</p><p>are</p><p>of</p><p>dual</p><p>polarization</p><p>and have</p><p>complex</p><p>amplitudes.</p><p>The</p><p>electromagnetic</p><p>field</p><p>radiated</p><p>by</p><p>each</p><p>of the</p><p>periodic</p><p>patch</p><p>currents</p><p>isexpressed</p><p>as</p><p>a double</p><p>series</p><p>of Floquet</p><p>modes.</p><p>The</p><p>complex</p><p>amplitudes</p><p>of the</p><p>fictitious</p><p>patch</p><p>currents</p><p>are</p><p>adjusted</p><p>to render</p><p>the</p><p>tangential</p><p>electric</p><p>field</p><p>zero</p><p>at</p><p>a selected</p><p>set</p><p>of points</p><p>on</p><p>the</p><p>surface</p><p>of</p><p>any</p><p>of the</p><p>scatterers.</p><p>The</p><p>procedure</p><p>is</p><p>simple</p><p>to</p><p>implement</p><p>and</p><p>is</p><p>applicable</p><p>to</p><p>arrays</p><p>composed</p><p>of</p><p>smooth</p><p>but</p><p>otherwise</p><p>arbitrary perfectly conducting scatterers. Results are givenand compared with an analytic approximation.</p><p>1.</p><p>INTRODUCTION</p><p>The</p><p>study</p><p>of</p><p>diffraction</p><p>of</p><p>a plane</p><p>wave</p><p>from</p><p>periodic</p><p>struc-</p><p>tures</p><p>is</p><p>long</p><p>standing.</p><p>It</p><p>has</p><p>been</p><p>motivated</p><p>by</p><p>academic</p><p>curiosity</p><p>as</p><p>well</p><p>as many</p><p>engineering</p><p>applications.</p><p>It is</p><p>of</p><p>practical</p><p>importance</p><p>in</p><p>designing</p><p>reflection</p><p>and</p><p>transmis-</p><p>sion</p><p>gratings</p><p>often</p><p>used</p><p>as</p><p>filters,</p><p>broadband</p><p>absorbers,</p><p>polarizers,</p><p>and</p><p>frequency</p><p>scanned</p><p>reflectors.</p><p>While</p><p>singly</p><p>periodic</p><p>gratings</p><p>have</p><p>been</p><p>treated</p><p>extensively,</p><p>doubly</p><p>peri-</p><p>odic</p><p>gratings,</p><p>being</p><p>in</p><p>general</p><p>more</p><p>difficult</p><p>not</p><p>only</p><p>to</p><p>analyzebut also to fabricate, have receivedconsiderably less</p><p>attention.</p><p>One</p><p>type of</p><p>doubly</p><p>periodic</p><p>structure</p><p>that</p><p>has</p><p>been</p><p>investigated</p><p>by many</p><p>researchers</p><p>comprises</p><p>infinitesi-</p><p>mally</p><p>thin</p><p>planar</p><p>doubly</p><p>periodic</p><p>screens</p><p>in various</p><p>configu-</p><p>rations,</p><p>often</p><p>referred</p><p>to</p><p>as</p><p>frequency-selective</p><p>surfaces.</p><p>-</p><p>3</p><p>Perfectly</p><p>conducting</p><p>screens</p><p>of</p><p>finite</p><p>thickness</p><p>consisting</p><p>of</p><p>doubly</p><p>periodic</p><p>arrays</p><p>of</p><p>apertures,</p><p>known</p><p>as inductive</p><p>grids,</p><p>have</p><p>also</p><p>been</p><p>extensively</p><p>studied.4</p><p>5</p><p>While</p><p>the</p><p>structures</p><p>in</p><p>Refs.</p><p>1-5</p><p>may</p><p>be</p><p>different,</p><p>a common</p><p>ingredi-</p><p>ent</p><p>of</p><p>essentially</p><p>all of</p><p>them</p><p>is</p><p>that</p><p>the</p><p>fields</p><p>in</p><p>the</p><p>various</p><p>regions</p><p>can</p><p>be</p><p>represented</p><p>by modal</p><p>expansions</p><p>relative</p><p>to</p><p>the axis</p><p>normal</p><p>to</p><p>the</p><p>grating.</p><p>These</p><p>modal</p><p>representa-</p><p>tions</p><p>are then</p><p>matched</p><p>by</p><p>using</p><p>appropriate</p><p>boundary</p><p>con-</p><p>ditions, and the unknown modal coefficients are readily de-</p><p>termined.</p><p>In</p><p>contrast</p><p>to</p><p>the</p><p>above</p><p>discussion,</p><p>rigorous</p><p>studies</p><p>of</p><p>diffraction</p><p>from</p><p>doubly</p><p>periodic</p><p>arrays</p><p>of</p><p>finite-</p><p>sized</p><p>perfectly</p><p>conducting</p><p>scatterers,</p><p>referred</p><p>to</p><p>as</p><p>capaci-</p><p>tive</p><p>grids,</p><p>have</p><p>not</p><p>been</p><p>reported</p><p>in</p><p>the</p><p>literature.</p><p>In this</p><p>latter</p><p>case</p><p>the fields</p><p>in</p><p>the</p><p>space</p><p>gap</p><p>between</p><p>the</p><p>scatterers</p><p>forming</p><p>the array</p><p>cannot</p><p>be</p><p>represented</p><p>in</p><p>terms</p><p>of</p><p>analyti-</p><p>cally</p><p>known</p><p>modal</p><p>functions.</p><p>5</p><p>Therefore</p><p>even</p><p>the</p><p>method</p><p>devised</p><p>in</p><p>Ref.</p><p>6 for</p><p>the</p><p>analysis</p><p>of</p><p>diffraction</p><p>by</p><p>doubly</p><p>periodic</p><p>surfaces</p><p>falls</p><p>short,</p><p>while</p><p>differential</p><p>method</p><p>pro-</p><p>cedures</p><p>might</p><p>be</p><p>infeasible</p><p>with</p><p>present-time</p><p>computer</p><p>storage</p><p>and</p><p>speed</p><p>limitations.</p><p>In</p><p>this</p><p>paper</p><p>we</p><p>present</p><p>a</p><p>new</p><p>method</p><p>for</p><p>analyzing</p><p>three-</p><p>dimensional electromagnetic scattering from doubly period-</p><p>ic</p><p>arrays.</p><p>The</p><p>technique</p><p>is</p><p>applicable</p><p>to</p><p>arrays</p><p>composed</p><p>of</p><p>perfectly</p><p>conducting</p><p>bodies</p><p>of smooth,</p><p>but</p><p>otherwise</p><p>arbi-</p><p>trary,</p><p>shapes.</p><p>An</p><p>example</p><p>of</p><p>an</p><p>array</p><p>is</p><p>depicted</p><p>in</p><p>Fig.</p><p>1.</p><p>We</p><p>follow</p><p>the</p><p>approach</p><p>outlined</p><p>in</p><p>Ref.</p><p>7</p><p>for</p><p>analyzing</p><p>scattering</p><p>by</p><p>smooth</p><p>homogeneous</p><p>scatterers.</p><p>The</p><p>basic</p><p>idea</p><p>in</p><p>Ref.</p><p>7 is</p><p>as</p><p>follows:</p><p>Instead</p><p>of</p><p>employing</p><p>surface</p><p>integral</p><p>equations</p><p>in</p><p>solving</p><p>for</p><p>conventional</p><p>electric</p><p>and</p><p>magnetic</p><p>surface</p><p>currents,</p><p>we</p><p>solve</p><p>for</p><p>fictitious</p><p>source</p><p>cur-</p><p>rents</p><p>that</p><p>lie</p><p>a</p><p>distance</p><p>away</p><p>from</p><p>the</p><p>surface.</p><p>This</p><p>idea</p><p>has</p><p>been</p><p>applied</p><p>successfully</p><p>to</p><p>two-dimensional</p><p>diffraction</p><p>from</p><p>gratings</p><p>of</p><p>cylinders,</p><p>8</p><p>as</p><p>well</p><p>as</p><p>to</p><p>sinusoidal</p><p>and</p><p>ech-</p><p>elette</p><p>gratings.</p><p>9</p><p>-</p><p>10</p><p>In</p><p>Refs.</p><p>8-10,</p><p>an</p><p>expansion</p><p>of</p><p>periodic</p><p>strip</p><p>currents</p><p>is used</p><p>for</p><p>the</p><p>unknown</p><p>fictitious</p><p>currents</p><p>that simulate the periodic scattered field, and point match-</p><p>ing</p><p>is</p><p>used</p><p>for</p><p>testing.</p><p>Here,</p><p>we</p><p>employ</p><p>the</p><p>basic</p><p>strategy</p><p>of</p><p>Refs.</p><p>7-10</p><p>for</p><p>facili-</p><p>tating</p><p>the</p><p>analysis</p><p>of</p><p>three-dimensional</p><p>scattering</p><p>from</p><p>doubly</p><p>periodic</p><p>arrays</p><p>of</p><p>isolated</p><p>perfectly</p><p>conducting</p><p>scat-</p><p>terers.</p><p>We</p><p>set</p><p>up</p><p>a simulated</p><p>equivalent</p><p>situation</p><p>to</p><p>the</p><p>original</p><p>one</p><p>in</p><p>the</p><p>region</p><p>surrounding</p><p>the</p><p>scatterers.</p><p>The</p><p>scattered</p><p>field</p><p>must</p><p>be</p><p>a</p><p>source-free</p><p>Maxwellian</p><p>field</p><p>satis-</p><p>fying</p><p>the</p><p>radiation</p><p>condition</p><p>at</p><p>z</p><p>-</p><p>,</p><p>the</p><p>periodicity</p><p>conditions</p><p>of</p><p>the Floquet</p><p>theorem,</p><p>and</p><p>the</p><p>boundary</p><p>con-</p><p>dition</p><p>on</p><p>the</p><p>surfaces</p><p>of the</p><p>scatterers.</p><p>Instead</p><p>of express-</p><p>ing</p><p>the</p><p>scattered</p><p>field</p><p>as</p><p>a conventional</p><p>integral</p><p>in</p><p>terms</p><p>of</p><p>the</p><p>physical</p><p>surface</p><p>currents,</p><p>we</p><p>simulate</p><p>the</p><p>actual</p><p>field</p><p>by</p><p>the fields of fictitious sources of yet unknown amplitudes</p><p>that</p><p>lie</p><p>a distance</p><p>from</p><p>the</p><p>surface.</p><p>Hence,</p><p>in the</p><p>simulated</p><p>equivalence</p><p>the physical</p><p>bodies</p><p>are</p><p>removed</p><p>and</p><p>the</p><p>period-</p><p>ic</p><p>field</p><p>that</p><p>they</p><p>scatter</p><p>is</p><p>simulated</p><p>by</p><p>the</p><p>field</p><p>of</p><p>a set</p><p>of</p><p>fictitious</p><p>doubly</p><p>periodic</p><p>patches</p><p>of</p><p>currents</p><p>satisfying</p><p>the</p><p>Floquet</p><p>periodicity</p><p>conditions</p><p>and</p><p>situated</p><p>in the</p><p>region</p><p>originally</p><p>occupied</p><p>by</p><p>the</p><p>scatterers.</p><p>Each</p><p>periodic</p><p>patch</p><p>current</p><p>lies</p><p>in</p><p>a plane</p><p>parallel</p><p>with</p><p>the</p><p>xy plane</p><p>(the</p><p>plane</p><p>spanned</p><p>by</p><p>the</p><p>directions</p><p>of</p><p>the</p><p>periodicity).</p><p>All the</p><p>patch-</p><p>es</p><p>are</p><p>characterized</p><p>by a</p><p>common</p><p>Fourier-transformable</p><p>magnetic-current</p><p>density</p><p>profile,</p><p>which,</p><p>for</p><p>each</p><p>periodic</p><p>source,</p><p>is</p><p>multiplied</p><p>by</p><p>an as</p><p>yet</p><p>undetermined</p><p>constant</p><p>complex</p><p>amplitude.</p><p>They</p><p>are</p><p>assumed</p><p>to</p><p>radiate</p><p>in</p><p>an</p><p>unbounded homogeneousspace filledwith the same medium</p><p>as</p><p>that</p><p>surrounding</p><p>the</p><p>scatterers.</p><p>Patches</p><p>of</p><p>electric</p><p>cur-</p><p>rent</p><p>could</p><p>be</p><p>used</p><p>as</p><p>well.</p><p>Patches</p><p>of magnetic</p><p>current</p><p>were</p><p>chosen</p><p>because</p><p>the</p><p>electric</p><p>field</p><p>that</p><p>they</p><p>produce</p><p>is</p><p>0740-3232/90/091712-07$02.00</p><p> 1990</p><p>Optical</p><p>Society</p><p>of America</p></li><li><p>8/10/2019 Doubly Periodic - BLB</p><p> 2/7</p><p>Vol. 7,</p><p>No.</p><p>9/September</p><p>1990/J.</p><p>Opt.</p><p>Soc.</p><p>Am.</p><p>A</p><p>1713</p><p>Unbounded</p><p>Space</p><p>,</p><p>E)</p><p>magnitudes</p><p>are</p><p>specified</p><p>by</p><p>the</p><p>respective</p><p>periods.</p><p>It</p><p>is</p><p>assumed</p><p>that</p><p>any</p><p>inhomogeneity</p><p>is confined</p><p>between</p><p>the</p><p>z</p><p>=</p><p>-b</p><p>and</p><p>the</p><p>z = b</p><p>planes.</p><p>The problem</p><p>geometry</p><p>together</p><p>with</p><p>a</p><p>relevant</p><p>coordinate</p><p>system</p><p>is</p><p>shown</p><p>in</p><p>Fig.</p><p>1.</p><p>It</p><p>should</p><p>be noted</p><p>that,</p><p>according</p><p>to</p><p>our</p><p>convention,</p><p>the</p><p>z</p><p>axis</p><p>is oriented</p><p>downward.</p><p>The</p><p>medium</p><p>surrounding</p><p>the</p><p>scat-</p><p>terers is of permeability</p><p>As nd</p><p>permittivity</p><p>e.</p><p>The medium</p><p>can</p><p>be</p><p>dissipative;</p><p>thus</p><p>g and</p><p>e</p><p>are</p><p>allowed</p><p>to</p><p>be</p><p>complex.</p><p>A plane</p><p>wave</p><p>given</p><p>by</p><p>Einc(r)</p><p>=</p><p>EinC</p><p>xp(-jkinc</p><p>r),</p><p>(1)</p><p>Fig.</p><p>1.</p><p>General</p><p>problem</p><p>of plane-wave</p><p>scattering</p><p>periodic</p><p>grating</p><p>of</p><p>finite-sized</p><p>scatterers.</p><p>from</p><p>a doubly</p><p>easier</p><p>to</p><p>compute.</p><p>Locating</p><p>the</p><p>sources</p><p>some</p><p>distance</p><p>away</p><p>from</p><p>the</p><p>surface</p><p>permits</p><p>us</p><p>to</p><p>use</p><p>periodic</p><p>patch</p><p>currents</p><p>with smooth current density profile that lie in planes parallel</p><p>with</p><p>the</p><p>xy</p><p>plane</p><p>spanned</p><p>by the</p><p>two</p><p>directions</p><p>of</p><p>periodici-</p><p>ty.</p><p>This</p><p>feature</p><p>is</p><p>attractive</p><p>because</p><p>it</p><p>enables</p><p>the</p><p>representation</p><p>of</p><p>the</p><p>field</p><p>produced</p><p>by</p><p>each</p><p>periodic</p><p>cur-</p><p>rent</p><p>patch</p><p>by</p><p>uniformly</p><p>convergent</p><p>series</p><p>of</p><p>z-directed</p><p>out-</p><p>going</p><p>and</p><p>decaying</p><p>plane</p><p>waves</p><p>known</p><p>as Floquet</p><p>modes.</p><p>It follows</p><p>hat</p><p>outside</p><p>the</p><p>grating</p><p>region</p><p>the</p><p>total</p><p>field</p><p>radi-</p><p>ated</p><p>by the</p><p>patches</p><p>can</p><p>also</p><p>be represented</p><p>analytically</p><p>by</p><p>means</p><p>of</p><p>these</p><p>Floquet</p><p>modes.</p><p>Thus</p><p>the</p><p>fields</p><p>can</p><p>be</p><p>deter-</p><p>mined</p><p>anywhere</p><p>by</p><p>summations</p><p>of</p><p>analytic</p><p>terms.</p><p>This</p><p>is</p><p>a</p><p>desirable</p><p>attribute</p><p>as</p><p>one</p><p>avoids</p><p>the</p><p>surface</p><p>integrations</p><p>associated</p><p>with</p><p>the</p><p>field</p><p>computation</p><p>at the</p><p>three</p><p>principal</p><p>stages</p><p>of</p><p>the</p><p>solution.</p><p>The</p><p>first</p><p>stage</p><p>is that</p><p>of</p><p>constructing</p><p>matrix equations for the problem, the second is that of test-</p><p>ing</p><p>the</p><p>solution</p><p>by</p><p>checking</p><p>the</p><p>degree</p><p>to</p><p>which</p><p>the</p><p>bound-</p><p>ary</p><p>conditions</p><p>are</p><p>satisfied</p><p>over</p><p>a denser</p><p>set</p><p>of points</p><p>on the</p><p>boundaries,</p><p>and</p><p>the</p><p>third</p><p>is</p><p>that</p><p>of</p><p>computing</p><p>the</p><p>scattered</p><p>field</p><p>and the</p><p>reflection</p><p>and</p><p>transmission</p><p>coefficients</p><p>of</p><p>vari-</p><p>ous</p><p>Floquet</p><p>modes</p><p>after</p><p>the</p><p>solution</p><p>has</p><p>been</p><p>established.</p><p>The</p><p>patch-current</p><p>sources</p><p>lying</p><p>a</p><p>distance</p><p>away</p><p>from</p><p>the</p><p>boundary</p><p>surfaces</p><p>produce</p><p>a set</p><p>of</p><p>smooth</p><p>field</p><p>functions</p><p>on</p><p>the</p><p>surfaces</p><p>that</p><p>may</p><p>be well</p><p>suited</p><p>for</p><p>spanning</p><p>the</p><p>actual</p><p>smooth</p><p>field</p><p>on</p><p>the</p><p>boundaries.</p><p>Furthermore,</p><p>since</p><p>we</p><p>are</p><p>actually</p><p>using</p><p>a</p><p>basis</p><p>of smooth</p><p>field</p><p>functions</p><p>for</p><p>repre-</p><p>senting</p><p>fields</p><p>on</p><p>the</p><p>boundary,</p><p>the</p><p>boundary</p><p>condition</p><p>can</p><p>be</p><p>enforced</p><p>by</p><p>a</p><p>simple</p><p>point-matching</p><p>testing</p><p>procedure</p><p>and the unknown source amplitudes are readily determined.</p><p>The</p><p>paper</p><p>is organized</p><p>in</p><p>the</p><p>following</p><p>manner.</p><p>The</p><p>problem</p><p>under</p><p>study</p><p>is</p><p>specified</p><p>in Section</p><p>2.</p><p>The</p><p>solution</p><p>is</p><p>formulated</p><p>in</p><p>Section</p><p>3.</p><p>Results</p><p>of several</p><p>numerical</p><p>simulations</p><p>are presented</p><p>in</p><p>Section</p><p>4 and</p><p>compared</p><p>with</p><p>an</p><p>analytic</p><p>approximation</p><p>in order</p><p>to</p><p>demonstrate</p><p>the</p><p>efficien-</p><p>cy</p><p>and</p><p>accuracy</p><p>of</p><p>the</p><p>proposed</p><p>technique.</p><p>Finally,</p><p>a few</p><p>conclusions</p><p>summarize</p><p>the</p><p>paper.</p><p>2.</p><p>PROBLEM</p><p>SPECIFICATION</p><p>Consider</p><p>a</p><p>doubly</p><p>periodic</p><p>array</p><p>of scatterers.</p><p>The</p><p>array</p><p>is</p><p>composed</p><p>of</p><p>an</p><p>infinite</p><p>set</p><p>of</p><p>identical</p><p>perfectly</p><p>conducting</p><p>scatterers arranged in a doubly periodic lattice. The lattice</p><p>is</p><p>described</p><p>by</p><p>two</p><p>vectors</p><p>d,</p><p>and</p><p>d</p><p>2</p><p>lying</p><p>in the</p><p>xy</p><p>plane.</p><p>The</p><p>vectors</p><p>d, and</p><p>d</p><p>2</p><p>are</p><p>referred</p><p>toas</p><p>lattice</p><p>vectors.</p><p>They</p><p>are</p><p>aligned</p><p>with</p><p>the</p><p>two</p><p>directions</p><p>of</p><p>periodicity,</p><p>and</p><p>their</p><p>with</p><p>harmonic</p><p>exp(jwt)</p><p>time</p><p>dependence</p><p>assumed</p><p>and</p><p>sup-</p><p>pressed,</p><p>is incident</p><p>on</p><p>the</p><p>grating.</p><p>Here,</p><p>kinc</p><p>and</p><p>E</p><p>0</p><p>c de-</p><p>note,</p><p>respectively,</p><p>the</p><p>wave</p><p>vector</p><p>and</p><p>the</p><p>amplitude</p><p>of</p><p>the</p><p>incident</p><p>field.</p><p>Our</p><p>objective</p><p>is</p><p>to</p><p>determine</p><p>the</p><p>field</p><p>scat-</p><p>tered</p><p>by</p><p>the</p><p>grating</p><p>(Es,</p><p>Hs)</p><p>(i.e.,</p><p>the</p><p>actual</p><p>field</p><p>minus</p><p>the</p><p>incident</p><p>field).</p><p>The</p><p>field</p><p>should</p><p>be</p><p>a source-free</p><p>solution</p><p>of</p><p>the</p><p>Maxwell</p><p>equations</p><p>and</p><p>obey</p><p>the</p><p>Floquet</p><p>periodicity</p><p>conditions</p><p>Es(r</p><p>+ dp) = exp(-jkinc -</p><p>dp)Es(r),</p><p>p = 1, 2.</p><p>(2)</p><p>In</p><p>addition,</p><p>(Es,</p><p>Hs) should</p><p>satisfy</p><p>the</p><p>boundary</p><p>condition</p><p>n</p><p>X ES</p><p>=</p><p>-h</p><p>X</p><p>Einc</p><p>(3)</p><p>where</p><p>S is</p><p>the boundary</p><p>of</p><p>an arbitrary</p><p>selected</p><p>scatterer</p><p>and</p><p>h is a</p><p>unit</p><p>vector</p><p>that</p><p>is</p><p>normal</p><p>to</p><p>S.</p><p>3.</p><p>FORMULATION</p><p>A.</p><p>Simulated</p><p>Equivalent</p><p>Situation</p><p>We</p><p>now</p><p>describe</p><p>how</p><p>the</p><p>simulated</p><p>equivalent</p><p>situation</p><p>to</p><p>the original one in the region surrounding the scatterers</p><p>is</p><p>set up.</p><p>According</p><p>to</p><p>our</p><p>general</p><p>idea,</p><p>in the</p><p>simulated</p><p>equivalence</p><p>that</p><p>is shown</p><p>in</p><p>Fig.</p><p>2, the</p><p>scattered</p><p>field</p><p>(Es,</p><p>Hs) is</p><p>simulated</p><p>by</p><p>a</p><p>field</p><p>of a</p><p>set</p><p>of</p><p>doubly</p><p>periodic</p><p>ficti-</p><p>tious</p><p>patches</p><p>of</p><p>magnetic</p><p>current</p><p>Mqi,</p><p>q =</p><p>1, 2, i</p><p>=</p><p>1,</p><p>2, ... ,</p><p>N.</p><p>These</p><p>sources</p><p>are</p><p>located</p><p>in</p><p>the</p><p>region</p><p>occupied</p><p>by the</p><p>scatterers</p><p>in</p><p>the original</p><p>situation</p><p>and</p><p>are treated</p><p>as</p><p>sources</p><p>Unbounded</p><p>Homogeneous</p><p>Space</p><p>(Einc</p><p>,Hinc)</p><p>kinc</p><p>(gia)</p><p>(Es+Einc</p><p>Hs</p><p>+Hinc)</p><p>Periodic</p><p>Patch</p><p>Currents</p><p>/</p><p>Mathematical</p><p>Boundary</p><p>C</p><p>Fig.</p><p>2.</p><p>Simulated</p><p>equivalence</p><p>for</p><p>the</p><p>region</p><p>surrounding</p><p>the</p><p>scat-</p><p>terers.</p><p>(EH )</p><p>\k</p><p>kin'</p><p>Boag</p><p>et</p><p>al.</p></li><li><p>8/10/2019 Doubly Periodic - BLB</p><p> 3/7</p><p>1714</p><p>J.</p><p>Opt.</p><p>Soc.</p><p>Am.</p><p>A/Vol.</p><p>7,</p><p>No.</p><p>9/September</p><p>1990</p><p>radiating</p><p>in</p><p>an</p><p>unbounded</p><p>space</p><p>filled</p><p>with</p><p>homogeneous</p><p>material</p><p>that is</p><p>identical</p><p>to that</p><p>surrounding</p><p>the</p><p>scatterers</p><p>in</p><p>the</p><p>original</p><p>situation.</p><p>They</p><p>have</