A Coupled-Field Expansion Method for Single-Layer and Multilayer Planar Periodic Structures

<table>(a) A unit cell. (b) Separation of a Type B unit cell into two regions. Region <svg height="15.4" id="M113" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 17.4 15.4" width="17.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
<g transform="matrix(.012,0,0,-.012,11.013,15.188)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg> is open at the top and has a PEC at the bottom. Equivalent magnetic currents (shown in light color at <svg height="12.5625" id="M114" style="vertical-align:-0.273pt" version="1.1" viewbox="0 0 36.787498 12.5625" width="36.787498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.163)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
t-47.5 -67l-25 13q15 39 38 78q40 66 83 66q57 0 120 -12q52 -10 80 -10q29 0 67 42z" id="x1D467"></path></g><g transform="matrix(.017,0,0,-.017,13.237,12.163)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,0,0,-.017,27.94,12.163)"><path d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" id="x210E"></path></g>
</svg>) exist just above the bottom PEC. Region <svg height="15.25" id="M115" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 17.4 15.25" width="17.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
<g transform="matrix(.012,0,0,-.012,11.013,15.188)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> has a PEC surface at the top and equivalent magnetic currents (shown in light color at <svg height="12.5625" id="M116" style="vertical-align:-0.273pt" version="1.1" viewbox="0 0 36.787498 12.5625" width="36.787498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.163)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
t-47.5 -67l-25 13q15 39 38 78q40 66 83 66q57 0 120 -12q52 -10 80 -10q29 0 67 42z" id="x1D467"></path></g><g transform="matrix(.017,0,0,-.017,13.237,12.163)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,0,0,-.017,27.94,12.163)"><path d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" id="x210E"></path></g>
</svg>) exist just below the top PEC. For Type A structures, the bottom of Region <svg height="15.25" id="M117" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 17.4 15.25" width="17.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
<g transform="matrix(.012,0,0,-.012,11.013,15.188)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> is open (not shown here). For Type B structures, it is a PEC (as shown).</table>

International Journal of Antennas and Propagation

fig4

Figure 4

Figure 4: A Coupled-Field Expansion Method for Single-Layer and Multilayer Planar Periodic Structures 