Inverse Diffraction Theory and Computation of Minimum Source Regions of Far Fields

<table>Minimum source region <svg height="15.25" id="M16" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 32.799999 15.25" width="32.799999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,12.675,15.187)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
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</svg> and the minimum ball <svg height="16.275" id="M17" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 55.5 16.275" width="55.5" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.137)"><path d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
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<g transform="matrix(.012,-0,0,-.012,10.55,16.212)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,20.295,16.212)"><path d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" id="x69"></path></g><g transform="matrix(.012,-0,0,-.012,23.484,16.212)"><path d="M524 0h-209v26q44 5 55 18t11 65v167q0 114 -91 114q-50 0 -109 -51v-235q0 -50 10 -61.5t55 -16.5v-26h-217v26q51 5 62 17t11 61v206q0 47 -9.5 58.5t-50.5 19.5v23q82 15 139 40v-79l67 47q47 30 83 30q60 0 94.5 -40t34.5 -112v-193q0 -50 10 -61.5t54 -16.5v-26z
" id="x6E"></path></g>
<g transform="matrix(.017,-0,0,-.017,30.613,12.137)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,36.494,12.137)"><path d="M745 361q0 -166 -116 -272t-291 -106q-136 0 -225.5 83t-89.5 218q0 161 115.5 272t288.5 111q137 0 227.5 -83t90.5 -223zM643 359q0 126 -56 199.5t-167 73.5q-130 0 -213.5 -104t-83.5 -246q0 -117 58 -190.5t165 -73.5q91 0 160.5 52t103 128.5t33.5 160.5z" id="x1D442"></path></g><g transform="matrix(.017,-0,0,-.017,49.549,12.137)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> for a given origin <svg height="11.75" id="M18" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 13.175 11.75" width="13.175" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M745 361q0 -166 -116 -272t-291 -106q-136 0 -225.5 83t-89.5 218q0 161 115.5 272t288.5 111q137 0 227.5 -83t90.5 -223zM643 359q0 126 -56 199.5t-167 73.5q-130 0 -213.5 -104t-83.5 -246q0 -117 58 -190.5t165 -73.5q91 0 160.5 52t103 128.5t33.5 160.5z" id="x1D442"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Inverse Diffraction Theory and Computation of Minimum Source Regions of Far Fields 