Negative Propagation Effects in Two-Dimensional  Silicon Photonic Crystals

<table>The EFC of frequency <svg height="15.0875" id="M32" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 119.375 15.0875" width="119.375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D714"></use></g><g transform="matrix(.017,-0,0,-.017,15.616,12.138)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,30.32,12.138)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,38.48,12.138)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,42.355,12.138)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g><g transform="matrix(.017,-0,0,-.017,50.515,12.138)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,58.674,12.138)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g><g transform="matrix(.017,-0,0,-.017,66.834,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,72.715,12.138)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,80.875,12.138)"><use xlink:href="#x1D70B"></use></g><g transform="matrix(.017,-0,0,-.017,90.921,12.138)"><use xlink:href="#x1D450"></use></g><g transform="matrix(.017,-0,0,-.017,97.822,12.138)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,104.826,12.138)"><use xlink:href="#x1D44E"></use></g><g transform="matrix(.017,-0,0,-.017,113.427,12.138)"><use xlink:href="#x29"></use></g>
</svg>. The air-PhC interface is normal to the Γ-M direction. <svg height="16.450001" id="M33" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 12.4625 16.450001" width="12.4625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123
t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" id="x1D703"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.525,16.25)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
</svg> is the incident angle of the plane wave. A Bloch wave denoted by A is excited in the PhC. <svg height="14.925" id="M34" style="vertical-align:-5.76984pt" version="1.1" viewbox="0 0 15.675 14.925" width="15.675" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M457 332q0 -81 -41 -161.5t-105.5 -131.5t-129.5 -51q-46 0 -79 30.5t-33 86.5q0 18 7 51q18 95 46 187q6 19 6 33q0 7 -7 7q-24 0 -78 -64l-20 23q32 48 73 77t78 29q33 0 33 -46q0 -38 -10 -70q-28 -92 -39 -152q-7 -38 -7 -57q0 -78 69 -78q68 0 118 75.5t50 194.5
q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" id="x1D463"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><path d="M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30
t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z" id="x1D454"></path></g>
</svg> is the group velocity of the Bloch wave, which has a refraction angle of <svg height="16.450001" id="M35" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.0375 16.450001" width="14.0375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D703"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.525,16.25)"><use xlink:href="#x1D450"></use></g>
</svg> against the normal of the interface.</table>

International Journal of Photoenergy

fig7

Figure 7

Figure 7: Negative Propagation Effects in Two-Dimensional  Silicon Photonic Crystals 