Tight Focusing of Partially Coherent Vortex Beams

<table>Degree of coherence along <i>π</i>/4 in focal plane  of aplanatic lens with  <svg height="11.0375" id="M95" style="vertical-align:-0.27588pt" version="1.1" viewbox="0 0 61.337502 11.0375" width="61.337502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">N</tspan>
<tspan style="font-size: 12.50px; " x="8.8021116" y="0">A</tspan>
<tspan style="font-size: 12.50px; " x="21.292608" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="33.332996" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="39.584499" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="42.710247" y="0">9</tspan>
</text>
</g>
</g>
</svg> for uniformly polarized vortex beams with <svg height="14.9" id="M96" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 102.925 14.9" width="102.925" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.9)">
<g transform="translate(72,-60.08)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
<tspan style="font-size: 12.50px; " x="16.516462" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="28.54435" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="38.971851" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-25.36,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑜</tspan>
</text>
<text transform="matrix(1,0,0,-1,-17.38,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">5</tspan>
</text>
</g>
</g>
</svg>, <svg height="13.55" id="M97" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 49.900002 13.55" width="49.900002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.55)">
<g transform="translate(72,-61.16)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
<tspan style="font-size: 12.50px; " x="10.440005" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.480394" y="0">𝜋</tspan>
<tspan style="font-size: 12.50px; " x="30.082218" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="33.558052" y="0">2</tspan>
</text>
</g>
</g>
</svg>; (a) <svg height="12.725" id="M98" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 26.9125 12.725" width="26.9125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.725)">
<g transform="translate(72,-61.82)">
<text transform="matrix(1,0,0,-1,-71.95,66.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,63.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan>
<tspan style="font-size: 8.75px; " x="4.8136001" y="0">𝑦</tspan>
</text>
</g>
</g>
</svg>(b) <svg height="10.875" id="M99" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 21.737499 10.875" width="21.737499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.875)">
<g transform="translate(72,-63.3)">
<text transform="matrix(1,0,0,-1,-71.95,66.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,63.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan>
<tspan style="font-size: 8.75px; " x="4.8136001" y="0">𝑧</tspan>
</text>
</g>
</g>
</svg>; for <svg height="14.7125" id="M100" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 59.412498 14.7125" width="59.412498" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑜</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.36,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.291889" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.417639" y="0">8</tspan>
</text>
</g>
</g>
</svg>(c) <svg height="12.725" id="M101" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 26.9125 12.725" width="26.9125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.725)">
<g transform="translate(72,-61.82)">
<text transform="matrix(1,0,0,-1,-71.95,66.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,63.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan>
<tspan style="font-size: 8.75px; " x="4.8136001" y="0">𝑦</tspan>
</text>
</g>
</g>
</svg>(d) <svg height="10.875" id="M102" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 21.737499 10.875" width="21.737499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.875)">
<g transform="translate(72,-63.3)">
<text transform="matrix(1,0,0,-1,-71.95,66.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,63.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan>
<tspan style="font-size: 8.75px; " x="4.8136001" y="0">𝑧</tspan>
</text>
</g>
</g>
</svg>.</table>

International Journal of Optics

Tight Focusing of Partially Coherent Vortex Beams

Figure 5