پرتوهای گرداب نوری - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle E_{l.p}(r,\phi ,z)=C_{pl}^{LG}{\frac {w_{0}}{w(z)}}\left({\frac {{\sqrt {2}}r}{w(z)}}\right)^{\!p}\exp \!\left(\!-{\frac {r^{2}}{w^{2}(z)}}\right)L_{l}^{p}\!\left({\frac {2r^{2}}{w^{2}(z)}}\right)\exp \!\left(\!-ik{\frac {r^{2}}{2R(z)}}\right)\exp(-ip\phi )\exp(i\psi _{l,p}(z))}$

${\displaystyle C_{pl}^{LG}={\sqrt {\frac {2l!}{\pi (p+l)!}}}\Rightarrow \int _{0}^{2\pi }d\phi \int _{0}^{\infty }rdr|u(r,\phi ,z)|^{2}=1}$

${\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}}\ }$

${\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}n}{\lambda }}}$

${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle {\begin{aligned}L_{l}^{p}(x)&={x^{-p}e^{x} \over l!}{d^{l} \over dx^{l}}\left(e^{-x}x^{l+p}\right)\\[4pt]\end{aligned}}}$

${\displaystyle k\;=\;{\frac {2\pi }{\lambda }}}$

${\displaystyle {\frac {1}{R(z)}}={\frac {z}{z^{2}+z_{\mathrm {R} }^{2}}}\ }$

${\displaystyle \psi _{l,p}(z)=(p+2l+1)\,\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right)}$