We adopt cylindrical coordinates and choose our observation point to lie on the $x$-axis. Since the absorbing layers are thin, we make the approximation that the heat source term does not depend on $z$, so the temperature rise becomes Display Formula
$\Delta T(r,\theta =0,t)=\u222b0tds\u222b02\pi d\theta \u2032\u222b0\u221er\u2032dr\u2032G(r,0,r\u2032,\theta \u2032,t,s)\epsilon (r\u2032,\theta \u2032,s)$
Display Formula$G=[4\pi \kappa (t\u2212s)]\u22123/2Exp[(r2+r\u20322\u22122\u2009rr\u2032cos\u2009\theta \u2032)/4\kappa (t\u2212s)]$
Display Formula$\epsilon rot(r\u2032,\theta \u2032,s)=\alpha P\pi w2\rho CpExp[\u2212r02+r\u20322\u22122r0r\u2032cos(\theta \u2032\u2212\omega s)w2]$
Display Formula$\epsilon avg(r\u2032,s)=\alpha P\pi w2\rho Cp12\pi \u222b02\pi Exp[\u2212r02+r\u20322\u22122r0r\u2032cos(\phi )w2]d\phi =\alpha P\pi w2\rho CpExp[\u2212r02+r\u20322w2]BesselI0(2r0r\u2032w2),$
where $w$ is the $1/e$ beam radius, $r0$ is the circle radius, $\omega $ is the angular frequency of the rotating beam, and $\alpha $ is the absorption coefficient.