ANGULAR ANISOPLANATISM

When the propagation beam is offset by a constant angle from the direction along which turbulence is measured, the effect is called angular anisoplanatism.[#!4!#] It arises naturally when one is tracking a satellite target and directing a laser beam at it. Because of the finite speed of light, the laser beam has to lead the tracking direction, resulting in an angular difference between the direction along which the target is tracked and the one along which the laser beam is directed. This error can be eliminated if the target has a reflector for the beacon that extends a suitable distance in the point-ahead direction. For the case of an angular error
d (z) = θ z, (30)
d 2 = 2.91 k02 μ2 θ2, (31)
E = 6.88 $\displaystyle {{\mu _2} \over {\mu _0}}$$\displaystyle \left(\vphantom{ {{\theta \over D}} }\right.$$\displaystyle {{\theta \over D}}$$\displaystyle \left.\vphantom{ {{\theta \over D}} }\right)^{2}_{}$$\displaystyle \left(\vphantom{ {{D \over {r_o}}} }\right.$$\displaystyle {{D \over {r_o}}}$$\displaystyle \left.\vphantom{ {{D \over {r_o}}} }\right)^{{5/3}}_{}$, (32)
σ$\scriptstyle \varphi$2 = 2.91 k02 θ5/3$\displaystyle \int\limits_{{\,\,\, 0}}^{{\,\,\,\,\,\, L}}$dz Cn2(z)z5/3 = $\displaystyle \left(\vphantom{ {\theta / \theta _o} }\right.$θ/θo$\displaystyle \left.\vphantom{ {\theta / \theta _o} }\right)^{{5/ 3}}_{}$, (33)

where the isoplanatic angle is defined by
θo = $\displaystyle \left(\vphantom{ {2.91\,k_0^2\,\mu _{5/ 3}} }\right.$2.91 k02 μ5/3$\displaystyle \left.\vphantom{ {2.91\,k_0^2\,\mu _{5/ 3}} }\right)^{{-3/ 5}}_{}$.     (34)

...