sec_psf_formulation_C.tex

A second possible approach lies in measuring the structure functions
$D_{\rm \phi_{inst,sci}}(\vec{s}, \vec{r} +\vec{s} )$ 
and
$D_{\rm \phi_{ao}}(\vec{s}, \vec{r} + \vec{s})$
and using the measured values directly in Equation
\ref{eqn:lgsotf}.  The static, field-dependent instrumental wavefront
error, $\phi_{\rm inst,sci}(\vec{r})$, may be measured using a technique 
such as phase diversity \cite{Fitzgerald:2014,Sitarski:2014}.  The
structure function may then be formed by brute force numerical
calculation via
\begin{equation}
\label{eqn:strfn_inst}
D_{\rm \phi_{inst,sci}}(\vec{r}_{1}, \vec{r}_{2})  = 
\left\langle \left\{\phi_{\rm inst,sci}\left(\vec{r}_{1}\right) - 
\phi_{\rm inst,sci}\left(\vec{r}_{2}\right)\right\}^{2}\right\rangle
\end{equation}
Similarly, the statistical properties of $\phi_{\rm
  ao}\left(\vec{r}\right)$ may be characterized via modelling
of the adaptive optics system.  Such a characterization would involve
understanding the contribution to the structure function $D_{\rm
  \phi_{ao}}(\vec{r}_{1},\vec{r}_{2})$ from measurement errors,
servo and fitting errors for specific hardware and under specific
observing conditions.  For % example, the statistical properties of 
% high-order fitting error are determined by the turbulence profile and
% actuator pitch on the deformable mirror
% \begin{notes}
% [CITE].
% \end{notes}
% Similarly, statistical
% properties of high-order measurement error are dictated by the
% geometry of the wavefront sensor, noise and diffusion properties of
% the wavefront sensor detector, and by the guide star brightness.  Finally,
% the statistical properties of high-order servo errors are dictated by the
% turbulence and wind profiles and by the latency in the real time
% controller.  If the four dimensional structure function $D_{\rm
%   \phi_{ao}}(\vec{r}_{1},\vec{r}_{2})$ may be modelled
% successfully, then the OTF may be computed
% directly from Equation \ref{eqn:lgsotf}.  
Modelling of $D_{\rm \phi_{ao}}(\vec{r}_{1},\vec{r}_{2})$ at Keck is beyond 
the scope of this project and is being pursued in a parallel effort
for Keck \cite{Jolissaint:2014,Ragland:2016}
The results of this modelling may demonstrate that 
$D_{\rm \phi_{ao}}(\vec{r}_{1},\vec{r}_{2})$ is nearly
stationary, which might be expected given that fitting,
servo, and measurement errors tend to decorrelate at separations of a
single subaperture, and do so uniformly over the pupil plane.  Were this
assumption to hold, one could write Equation \ref{eqn:lgsotf} as
\begin{equation}\label{eqn:lgsotfb}
\begin{aligned}
{\rm OTF}^{\rm LGS}_{\rm sci}(\vec{r}) = 
& \exp{
\left\{ -\frac{1}{2} \bar{D}_{\rm \phi_{\rm ao}}(\vec{r}) \right\}}  \\
& \int 
W \left( \frac{\vec{s}}{R} \right)
W \left( \frac{\vec{r} + \vec{s}}{R} \right)
\exp{ \left\{ -\frac{1}{2} \left[
D_{\rm \phi_{apl}}(\vec{s}, \vec{r} + \vec{s}) +
D_{\rm \phi_{inst,sci}}(\vec{s}, \vec{r} + \vec{s})
\right] \right\} }
\; d\vec{s} 
\end{aligned}
\end{equation}
Naturally this stationarity assumption must be validated against
performance of real hardware, and its validity is dictated by the
modelling accuracy required for the astronomical application.  

Finally, the approach adopted by AIROPA is a model that assumes the instrumental
structure function is also stationary, so that
\begin{equation}\label{eqn:lgsotfc}
\begin{aligned}
{\rm OTF}^{\rm LGS}_{\rm sci}(\vec{r}) = 
& \exp{ \left\{ -\frac{1}{2}
\left[ \bar{D}_{\rm \phi_{\rm ao}}(\vec{r}) \right] \right\} } 
\exp{ \left\{ -\frac{1}{2}
\left[ \bar{D}_{\rm \phi_{\rm inst,sci}}(\vec{r}) \right] \right\} }  \\
& \int 
W \left( \frac{\vec{s}}{R} \right)
W \left( \frac{\vec{r} + \vec{s}}{R} \right)
\exp{ \left\{ -\frac{1}{2} \left[
D_{\rm \phi_{apl}}(\vec{s},\vec{r} + \vec{s})
\right] \right\} }
\; d\vec{s} 
\end{aligned}
\end{equation}
Example PSF and OTFs are shown in Figure \ref{fig:lgs_atm_inst} for
the case of an on-axis LGS and TTS with both atmospheric
anisoplanatism and stationary instrumental 
aberrations. This approximation would be suitable when 
$\phi_{\rm inst,sci} \left( \vec{r} \right) \ll 
\phi_{\rm apl} \left( \vec{r} \right)$.