Vector Representation of Subaperture, Actuator and Wavefront Meshes

In the AO tools, the coordinates of subaperture centers, actuators, and points where wavefronts are evaluated or reconstructed are represented by 1-D vectors, where the 2-D (x,y) layout of the mesh points is organized for algorithm convenience into a 1-D vector.  This is a convenient format for matrix representation of the algebra, but the main motivation for the 1-D representation is generality:  subaperture (and mirror actuator) locations can have a rather arbitrary layout, with complications such as annular holes, irregular borders, rectangular or hexagonal meshes, and so forth.  Not all of these features are presently available in the AO tools, but to include them in the future (or now by user customization), the 1-D representation is the best approach.

Thus, we have the following vectors:

xsub  = a column vector of dim (nsub x1), containing the x-coordinates of the centers of the WFS subapertures.  The subaperture indexing (ordering) convention will be described later.

ysub = a column vector of dim (nsub x1), containing the y-coordinates of the centers of the WFS subapertures.

xact  = a column vector of dim (nact x1), containing the x-coordinates of the actuator locations. The actuator indexing (ordering) convention will be described later.

yact  = a column vector of dim (nact x1), containing the y-coordinates of the actuator locations.

These four data vectors (and the length scalars nsub and nact) appear as variables in the .mat file created by aogeom.  As noted previously, all distances in AO tools (including OPDs), and in LightLike generally, are expressed in meters.

Similarly, in our AO background discussion, we will use the following vectors:

s  = column vector of subaperture slopes, dim (2nsub x 1)

   = [sx(r1) ,..., sx(rnsub),  sy(r1), ..., sy(rnsub)]T ,  in units of radians of angle.

Note that:  

(a)  the ordering 1,...,nsub is the same as the xsub and ysub vectors;

(b)  the x and y slope components have been concatenated into one vector.

dact  = vector of actuator displacements, in m;  dim (nact x 1).

The ordering is the same as the xact and yact vectors.

dopd = vector of wavefront surface displacements, in m; dim (nopd x 1).

The displacements are defined with respect to a flat, 0-tilt reference wavefront.

nopd may be identical to nact (see, e.g., routine lsfptos), but may also represent a denser mesh.

The notation dopd to characterize the wavefront implies that a continuous surface of constant phase exists, so that a DM with a continuous facesheet can be shaped to compensate.  The extent to which this is valid depends on the integrated strength of turbulence, but the reconstruction procedures included in the AO tools are based on this concept.  

Note that in the LightLike propagation code, the 1-D representation of DM displacements is eventually converted to a 2-D array representing the corresponding phasor  exp[i*(2π/λ)*d(xi,yj)].  The correction applied to an incident wavefront is then based on manipulation of the 2-D complex-field arrays.