The data analyzed here are from Band 9 Scan 103, Beams 1 and 2,
in the linear combination (Beam1 - i Beam2)/2.  The following three
plots show the near - field amplitude in 3-D perspective, then in the
form of a color density plot and as a contour plot.    For the latter
the chosen contour levels are at (0, -3, -6, -9,  ...) dB, with the peak
amplitude having been normalized to unity.

Graphics:Near-Field Amplitude, 3-D Perspective


Graphics:Near-Field Amplitude (dB)


Graphics:Contour Plot of Near-Field Amplitude (dB)

Now a plot of the near - field phase:

Graphics:Near-Field Phase

       The following is a plot of the near - field phase after having
been unwrapped using an algorithm due to T. J. Flynn.    See
Cennis C. Ghiglia and Mark D. Pritt,  Two-Dimensional Phase
Unwrapping:  Theory, Algorithms, and Software,
Wiley, New York,
1998.   The C code of Appendix A9 was used here.    The original
publication the algorithm is (T. J. Flynn,  JOSA A, 14 (1997) 2692 - 2701).
     I call up the C code from within Mathematica.    The run time
is ordinarily on the order of a few seconds.  

Graphics:Unwrapped Near-Field Phase

     Here is a 3-D perspective plot of the unwrapped phase.   The
only apparent problems in the unwrapping are near the edges :

Graphics:Unwrapped Near-Field Phase, 3-D Perspective


       The following table  gives the result of a linear least - squares
fit to the unwrapped phase.    Here I fit for a constant term, the
x- slope, y- slope, and coefficients of "effcalc.annotated_11.gif", terms.     
Internally, the (x,y) units are meters, so here, e.g, the fitted x- slope
is 134.805 radians/meter with a standard error of 0.396 rad/m.      



       The NSI Band 9 Scan 103 measurements were taken over a 108×108
point grid.    The (x,y)=(0,0) coordinate origin is located midway
between the central grid points.  To achieve an effective 53.5 grid
point shift of origin in each coordinate I do a pre - FFT cyclic
leftward rotation by 53 grid units;  then, post - FFT, I apply a phase
shift  Exp(-i π (u Δx+v Δy)), to achieve the additional one - half
pixel shifts of origin.  I could equally well have omitted the pre - FFT
cyclic shift and multiplied post - FFT by a factor  Exp(-2 i π 53.5 (u Δx+v Δy)),
to thereby achieve the full shift of origin in just a single step.  
     I chose to zero pad to 512×512, same as NSI.   However, I can
zero - pad to any desired size, even or odd parity, since Mathematica
has fast built - in algorithms for the discrete Fourier transform
for arbitrary n (even including prime n).
      Here are plots of the computed far - field amplitude :

Graphics:Far-Field Amplitude, 3-D Perspective


Graphics:Far-Field Amplitude (dB)

      I use "effcalc.annotated_17.gif" to denote the computed first moments of the
far - field amplitude distribution.  This location is indicated
by the dot in the above figure and the one below.

Graphics:Far-Field Amplitude (dB)

     Here is a plot of the far - field phase, showing many phase
wraps and considerable noise.

Graphics:Far-Field Phase

               Below is an attempt at unwrapping the computed far - field
phase.   The result looks OK over the central region of the beam,
where indeed several wraps were apparent, but once into the
sidelobes there are evident 2π n mistakes in the unwrapped result.

Graphics:Unwrapped Far-Field Phase

          In this case, a 3-D perspective plot shows a better
picture of the result of the unwrapping algorithm.   Again, the
ordinate is radians of phase.


          The following table shows the the result of a linear
least - squares fit to the unwrapped far - field phase.   Here
I have fit for a constant term and the coefficients of  the "effcalc.annotated_22.gif"
"effcalc.annotated_23.gif" and  "effcalc.annotated_24.gif"  terms.    The fit is to all the data
within a radius of 3.58° about "effcalc.annotated_25.gif", "effcalc.annotated_26.gif"), and here I have weighted
by amplitude.    



        From the fit I derive  (x,y,z) offsets, in the frame of the
nominal beam - and a corresponding phase efficiency - which
are in reasonable accord with the results from Richard' s
spreadsheet.    Fitting by weighted linear - least squares to
the unwrapped phase, and weighting the data by amplitude,
yields a phase efficiency essentially identical to that given by
Richard's efficiency maximization procedure.







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