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.

Now a plot of the 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.

Here is a 3-D perspective plot of the unwrapped phase. The

only apparent problems in the unwrapping are near the edges :

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 , 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 :

I use 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.

Here is a plot of the far - field phase, showing many phase

wraps and considerable noise.

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.

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

and terms. The fit is to all the data

within a radius of 3.58° about , ), 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.