# ALMA polarization: XY-phase solver avoids +-45 deg solution

When using the task gaincal with parameter gaintype='XYf+QU', the solution to solve is a fit for a slope in 2D data (imag vs. real) from data that has noise is both dimensions.  In such cases, it is always better to fit a shallow (not steep) slope, so if the slope (for imag vs. real) comes out >1.0, it flips the axes (to real vs. imag) and re-fits it (and inverts the resulting value to account for the swap).  This minimizes the effect of the real axis noise on the slope calculation. This yields far more accurate solutions when the nominal slope is very large (>>1.0, e.g., the data nearly parallel to the imag axis == cross-hand phase approaching +/-90 deg).

The case of slope = 1.0 (which is cross-hand phase of 45 deg) corresponds to the slope at which to pivot the axis swap decision. When plotting the cross-hand phase solutions, a gap appears at +-45 deg (see figure). This gap is a property of the typical spacings between values in the statistical distribution of slope values.  I.e., for a sample of N points filling a distribution with a finite width, there is a characteristic minimum spacing between values that is some small fraction of the width of the distribution.  Of course, smaller spacings are not forbidden, but they are rare.  The axis swap reveals this property since all of the (nominal) slopes that are >1.0 (cross-hand phases >45.0 deg) are fit with swapped data and yield the inverse slope (<1.0), and than inverted to be >1.0 again.  The typical slopes mostly do not get arbitrarily close to exactly 1.0, so the gap appears.  This is essentially an extension of the fact that (for any slope), the precise exact value need not be realized in any instance in a sample of solutions for it.  E.g., in a sample of N gaussian-distributed values centered on some specific value the likelihood of any sample having that precise central value is vanishingly small.

The gap should become smaller when either the noise decreases, or the number of channels (for the same noise) increases.

This feature should be harmless for the calibration of ALMA full-polarization data.