SIGN CONVENTIONS USED FOR SHARP

April 13-17, 2006

***May 18, 2007: the signs of Q_sky and U_sky are inverted 
***...but the PHI_SKY is correct (see notes below)

(1) sharp_integ defines the Stokes' parameters as follows:

First we define the half-wave plate angles:

   theta_encoder is the hwp value read by the encoder

   theta_nom is the nominal hwp angle, which equals zero for the first 
   of the four half-wave plate positions

   For almost all data taken in Jan 2006, we had:

   theta_encoder = 35 + theta_nom

Then Stokes' parameters are defined as follows:

   POL_FLUX(theta) = H(theta) - V(theta)
   T_FLUX(theta) = H(theta) + V(theta)

   Q_raw = [POL_FLUX(0.0) - POL_FLUX(45.0)] * 0.5
   U_raw = [POL_FLUX(67.5) - POL_FLUX(22.5)] * 0.5
   I = [T_FLUX(0.0) + T_FLUX(22.5) + T_FLUX(45.0) + T_FLUX(67.5)] * 0.25


(2) Here's how to relate these raw Stokes' parameters to the angle of polarization relative 
the instrument:

   Q_raw = P cos (2 * Phi_Inst_raw) 
   U_raw = P sin (2 * Phi_Inst_raw)

   Phi_Inst_cor = -1.0 * Phi_Inst_raw + Instrument_Offset

   where Instrument_Offset = +60  (this was determined recently from grid tests)

   Phi_Inst_cor (the "cor" stands for "corrected") is the angle of the 
   E-vector at SHARP's input.  The angle is zero when the E-vector is 
   vertical, and increases as the E-vector is rotated counter-clockwise as 
   viewed by someone looking down the Nasmyth tube from the Nasmyth platform.

   These relationships can be visualized in Figure 1, which is a plot in "Q-U (raw) space".

   (Note that the above directional relationship between Phi_Inst_cor and the raw Stokes' parameters is 
   consistent with the direction of positive half-wave plate rotation that was recorded in the instrument 
   logbook.  This consistency check is described in HWP rotation sign check.)


(3) Here's how to relate all of this to the polarization angle on the sky:

Formulas relating angles:

   PHI_SKY = - Phi_Inst_cor + (PAR + EL)

   PHI_SKY is angle of E-vector on sky, increasing E from N
   EL is the Elevation
   PAR is the Parallactic Angle (see Definition of Parallactic angle)

Formulas used by John to relate raw Stokes' parameters to polarization angle on sky:

   sharpcombine uses the following formulas:

   Q_sky = Q_raw * cos(SH_COMB_ANG) - U_raw * sin(SH_COMB_ANG)
   U_sky = U_raw * cos(SH_COMB_ANG) + Q_raw * sin(SH_COMB_ANG)

   where SH_COMB_ANG = 2 * (PAR + EL + HWP)

***NOTE ADDED MAY 07: these definitions of Q_sky and U_sky are actually
***incorrect.  see John's memo of May 2007

   (SH_COMB_ANG stands for "sharp-combine angle")

   and HWP is a flag that takes into account the arbitrary HWP offset

   These relationships can be visualized in Figure 2

***NOTE ADDED MAY 07: the definitions of Q_sky and U_sky in Fig. 2 are actually
***incorrect.  see John's memo of May 2007

   Then John computes PHI_SKY as follows:

   Q_sky = P cos (2 * [PHI_SKY - 90]) 
   U_sky = P sin (2 * [PHI_SKY - 90])

***NOTE ADDED MAY 07: these formulas are also incorrect, but
***the PHI_SKY one obtains is actually OK.  see John's memo of 
***May 2007.

Consistency check:

   From Figure 3 we can see that:

***NOTE ADDED MAY 07: the definitions of Q_sky and U_sky in Fig. 3 are actually
***incorrect.  see John's memo of May 2007

   2(Phi_Inst_raw) + SH_COMB_ANG = 2(PHI_SKY - 90)
   2(Phi_Inst_raw) + 2(PAR + EL + HWP) = 2(-Phi_Inst_cor + PAR + EL - 90)
   Phi_Inst_raw + PAR + EL + HWP = -Phi_Inst_cor + PAR + EL - 90
   Phi_Inst_raw + PAR + EL + HWP = Phi_Inst_raw - Instrument_Offset + PAR + EL - 90
   HWP = -Instrument_Offset - 90 = -150 = +30    

   One way to visualize the relationship between PHI_SKY and Phi_Inst_cor is shown in Figure 4.

***NOTE ADDED MAY 07: the definitions of Q_sky and U_sky in Fig. 4 are actually
***incorrect.  see John's memo of May 2007