How to estimate S/N for ALFOSC polarimetry
High signaltonoise ratios (S/N) are required for a good polarisation
accuracy. Let E_{p} be the 1 sigma error in P (the
degree of polarization). The S/N required to obtain a given accuracy
E_{p}, assuming only photon shot noise, can be expressed
as:
S/N_{(fo +fe)} = sqrt(2) / (sqrt(n) * E_{p})
where n is the number of halfwaveplate position angles, and
S/N_{(fo +fe)} is the signal to noise ratio
of the flux of the combined components (ordinary + extraordinary) in one
of the n intensity images, cf.
Patat & Romaniello (2006, PASP 118, 146).
To better account for sky noise and readout noise through the use of our
Exposure Time Calculator, however, the above formula is here modified to refer
to the S/N of each dualbeam component. Assuming low polarization,
f_{o} is about equal to f_{e}, and for half stellar intensity
the relative sky noise is larger by a factor of 2 unless a focal plane mask
is used. This may become important for faint targets. In the following we
will thus refer to the signal to noise ratio of each dualbeam component as
follows:
S/N = 1 / (sqrt(n) * E_{p})
For linear polarimetry with ALFOSC/FAPOL with the default setup,
n = 4, this means:
S/N = 0.5/ E_{p}
Example: to obtain an accuracy of ±0.3 % polarisation,
i.e. E_{p} = 0.003, a signal to noise ratio of S/N = 167 is needed for
each dualbeam component in each of the 4 images.
When using the Exposure Time
Calculator for ALFOSC, one has to: 1) correct for the flux being split
in two components by the calcite plate, and 2) correct for the flux losses
by introducing the calcite and the 1/2 wave plate into the beam
(check FAPOL commissioning report ).
In practice, this means correcting the input magnitude of your target by
adding the following magnitude offsets: 1.4 (U), 1.05 (B,V), and 1.0 (RI).
The modified input magnitude is used in the Exposure Time Calculator to
find the exposure time you need (for each of the n halfwaveplate
positions) in order to reach the required S/N.
The uncertainty in the position angle (in degrees) goes as: 28.65 x
E_{p} / P whenever P >> E_{p}.
The above estimation has some caveats:

The limiting instrumental accuracy for FAPOL (lambda/2), as measured on bright
standards, was found to be 0.06 % using 4 angles. It is possible to increase
the accuracy using 8 or 16 angles. See
Measured standard star accuracy.
 Note that the image quality is not undisturbed through the calcites, and a
feature of the calcite is that both components can not be perfectly in focus at
the same time. This means that one should not be too optimistic about the input
FWHM in the ETC.
 at low degrees of polarization one has to take the nongaussian distribution
of the measurements into account, this will increase the effective errors.
Comments to Anlaug Amanda Djupvik
