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Photon noise calculation

It is notoriously difficult to get the photon noise calculation right for polarimetry. We therefore include a reasoned example which we believe to be correct.

In the standard staring mode of measuring linear polarization, one Stokes parameter is derived from four spectra, recorded in two CCD images, for halfwave plate position angles differing by 45 degrees. Let tex2html_wrap_inline1043 be the number of photons detected per spectral resolution element in the o and e spectra for the two CCD exposures. The Stokes parameter, for example tex2html_wrap_inline1023 , is then given by the formula in Section 3.2.

When integration times for both exposures are the same and the level of polarization is small, tex2html_wrap_inline1047 , where tex2html_wrap_inline1049 is the total number of photons (per spectral resolution element) in the two exposures of the two spectra. In that case R tex2html_wrap_inline1051 1 and Q tex2html_wrap_inline1051 0.

Photon statistics gives an uncertainty of tex2html_wrap_inline1055 for each of the N. The fractional error in each N is therefore tex2html_wrap_inline1057 . For multiplication and division, the fractional errors add in quadrature, hence the fractional error in R tex2html_wrap_inline1059 is tex2html_wrap_inline1061 , the fractional error in R is tex2html_wrap_inline1063 . Since R tex2html_wrap_inline1051 1, the absolute error in R-1 is also tex2html_wrap_inline1069 . Since R+1 tex2html_wrap_inline1071 R-1, the fractional error in tex2html_wrap_inline1023 is dominated by that in R-1, which is tex2html_wrap_inline1079 . Finally, this makes the absolute error in tex2html_wrap_inline1023 :
tex2html_wrap_inline1083 .

For example, in order to determine one Stokes parameter to a degree-of-polarization accuracy of 0.005 (e.g. tex2html_wrap_inline1023 = 2.5 tex2html_wrap_inline1087 0.5 % or tex2html_wrap_inline969 = 1.0 tex2html_wrap_inline1087 0.5 % or tex2html_wrap_inline1093 = 0.1 tex2html_wrap_inline1087 0.5 %), \ 4 tex2html_wrap_inline1097 photons per resolution element are required. To obtain both linear Stokes parameters with that accuracy, we need two such observations. With the present La Palma CCDs, maximum recorded output, which is close to saturation charge, is of order 6 tex2html_wrap_inline1097 electrons per pixel. It is advisable to stay somewhat below this in single exposures, and to average over pixels and/or repeated observations as necessary for the desired accuracy.


next up previous contents
Next: Statistics Up: Noise and statistics in Previous: Noise and statistics in

manuals store
Tue Oct 7 17:34:45 BST 1997