Issue 
A&A
Volume 537, January 2012



Article Number  A75  
Number of page(s)  13  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201117333  
Published online  12 January 2012 
Online material
Appendix A: Instrument simulation
Here we discuss the details of the instrument simulations. The instrument simulator was developed to simulate observations with the Extreme Polarimeter (ExPo), which is a visitor instrument of the 4.2 m WHT.
A.1. Wavefront generation
To generate wavefronts, we make use of a model of turbulence first proposed by Kolmogorov (1941) and later developed by Tatarski (1961) and Fried (1966). According to this model, a flat wavefront traveling through the turbulent atmosphere will modify its phase, while the change in its amplitude is negligible compared to the phase fluctuations. After it travels through the telescope, the amplitude of the wavefront is proportional to the telescope’s aperture. The power spectrum of the phase fluctuations of a wavefront traveling through the atmosphere is given by (A.1)where r_{0} is Fried’s Parameter and k the wave number. The parameter r_{0} represents the circular aperture over which the wavefront phase variance is equal to 1 rad^{2} for the case of Kolmogorov turbulence. Therefore, the higher r_{0}, the more stable the atmosphere, or, in other words, the better the seeing is. For instance, a value of r_{0} of about 10 cm produces 1 arcsec seeing at λ = 500 nm (i.e. Nightingale & Buscher 1991). The value of r_{0} depends on the wavelength, following a power law: r_{0} ∝ λ^{6/5}. Figure A.1 shows an example of an uncalibrated wavefront phase generated from Eq. (A.1).
The timescale at which one of these wavefronts remains constant, the coherence time, at optical wavelengths is only a few milliseconds (Vernin & MunozTunon 1994; Law 2006; GarcíaLorenzo et al. 2009). The single frame PSF has two main components:

Tilt, producing random motion of the whole image.

Roughness, producing the observed speckle pattern.
When the exposure time is similar to the coherence time, the image motion due to the tilt component can be removed in the data processing by proper centering of each frame. Noll (1976) showed that the variance σ^{2} of the wavefronts can be expressed in terms of the telescope diameter, D, and the Fried parameter, r_{0}(A.2)The standard deviation σ of the wavefront phase ϕ, expressed in wavelength units, can be obtained from the variance as σ_{λ} = 2πσ(A.3)In our simulation, D is the diameter of the WHT main mirror (4.2 m), and three different values of r_{0}, (5, 7, and 10 cm) are used to account for bad, medium, and good seeing at a reference wavelength of 500 nm, λ_{500}. From Eq (A.2) and the dependence of r_{0} with wavelength, σ_{λ} can be computed at different wavelengths from our reference as (A.4)Wavefronts at different wavelengths are calibrated according to Eqs. (A.2) and (A.4). A set of wavefronts at different wavelengths is then generated for each of the three different r_{0} values used in this simulation.
Simulation parameters.
A.2. PSF generation
To generate a PSF, we first simulate the aperture of the WHT, taking the spiders of the telescope and the central obscuration of the main mirror into account. Once the telescope aperture and the wavefront are computed, a monochromatic PSF is calculated according to Eq. (2). A broadband PSF (PSF_{bb}) is calculated as the sum of monochromatic PSFs calculated over the range of 400 nm to 700 nm, in steps of 10 nm: (A.5)This PSF produces the speckle pattern obtained when observing at exposure times that are close to the coherence time (a few milliseconds). The PSF measured when observing at longer exposure times will be the sum of the shortexposure broadband PSFs. To simulate ExPolike PSFs, we set the coherence time to 9.3 s and then generate a 28 s exposure PSF as the sum of three statistically independent shortexposure broadband PSFs: (A.6)These calculations are computed on a 2048 × 2048 pixels grid. The pixel size of this simulation is determined by the Nyquist frequency: N_{ν} = (λ/2D)·206265, which produces a pixel size of 0.0122 [′′/pixel], for a 4.2 m telescope and a central wavelength of 500 nm. The original 2048 × 2048 pixel grid is then binned to an smaller grid to produce simulated images at the ExPo pixel size (0.078 [′′/pixel]). Figure A.2 shows an example of a PSF for both simulated (left) and real (right) data (0.8′′ seeing, 0.028 s exposure time).
A total of 100 different PSF_{ExPo} are produced for each of the three different seeing conditions tested here. Figure A.3 shows an example of three different broadband PSF’s calculated for bad (left), normal (center), and good (right) seeing.
Fig. A.1
Phase of a wavefront following the Kolmogorov model of turbulence. 

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Fig. A.2
Left: Simulated PSF with seeing ≈ 0.8′′, 0.028 s exposure time. Right: observed PSF under the same conditions. 

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A.3. Dual beam simulation
As a dual beam instrument, ExPo produces two simultaneous images with opposite polarization states that are imaged onto a CCD. We refer to these two images as left and right images. These measurements are modulated by a Ferroelectric liquid crystal (FLC), which switches between two orthogonal polarization states (A and B). Two different frames, containing four different images are produced at the end of one FLC cycle: where neither instrumental polarization nor instrumental effects are considered for the sake of simplicity, and I_{0} and P_{0} represent the total intensity and polarization (Stokes Q or Stokes U) of the observed target, as seeing without atmosphere and diffraction
effects, respectively. PSF_{A} and PSF_{B} are the short exposure PSF for the A and B frames, respectively. These images are given in mJy units, and they are converted to counts by considering the telescope area, exposure time, filter transmission (Johnson V filter is used here), atmospheric + instrument absorption, and CCD efficiency.
Fig. A.3
Simulated speckle pattern for different values of r_{0}, given in centimeters. 

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The poidev function from the NASA IDL library is used to simulate the photon noise for each of these images. Real readout noise from ExPo measurements is added to the simulated data. Instrumental polarization is simulated by including two transmission coefficients for each of the beams (T_{L},T_{R}) and another two coefficients to account for the FLC transmission (T_{A},T_{B}). The value of these coefficients is listed in Table A.2.
Transmission coefficients measured from ExPo instrument.
To mimic guiding problems, a random shift with a maximum amplitude of ten pixels is applied to each of the simulated images. The final simulated images are then described by where M(x,y) represents the imageshifting function, Ph is the photon noise, and RO represents the readout noise.
We finally run the full data reduction pipeline for the simulated observations to obtain the final images presented in Figs. 7–9
© ESO, 2012
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