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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 r₀ is Fried’s Parameter and k the wave number. The parameter r₀ represents the circular aperture over which the wavefront phase variance is equal to 1 rad² for the case of Kolmogorov turbulence. Therefore, the higher r₀, the more stable the atmosphere, or, in other words, the better the seeing is. For instance, a value of r₀ of about 10 cm produces 1 arcsec seeing at λ = 500 nm (i.e. Nightingale & Buscher 1991). The value of r₀ depends on the wavelength, following a power law: r₀ ∝ 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 & Munoz-Tunon 1994; Law 2006; García-Lorenzo 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 σ² of the wavefronts can be expressed in terms of the telescope diameter, D, and the Fried parameter, r₀(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 r₀, (5, 7, and 10 cm) are used to account for bad, medium, and good seeing at a reference wavelength of 500 nm, λ₅₀₀. From Eq (A.2) and the dependence of r₀ 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 r₀ values used in this simulation.

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 short-exposure broadband PSFs. To simulate ExPo-like 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 short-exposure 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 Ferro-electric 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 I₀ and P₀ 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 r0, 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.

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 image-shifting 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