The Telescope Optics in .NET framework

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73 The Telescope Optics
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Stellar radiation collected by a telescope is brought to a focus before analysis by an attached polarimeter In the transmission, or re ection, by the various elements of the fore-optics, any collected radiation may have its polarization characteristics altered by small amounts Generally to the accuracies that are regularly achieved, the main considered effect is that of systematic instrumental polarization This shows itself by unpolarized radiation accruing a small polarization Its effect on the data
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is generally removed by calibrations, and, by applying the inverse of the Additivity Theorem of Small Polarizations (see Section 45), the raw measurements being considered as the simple addition of the true underlying value plus the instrumental polarization For very high accuracy measurements, this simple approach is inadequate and the source of the contamination needs to be identi ed more carefully Although the incoming radiation may carry only a small polarization, the second-order effects of the action on this by the small polarizance and phase shifts associated with the instrumental polarization need to be taken into account By summarizing the instrumental effects as a Mueller matrix, an inversion process should be applied in data reductions The two main effects of the telescope optics are that the mirrors may offer a weak polarizance and/or may introduce phase changes which affect the incoming radiation These effects are expressable in terms of Mueller matrices, with a reference axis that may be set at some arbitrary angle, , relative to the coordinate frame describing the Stokes parameters of the radiation Any polarizance effect may be expressed as 2 3 I0 07 6Q 6 7DR 4U 0 5 2 V0 2 M11 6 M21 6 4 0 0 M12 M22 0 0 0 0 M33 0 32 1 0 0 7 60 76 0 5 40 0 M44 0 cos 2 sin 2 0 0 sin 2 cos 2 0 32 3 I 0 07 6 Q 7 76 7 05 4U 5 1 V (72) where R is an average re ection coef cient It is expected that M11 ' M22 ' M33 ' M44 ' 1, and M12 ' M21 , these carrying very small values By multiplying out (72): 2 03 2 3 I M11 I C M12 (Q cos 2 C U sin 2 ) 07 6Q 6 7 6 7 D R 6 M21 I C M22 (Q cos 2 C U sin 2 )7 , (73) 4U 0 5 M33 (Q sin 2 C U cos 2 ) 5 2 4 V0 M44 V and this may be rewritten as 2 03 2 3 I M11 C M12 (q cos 2 C u sin 2 ) 6 Q0 7 6 7 6 7 D R I 6 M21 C M22 (q cos 2 C u sin 2 )7 4U 0 5 4 M33 (q sin 2 C u cos 2 ) 5 2 0 M44 v V
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The standard procedure to deal with this induced polarizance effect is to observe a selection of unpolarized standard stars to determine the value of M21 Its value is simply treated as an offset and subtracted from all subsequent measurements see 5 In the measurement of linear polarization, such an approach assumes that M11 D M22 D M33 D 1, and that the second-order effects associated with them not being exactly equal to unity are neglected When the high accuracy polarimetry is being pursued, their values need to be determined by observing a variety of precise polarized standard stars, and then used in an inversion procedure
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Stellar P larimetry
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to determine the true values of q and u of any other source Such a treatment is required to remove the polarizance fringes, described in 6, that can affect spectropolarimetry conducted by instruments using a wave-plate modulator For the case of telescope mirrors carrying phase changes, their effects may be included by considering values for M34 and M44 which are not zero The resultant changes to the incoming radiation may now be expressed as 2 3 3 M11 C M12 (q cos 2 C u sin 2 ) I0 07 6 7 6Q 6 7 D R I 6 M21 C M22 (q cos 2 C u sin 2 ) 7 4U 0 5 2 4 M33 (q sin 2 C u cos 2 ) C M34 v 5 M44 v M43 (q sin 2 C u cos 2 ) V0 2
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It can be seen that there is cross-talk within the U 0 and V 0 parameters as a result of both circular-to-linear and linear-to-circular conversions The latter effect is generally more troublesome than the former, as the incoming radiation is likely to carry linear polarization, which is much larger than any circular component being measured In principle, values for M34 and M44 can be determined from observations of standard stars and they can be applied in reduction procedures by inversion In the study by Miller (1963), the effects of telescopes on astronomical measurements have been formulated in terms of the existence of polarization eigenstates associated with the collector mirrors In addition to a discussion on the need to establish zero-polarization standard stars, the paper also discussed the proposal to develop a rotatable telescope (see further) to remove disturbing effects of the mirrors Attention was drawn to some of the problems that a rotatable telescope system cannot solve, particularly in relation to the measurement of circular polarization From measurements made using standard stars, Gehrels (1960) investigated the instrumental polarization of three different telescopes The polarization produced by the aluminized mirrors was wavelength dependent, generally increasing in the ultraviolet Some of the possible origins of the cause of the effects were discussed in terms of the aluminizing processes For axi-symmetric collectors such as a Cassegrain system, it might be expected that the telescope instrumental polarization would be zero It is suspected that in some aluminizing procedures, the sputtering process gives rise to crystalline-type growths with a preferred direction on the optical surface, this introducing an instrumental polarization To counteract such a problem, Gehrels & Meltzer (1966) suggested that, at the time of aluminizing telescope mirrors, the secondary might be mounted in the hole of the primary and on completion, it should be mounted after rotation by 90 with respect to the primary Even if the mirrors are in perfect condition, problems may arise with photomultiplier detectors In instruments employing these, an image of the telescope collector aperture is usually focussed on the cathode by means of a Fabry lens Different regions of the aperture may be given unequal weights because of non-uniform sensitivity of the photocathode over its area Polarization may also be introduced as a result of non-uniformities over the mirror surfaces Protective coatings can cause polarization contaminations as well
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