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a modulator comprising a rotatable half-wave plate prior to the xed polarizer and detector system Consider now applying the Pickering method to such an instrument The two signals may be represented by Sk ( ) D 1 Gk ( )T(t) I C Q cos 4 C U sin 4 , 2 1 S ( ) D G ( )T(t) I Q cos 4 U sin 4 2 (824) (825)
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By making records at angular settings of the wave plate of 0 and 45 , with ratios of the signals subsequently taken at each of these positions, the pair of measurements may be represented by R0 D
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1 T (t)Gk (I C Q) Gk (I C Q) 2 0 , D 1 G (I Q) T0 (t)G (I Q) 2 1 T45 (t)Gk (I Q) Gk (I Q) R45 D 12 D G (I C Q) T45 (t)G (I C Q) 2
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(826) (827)
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By taking the ratios of these determined values, the problem of the different sensitivities of the two detectors is then eliminated, i e Rq D R0 Gk (I C Q) Gk (I C Q) D D R45 G (I Q) G (I Q) I Q I CQ 2 , (828)
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from which the NSP may be calculated as p Rq 1 qD p Rq C 1
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(829)
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A similar expression for the u parameter may be developed by using the polarizer at settings of 22 5 and 67 5, leading to p Ru 1 (830) uD p , Ru C 1 R225 I CU 2 (831) D where Ru D R675 I U Thus, the linear polarization parameters are determinable without calibrating the relative sensitivities of the detectors; compensation for atmospheric effects is also automatically achieved This is of great importance to the application of CCDs to polarimetry Tinbergen & Rutten (1992) promoted the concept of this method in their User Guide for Spectropolarimetry for the William Herschel Telescope In their general use, CCDs require the time-consuming procedure of at- elding, which would also be necessary if single-beam polarimetry were to be attempted A strict requirement of the modulator system as described above is that the image pairs should not wander over the CCD pixels as the wave plate is rotated to the designated positions
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8 Polarimetric Principles and Instruments
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Various data reduction strategies are applied to the observations made by instruments which record intensity values at xed positions of the rotatable wave plate Statistically rigorous methods for doing this have been presented by von der Heide & Knoechel (1979)
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With records made at xed positional angles of the modulator, two-beam polarizers allow for the removal of atmospheric generated noise by taking the ratio of the simultaneously recorded signals Such noise can be overcome by rotating the modulator at high speed; the process can be enhanced further by taking ratios if orthogonal beams are available There are many techniques for recording the waveform with a continuously rotating modulator In the instrument (MINIPOL) designed by Frecker & Serkowski (1976), which has had extensive use at the University of Arizona, the sinusoid generated by a half-wave plate rotated at 20 Hz, was sampled over 12 equal angular spaced sections; least-squares sine ts were performed on the records to determine q and u Another successful recording scheme involves the integration of the signal over three angular sections as the modulator rotates For the case of a rotating polarizer modulator, a minimum of three channels are required to obtain a measurement Following Klare, Neckel & Schnur (1974), the rst channel records the signal continuously, while the the second integrates over the intervals of 0 ! /2 and ! 3 /2 and the third from /4 ! 3 /4 and 5 /4 ! 7 /4 Thus, the three recorded measurements over N complete cycles may by written as Z Sk1 D N Z Sk2 D N D 2N Z Sk3 D N D 2N
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