Polarimetric Principles and Instruments in Visual Studio .NET

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8 Polarimetric Principles and Instruments
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Most designs of polarimeter operate with converging, or diverging, beams passing through the modulator as the elements are usually capable of accepting reasonably large cone angles without losing ef ciency There may be concerns over the operation of wave plates with such arrangements, and it is sometimes better if the transmitted beams are rst collimated In the case of the design by Clarke & McLean (1975), sketched out in Figure 84, a collimated beam was employed to avoid spectral passband spread of the narrow-band interference lters placed after the polarizer The principles of polarimetric techniques are readily appreciated in terms of the signals generated by the modulator For some forms of detector, the information may be apparent in real time, the output keeping pace with the generated modulation Other detectors may accumulate the signal before presenting a read-out at appropriate times As described in the the following sections, the forms of the signals are generally schematic and may need to be interpreted according to the design of particular equipment
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82 Rotation of a Polarizer or Wave Plate
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The most obvious polarimetric technique is to use a modulator which contains rotatable elements whose transmittances are directly sensitive to the vibration directions of the electric disturbances within the incoming beam In the early days of experimentation, this was done by introducing sheet Polaroid into the conventional photoelectric photometers and taking brightness readings at speci c orientations of its principal axis The observational scheme is essentially the same as that used for multi- lter photometry and, for that reason, early reported measurements of the degree of polarization were expressed as magnitude differences Several of the early catalogues (e g Hiltner, 1951a, Behr, 1959) list p in magnitudes and these require conversion to fractional, or percentage, values to correspond with modern convention (see Section 47) The behaviour of an instrument based on the rotation of a polarizer may be expressed in terms of Mueller matrices, set out in their correct order, operating on the input Stokes vector fI, Q, U, V g of the incoming light According to the orientation, , of the principal axis of the polarizer with respect to the axial frame describing the original Stokes parameters, as in Figure 82, the output Stokes vector fI 0 , Q0 , U 0 , V 0 g may be determined from 2 3 1 I0 6 Q0 7 1 61 6 7D 6 4U 0 5 2 40 0 0 V 2 1 1 0 0 0 0 0 0 32 1 0 07 60 76 05 40 0 0 0 cos 2 sin 2 0 0 sin 2 cos 2 0 32 3 I 0 07 6 Q 7 76 7 05 4 U 5 V 1
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(81)
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Stellar P larimetry
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Fig 82 The rotation of a polarizer through a positive angle, , relative to the reference axis de ning Q
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It is the parameter, I 0 , and its dependence on that is measured by the detector, and, following the matrix multiplication, it may be expressed as I 0 ( ) D 1 I C Q cos 2 C U sin 2 2 (82)
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Remembering that Q/I D q D p cos 2 , and U/I D u D p sin 2 (see (426)), the above equation may be rewritten as I 0 ( ) D 1 I 1 C p cos 2( 2 ) (83)
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Generally the reference axis of the polarimeter is described with respect to equatorial coordinates, i e the duciary of the instrument is such that, when is zero, the polarizer s transmission axis runs N/S, and aligned with Declination Thus, it can be seen that the variation of intensity produced by a rotating polarizer modulator comprises a mean level, on which a modulation is superimposed, with amplitude dependent on the degree of polarization, and with the phase dependent on the direction of vibration of the incoming light Essentially, there are three unknowns requiring quanti cation In fact, the actual signal, Sk ( ), involves additional parameters to describe it, and should be written as Sk ( ) D 1 Gk ( )T(t) I C Q cos 2 C U sin 2 , 2 (84)
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where Gk ( ) expresses the detector s response to unit intensity, also showing that it might perhaps be affected according to the rotational position of the polarizer as a result of beam wobble, or to the detector s sensitivity with respect to the direction of vibration of the polarized light which illuminates it; the symbol, k, refers to the fact that the intensity with vibration parallel to the polarizer s principal axis is being measured The coef cient, T(t), describes the variation of the signal with time caused either by the Earth s atmosphere as a result of intensity scintillation, or by the slower, secular changes in atmospheric extinction
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