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in (234), (238) and (239) The matrices describing various polarimetric elements are summarized in Table 31 Clearly from its formulation, the Jones calculus deals with situations involving radiation with waves which have long-term phase coherence It is important to remember that the algebra can deal with the combination of beams of radiation only if the phases of the component beams are taken into account Thus, suppose that the disturbances within a beam of radiation can be expressed in the form of (213) and (214), i e E1x D E1x 0 cos (2 t C 1x ) , E1y D E1y 0 cos 2 t C 1y , (317) (318)
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these giving rise to a particular polarization form Suppose that a second polarized beam is added with classical waves expressed as E2x D E2x 0 cos (2 t C 2x ) , E2y D E2y 0 cos 2 t C 2y (319) (320)
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The combination gives rise to disturbances which are described as E x D E1x 0 cos (2 t C 1x ) C E2x 0 cos (2 t C 2x ) , E y D E1y 0 cos 2 t C 1y C E2y 0 cos 2 t C 2y These latter two equations can be contracted to E x D E x 0 cos 2 t C x , E y D E y 0 cos 2 t C y , where, (E x 0 )2 D fE1x 0 cos 1x C E2x 0 cos 2x g2 C fE1x 0 sin 1x C E2x 0 sin 2x g2 , 2 2 (E y 0 )2 D E1y 0 cos 1y C E2y 0 cos 2y C E1y 0 sin 1y C E2y 0 sin 2y , tan x D fE1x 0 sin 1x C E2x 0 sin 2x g / fE1x 0 cos 1x C E2x 0 cos 2x g , tan y D E1y 0 sin 1y C E2y 0 sin 2y / E1y 0 cos 1y C E2y 0 cos 2y Thus, as can be seen from (323) and (324), combinations of orthogonally resolved classical coherent waves, with particular polarization forms, lead to a pair of classical waves with a new resultant polarization form (323) (324) (321) (322)
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33 The Description of Scattering
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It is impossible here to provide all the material required to appreciate fully the physics and mathematics associated with scattering within astrophysical situations There are several texts available which comprehensively cover these matters However, it is important that scattering mechanisms are understood to some degree in respect of their polarigenic potential As it turns out, some of the algebra used to formulate scattering processes has overtones with the Jones calculus When radiation encounters assemblies of small particles such as dust within circumstellar envelopes, electron clouds in extended dissociated stellar atmospheres, or dust in the interstellar medium, the interaction causes it to be scattered Generally the scattering is not isotropic with the amplitudes of the waves being dependent on the angle of emergence with respect to the original direction of incidence In addition, the scattering is also sensitive to polarization By de ning some plane associated with any incoming radiation, the waves may be resolved in directions perpendicular ( ) and parallel (k) to this plane Thus, the orthogonal vibrations may be written as E D E o cos (2 t C ) , Ek D Ek o cos 2 t C k (325) (326)
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3 The Algebra of Polarization
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Fig 32 Radiation travelling along the z-axis hits an assembly of scattering particles positioned in the x y-plane and is scattered in a direction given by ,
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Following the scattering process, the magnitudes of the perpendicular and parallel vibrations may be represented by E o ( , ) and Ek o ( , ), where and are the polar angles describing the direction of the emergence of the radiation (see Figure 32) These waves may be simply represented in terms of a linear transformation of the incident wave, and consequently the relationship can be described in terms of an interaction, much in the same way as is done by the Jones calculus Hence the description of scattering may be written as " # 0 E o ( , ) S ( , ) S2k ( , ) E o , (327) D 1 0 Ek o ( , ) S2 ( , ) S1k ( , ) Ek o
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0 0 where E o , Ek o are the resulting vibrations and S1 , S2 , S1k and S2k are the amplitude scattering functions, dependent on and They comprise real and imaginary parts, in turn dependent on the refractive index, m D m 0 im 00 , of the Q scattering particle, and on its size, a, in relation to the wavelength of the radiation being scattered; normally the size/wavelength relationship is expressed as x D 2 a/ D k a, with 2 / k de ning the wavenumber Generally in an astrophysical situation, the scattering is simple, corresponding to the equivalent of linear dichroism and linear birefringence, there being no circular dichroism and no circular birefringence For mathematical simplicity, it is also convenient to assume that the principal axes of the tensor describing the particle anisotropy coincides with the frame de ned by the and k directions With these assumptions S2 and S2k D 0, and the unity subscript may be dropped leading to a description represented by " # 0 E o ( , ) 0 S ( , ) E o (328) D 0 Ek o ( , ) 0 Sk ( , ) Ek o
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The amplitude scattering functions may comprise real and imaginary parts so that S D a Sk D a k i b i bk , (329) (330)
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where the terms a and a k dictate the absorptive effects, their combination describing the extinction; the terms b and b k also describe the linear birefringence According to the type of scattering particle, the values of a , a k , b and b k may be assigned
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