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Fig 41 The development of the polarization ellipse showing rotation of the major axis over a few cycles of the progression of an electromagnetic wave
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of the detected radiation The values of the above terms may be expressed as
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2 2 I D hE x0 C E y 0 i ,t 2 Q D hE x0 2 E y 0 i ,t
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(41) (42) x i ,t x i ,t (43) (44)
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U D h2E x0 E y 0 cos y V D h2E x0 E y 0 sin y
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These de nitions for I, Q, U, V are referred to as the Stokes parameters, following the classical paper of Stokes (1852) They may be written in the form of a column vector or Stokes vector The elements of the vector are described in terms of speci c intensity, when discussing theoretical ideas or astrophysical modelling problems, or ux units, when describing their measurement According to Collett (1971), the impasse to a complete understanding of the Fresnel Arago laws (see 1) remained for nearly 35 years until Stokes considered the problem, and successfully introduced a mathematical description of unpolarized light as contained within the concept of the parameters named after him It may be noted that some workers prefer to express the Stokes parameters in an alternative mathematical form as D E I D E x0 E x0 C E y 0 E y 0 (45) E ,t D Q D E x0 E x0 E y 0 E y 0 (46) D E ,t U D E x0 E y 0 C E y 0 E x0 (47) ,t E D V D i(E x0 E y 0 E y 0 E x0 ) , (48)
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where the superscript
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refers to the complex conjugate of the function
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4 The Stokes Parameters
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It is interesting to note that, across a range of science disciplines, Stokes parameters have been reinvented or rediscovered at various times In the eld of chemistry, they were restated by Perrin (1942), and they were also described by Walker (1954) In his original paper, Stokes identi ed the parameters as fA, B, C, Dg; Perrin (1942) used the variables fI, M, C, S g, while Walker (1954) referred to them as fI, Q, U, V g They were nally introduced to Astronomy by Chandrasekhar (1947), de ning them in relation to his studies on radiative transfer theory associated with stellar atmospheres A disguised use of them was also made by Wesselink (1958) Chandrasekhar (1947) used the notation of fI, Q, U, V g and this practice will be continued in this work If the makeup of the radiation comprises electrical uctuations from waves appearing and dying with complete lack of time coherence, then Q D U D V D 0, and the light is said to be unpolarized According to the degree of coherence that is maintained over the experimental time, Q, U and V may take values up to a maximum equal to I For reference, in relation to the description of perfectly polarized light in 2 (see (240)), it will be noted that
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2 2 2 I D (Idiff C Icos C Isin )1/2 ,
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(49)
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or I D (Q2 C U 2 C V 2 )1/2 (410)
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Partial randomization of the phases of the contributing waves with the progress of time provides the condition of I Q2 C U 2 C V 2
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(411)
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Part of the contribution to the inequality may also arise from the fact that the waves are non-classical, comprising components with a range of frequencies, the averaging process taking this into account In the classical paper of Stokes (1852), it was demonstrated that when beams of radiation are combined, the parameters describing their individual polarizations are simply additive to provide a description of the resultant beam; there is no phase coherence that needs to be taken into account By representing the parameters as column vectors, this additivity rule may be represented as 2 3 2 3 2 3 IT I1 I2 6 Q T 7 6Q 17 6Q 27 6 7D6 7C6 7C 4 UT 5 4 U1 5 4 U2 5 VT with IT D
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