A The Fresnel Laws in Visual Studio .NET

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Appendix A The Fresnel Laws
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Fig A5 The variation of the phase difference, internal re ection for a dielectric with n D 2/3
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k , at total
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angle of incidence given by s cos
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1 n2 1 C n2
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(A46)
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The value of this minimum phase difference is given by ( k )min D 4 arctan(n) (A47)
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The phase difference ( k ) is greater than /2 for all values of n greater than p p 2 1 , i e for all dielectrics with refractive indices less than 2 C 1 It is interesting to note that Lord Kelvin (1904) (see p 388) commented on this as follows:
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We see that the phasal differences for internal re ection in glasses and all known transparent bodies of refractive index less than 2414, are obtuse for all angles of incidence through the whole range of total internal re ection This conclusion was very startling to myself, because for eighty years we have been taught that, for total internal re ection in glass, the phasal difference was an acute angle in a single re ection
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Inspection of the current literature frequently shows that some teaching still carries this error and lessons have not been learned
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Stellar P larimetry
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A4 Metallic Re ection
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To describe metal re ection, the optical properties of the material require its refractive index to be considered as being complex The classical texts show that alternative expressions are used with Cve and being applied to the complex component The two forms may be summarized by the expression: n D n(1 i ) , O (A48)
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where n is the real part of the complex index, and is the attenuation index, or extinction coef cient Rather than presenting all the arcane and laborious details for the determination of the amplitude re ection coef cients, with either the plus or minus sign being used, it may be assumed that their behaviour must be similar to that for a dielectric surface, particularly for small values of Values of r and rk are always negative for the complete range of i , from 0 to /2 Unlike the behaviour of a dielectric, rk is never zero, although its modulus displays a minimum at some angle of incidence, p , referred to as the principal angle of incidence In addition, each resolved component suffers a phase change from to zero, but with differing values, according to the angle of incidence; a phase difference, D ( k ), grows from zero to as i increases from 0 to /2 At the principal angle, we have D /2 Unless interference effects are under consideration, it is simply the intensity coef cients that need to be determined, together with the generated phase difference, to describe the behaviour of a metallic re ection Overall a metallic re ection may be represented by 2 1 0 0 60 1 0 6 40 0 1 0 0 0 2 (R C Rk ) 6 (R Rk ) 6 4 0 0 3 0 07 7 05 1 (R Rk ) (R C Rk ) 0 0 0 0 p 2 p Rk cos R 2 R Rk sin
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3 0 7 0 7 , (A49) p 2p R Rk sin 5 2 R Rk cos
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where the rst matrix relates to the re ection of the axes, and the second describes the basic electromagnetic behaviour of the metallic re ection in the original coordinate frame By multiplying out (A49), the nal matrix may be written as 2 (R C Rk ) 6 (R Rk ) 6 4 0 0 (R Rk ) (R C Rk ) 0 0 0 0 p 2 pR Rk cos C2 R Rk sin 3 0 7 0 7 p 2p R Rk sin 5 2 R Rk cos (A50)
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Appendix A The Fresnel Laws
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It may be noted that as r and rk both carry the same sign ( ve), the numerical p value of R Rk is always positive As takes values from zero to /2, cos and sin are also always positive Elements [3,3], [3,4] and [4,4] of the matrix in (A49) therefore carry negative values, while [4,3] is always positive According to Ditchburn (1952) the values of R and Rk , and their dependence on the angle of incidence, may be written as R D n 2 (1 C 2 ) cos2 n 2 (1 C 2 ) cos2 n 2 (1 C 2 ) cos2 n 2 (1 C 2 ) cos2
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