2 2 (r C rk ) 6 2 2 rk ) 6 (r D6 4 0 0

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2 2 (r rk ) 2 2 (r C rk ) 0 0

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0 0 2r rk 0

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3 0 7 0 7 7 0 5 2r rk

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(A28)

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By considering the signs of the values of r and rk , for insertion in this matrix, it is clearly apparent that, for angles of incidence from zero up to Brewster s angle, there is a handedness change for re ected circularly polarized light For angles of incidence greater than Brewster s angle, there is no handedness reversal By using intensity re ection coef cients, R , Rk , this Mueller matrix may be written as 2 3 (R C Rk ) (R Rk ) 0 0 6 (R Rk ) (R C Rk ) 7 0 6 7, p0 (A29) 4 5 0 0 2 R Rk p0 0 0 0 2 R Rk

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Appendix A The Fresnel Laws

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100 80

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Re ectivity (%) Degree of polarization (%)

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60 40

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Brewster s angle

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30 40 50 60 Angle of incidence

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30 40 50 60 Angle of incidence

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Fig A3 The intensity re ection coef cients, R , Rk , for the directions of vibration perpendicular and parallel to the plane of incidence are plotted as a function of the angle of incidence for a dielectric interface with a refractive

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index ratio n t / n i D 15 (a); (b) displays the degree of polarization produced by re ection At Brewster s angle the re ected light is 100 per cent polarized, with a direction of vibration perpendicular to the plane of incidence

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p but p it must be remembered that the value of R carries a negative sign, and for Rk the sign may be either Cve , or ve , depending on the angle of incidence The variation with angle of incidence of the numerical values of the resolved intensity coef cients, together with the engendered degree of polarization given by p D (R Rk )/(R C Rk ) is depicted in Figure A3 for a refractive index ratio of n t /n i D 15 The Mueller matrix for the change in polarization affecting the transmitted beam may be written as 2 3 2 2 2 2 (t C tk ) (t tk ) 0 0 6 2 7 2 2 2 0 0 7 6 (t tk ) (t C tk ) (A30) 6 7 4 0 0 2t tk 0 5 0 0 0 2t tk

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A3 Re ection at a Dense- to Less-Dense Medium

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If the refractive index of the rst medium is higher than that of the second (n i > n t ), then the behaviour of the re ected wave becomes complicated above a certain angle of incidence referred to as the critical angle Rewriting (A4) and (A9) in terms of the relative refractive index n D n t /n i gives r D rk D cos cos cos cos n cos C n cos i

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i t t t t i i

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(A31)

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n cos C n cos

(A32)

Stellar P larimetry

Using the law of refraction, namely: sin i D n sin t , the term cos t may be q expressed as (1/n) n 2 sin2 i , allowing (A31) and (A32) to be rewritten as q ( n 2 D q cos i C ( n 2 cos

sin2 sin2 n 2 cos

i) i)

(A33)

q n2 rk D q n2

sin2 sin2

(A34)

C n 2 cos

Since n < 1, for values of i 6 arcsin(n), the square root is real, and there is no dif culty in deciding which of the alternative signs should be chosen, as t is less than /2, and cos t must be positive For i > arcsin(n), however, the square root, and hence cos t , becomes imaginary, and care must be taken to resolve the sign ambiguity This is a troublesome point as the literature bears testament Many workers simply choose the positive sign, without considering the problem, and this has led to incorrect assessment of the phase changes on total internal re ection In fact, Astronomer Royal Airy (1831), using an incorrect Cve sign for the Fresnel tangent formula, accidentally obtained the correct answer to the Fresnel rhomb ( 6) by arbitrarily and, as we shall see, incorrectly by choosing the Cve sign for the above square root The problem can be solved by considering the disturbance in the second medium At some point, (y, z), with both values being positive (see Figure A1) in the medium of the refractive index, n t , the transmitted wave may be written in the form i h nt (y sin t C z cos t ) , (A35) Et D Et0 exp i t c where Et0 is the amplitude of the transmitted disturbance Since cos nary, and sin t real, but greater than one, (A35) can be rewritten as q 1 sin2 t n 2 z i Et D Et0 exp i n t c n i h nt sin t y , exp i t c i e a disturbance oscillatory in y, and varying with z, according to q nt sin2 t n 2 z Et (z) D Et0 exp c n

is imagi-

(A36)

(A37)

Depending on whether the positive or negative sign for the square root is selected, there will be an exponential increase, or decrease, in the amplitude of the disturbance As it turns out, there is no energy in this transmitted wave, as the re ection coef cients for both parallel and perpendicular components are unity The transmitted E and H vectors are in quadrature, in fact, and the Poynting vector is