Towards a covariant formulation of electromagnetic wave polarization

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36CHAPTER 4. IRREDUCIBLE REPRESENTATION OF POLARIZATION The condition of a divergence free magnetic field is easily seen from this equation. By taking the scalar product with ⃗n, from the left we obtain ⃗n · (⃗n × ⃗ E) = c⃗n · ⃗B ⇒ ⃗n · ⃗B = 0, (4.25) where we have used the vector relations ⃗a · ( ⃗ b × ⃗c) = (⃗a × b) · ⃗c and ⃗n × ⃗n = 0. We also multiply with ⃗n † , from the left, hence ⃗n † · (⃗n × ⃗ E) = c⃗n † · ⃗B ⇒ ⃗n † · ⃗B = 0. (4.26) The condition for the last equation to be zero is that ⃗n † ×⃗n = 0. This is fulfilled if ⃗n can be written ⃗n = n⃗r, where n is a complex scalar and ⃗r is a real vector. This means that the refractive index is the same in all direction, which is true for a homogeneous medium. To get an expression for ⃗n we use Faraday’s law cross multiplied from the right with ⃗ B † (⃗n × ⃗ E) × ⃗ B † = c ⃗ B × ⃗ B † , (4.27) which leads to −⃗n( ⃗ B † · ⃗E) + ⃗ E( ⃗ B † · ⃗n) = c ⃗ B × ⃗ B † . (4.28) If we restrict ourself to the case ⃗n × ⃗n † = 0 (homogeneous medium), the second term on the left hand side vanishes and we are left with which for ⃗ E · ⃗B † ≠ 0 can be rewritten as −⃗n( ⃗ E · ⃗B † ) = c ⃗ B × ⃗ B † , (4.29) B ⃗n = −c ⃗ × B ⃗ † . (4.30) ⃗E · ⃗B † If we compare this expression with the parameters defined earlier in this chapter we have ⃗B × ⃗ B † = i c 2 ( ⃗V Σ − ⃗ V ∆) , (4.31) and Inserting this in the expression for ⃗n we get ⃗E · ⃗B † = 1 2c( S χ + S ψ) . (4.32) ⃗n = −2 i( ⃗ V Σ − ⃗ V ∆) S χ + S ψ . (4.33) Given the S, V and T parameters this is a simple calculation. 4.4 Stokes parameters from the S, V, T parameters In the two dimensional case, with E 3 = 0 and only considering the electric field (putting ⃗ B = 0), we have Stokes parameters. In terms of the S, V, T parameters
4.4. STOKES PARAMETERS FROM THE S, V, T PARAMETERS 37 they can be written I = F 2 x + F 2 y = S Σ , (4.34) Q = F 2 x − F 2 y = T Σ 11 − T Σ 22, (4.35) U = 2Re[F y F ∗ x ] = 2T Σ 12, (4.36) V = 2Im[F y F ∗ x ] = −2V Σ 3 , (4.37) where Σ can be replaced by ∆, since the magnetic field, ⃗ B , is zero. It is similar for the polarization of the magnetic field. The two dimensional spectral density matrix (Eq. (2.53)) becomes ( ) Fx Fx S d = ∗ F x Fy ∗ F y Fx ∗ F y Fy ∗ = 1 2 ( ) S Σ + T11 Σ − T22 Σ 2T12 Σ + i2V3 Σ 2T12 Σ − i2V3 Σ S Σ − T11 Σ + T22 Σ , (4.38) where we have put in the Stokes parameters in the S, V, T parameter form. We can simplify this to ( ) T Σ S d = 11 T12 Σ + iV3 Σ T12 Σ − iV3 Σ T22 Σ . (4.39)
Page 1: Towards a covariant formulation of
Page 4 and 5: ii Preface This report is a Master
Page 6 and 7: iv CONTENTS 4.4 Stokes parameters f
Page 8 and 9: 2 CHAPTER 1. INTRODUCTION according
Page 10 and 11: 4 CHAPTER 2. BACKGROUND form a subg
Page 12 and 13: 6 CHAPTER 2. BACKGROUND For a tenso
Page 14 and 15: 8 CHAPTER 2. BACKGROUND 2.1.3 Const
Page 16 and 17: 10 CHAPTER 2. BACKGROUND With the e
Page 18 and 19: 12 CHAPTER 2. BACKGROUND Karlsson [
Page 20 and 21: 14 CHAPTER 2. BACKGROUND and ⎛ 0
Page 22 and 23: 16 CHAPTER 2. BACKGROUND real and 1
Page 24 and 25: 18 CHAPTER 2. BACKGROUND and if the
Page 26 and 27: 20 CHAPTER 2. BACKGROUND Maxwell st
Page 28 and 29: 22 CHAPTER 2. BACKGROUND 2.4 Group
Page 30 and 31: 24 CHAPTER 2. BACKGROUND using the
Page 32 and 33: 26 CHAPTER 2. BACKGROUND Using thes
Page 34 and 35: 28 CHAPTER 3. COVARIANT SPECTRAL DE
Page 36 and 37: 30 CHAPTER 3. COVARIANT SPECTRAL DE
Page 38 and 39: 32CHAPTER 4. IRREDUCIBLE REPRESENTA
Page 46 and 47: 40 CHAPTER 5. COVARIANT POLARIZATIO
Page 52 and 53: 46 CHAPTER 6. REDUCIBLE REPRESENTAT
Page 54 and 55: 48 CHAPTER 6. REDUCIBLE REPRESENTAT
Page 56 and 57: 50 CHAPTER 7. CONCLUSIONS The new p
Page 58 and 59: 52 APPENDIX A. GENERALIZED POLARIZA
Page 60 and 61: 54 APPENDIX B. PARAMETERS FROM DIRA
Page 62 and 63: 56 APPENDIX B. PARAMETERS FROM DIRA
Page 64 and 65: 58 APPENDIX C. ⃗ E, ⃗ B COMPONE
Page 66 and 67: 60 APPENDIX C. ⃗ E, ⃗ B COMPONE
Page 68 and 69: 62 APPENDIX D. COVARIANT POLARIZATI
Page 74: 68 BIBLIOGRAPHY [16] Jerry B. Mario

Towards a covariant formulation of electromagnetic wave polarization

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