Towards a covariant formulation of electromagnetic wave polarization

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42 CHAPTER 5. COVARIANT POLARIZATION TENSORS ∆ Y µν = 1 2 [−E 1 E ∗ 1−E 2 E ∗ 2−E 3 E ∗ 3+cB 1 cB ∗ 1+cB 2 cB ∗ 2+cB 3 cB ∗ 3, cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 3 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1] [cB 2 E ∗ 3−E 3 cB ∗ 2−cB 3 E ∗ 2+E 2 cB ∗ 3, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, − cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, −E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] [cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2, −cB 2 cB ∗ 3+cB 3 cB ∗ 2−E 3 E ∗ 2+E 2 E ∗ 3] [cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, E 3 E ∗ 2−E 2 E ∗ 3+cB 2 cB ∗ 3−cB 3 cB ∗ 2, E 1 E ∗ 1−cB 3 cB ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+E 3 E ∗ 3+E 2 E ∗ 2] (5.17) ψ Y µν = 1 2 [−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3+E 1 cB ∗ 1+E 2 cB ∗ 2+E 3 cB ∗ 3, E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2] [E 3 E ∗ 2−E 2 E ∗ 3−cB 2 cB ∗ 3+cB 3 cB ∗ 2, cB 1 E ∗ 1+E 3 cB ∗ 3+E 2 cB ∗ 2−E 1 cB ∗ 1−cB 3 E ∗ 3−cB 2 E ∗ 2, cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1] [cB 1 cB ∗ 3 − cB 3 cB ∗ 1 − E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2 + cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 2 E ∗ 2+E 3 cB ∗ 3+E 1 cB ∗ 1−cB 3 E ∗ 3−E 2 cB ∗ 2−cB 1 E ∗ 1, cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2] [−cB 1 cB ∗ 2 + cB 2 cB ∗ 1 + E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3 + cB 1 E ∗ 3 − E 3 cB ∗ 1, cB 3 E ∗ 2−E 2 cB ∗ 3+cB 2 E ∗ 3−E 3 cB ∗ 2, cB 3 E ∗ 3+E 2 cB ∗ 2+E 1 cB ∗ 1−cB 2 E ∗ 2−E 3 cB ∗ 3−cB 1 E ∗ 1] (5.18) 5.3 The K and K tensors The K and K tensors are formed by removing the trace from ∆ Y µν and χ Y µν , respectively, and are given by K µν = 1 2 [0, cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1] [cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3, 0, −cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2, − E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1] [cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, 0, − cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3] [cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0] (5.19)
5.3. THE K AND K TENSORS 43 and K µν = 1 2 [0, −cB 2 cB ∗ 3 + cB 3 cB ∗ 2 − E 3 E ∗ 2 + E 2 E ∗ 3, cB 1 cB ∗ 3 − cB 3 cB ∗ 1 + E 3 E ∗ 1 − E 1 E ∗ 3, − cB 1 cB ∗ 2 + cB 2 cB ∗ 1 − E 2 E ∗ 1 + E 1 E ∗ 2] [E 3 E ∗ 2 − E 2 E ∗ 3 + cB 2 cB ∗ 3 − cB 3 cB ∗ 2, 0, cB 1 E ∗ 2 − E 2 cB ∗ 1 − cB 2 E ∗ 1 + E 1 cB ∗ 2, cB 1 E ∗ 3 − E 3 cB ∗ 1 − cB 3 E ∗ 1 + E 1 cB ∗ 3] [−E 3 E ∗ 1 + E 1 E ∗ 3 − cB 1 cB ∗ 3 + cB 3 cB ∗ 1, cB 2 E ∗ 1 − E 1 cB ∗ 2 − cB 1 E ∗ 2 + E 2 cB ∗ 1, 0, cB 2 E ∗ 3 − E 3 cB ∗ 2 − cB 3 E ∗ 2 + E 2 cB ∗ 3] [E 2 E ∗ 1 − E 1 E ∗ 2 + cB 1 cB ∗ 2 − cB 2 cB ∗ 1, cB 3 E ∗ 1 − E 1 cB ∗ 3 − cB 1 E ∗ 3 + E 3 cB ∗ 1, respectively. cB 3 E ∗ 2 − E 2 cB ∗ 3 − cB 2 E ∗ 3 + E 3 cB ∗ 2, 0], (5.20)
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 50 and 51: 44 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

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