Towards a covariant formulation of electromagnetic wave polarization

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40 CHAPTER 5. COVARIANT POLARIZATION TENSORS and ⎛ ⎞ 0 cB 1 cB 2 cB 3 F µν = ⎜−cB 1 0 E 3 −E 2 ⎟ ⎝−cB 2 −E 3 0 E 1 ⎠ (5.9) −cB 3 E 2 −E 1 0 are the corresponding dual tensors. 5.1 The X tensors In E, B-component form the n X µν tensors are given by 1 X µν = [−E 1 E ∗ 1 − E 2 E ∗ 2 − E 3 E ∗ 3, −E 2 cB ∗ 3 + E 3 cB ∗ 2, E 1 cB ∗ 3 − E 3 cB ∗ 1, −E 1 cB ∗ 2 + E 2 cB ∗ 1] [−cB 3 E ∗ 2 + cB 2 E ∗ 3, E 1 E ∗ 1 − c 2 B 3 B ∗ 3 − c 2 B 2 B ∗ 2, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1] [cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 2 E ∗ 2 − c 2 B 3 B ∗ 3 − c 2 B 1 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2] [−cB 2 E ∗ 1 +cB 1 E ∗ 2, E 3 E ∗ 1 +c 2 B 1 B ∗ 3, E 3 E ∗ 2 +c 2 B 2 B ∗ 3, E 3 E ∗ 3 −c 2 B 2 B ∗ 2 −c 2 B 1 B ∗ 1] (5.10) 2 X µν = [−c 2 B 1 B ∗ 1−c 2 B 2 B ∗ 2−c 2 B 3 B ∗ 3, −cB 3 E ∗ 2+cB 2 E ∗ 3, cB 3 E ∗ 1−cB 1 E ∗ 3, −cB 2 E ∗ 1+cB 1 E ∗ 2] [−E 2 cB ∗ 3 + E 3 cB ∗ 2, c 2 B 1 B ∗ 1 − E 3 E ∗ 3 − E 2 E ∗ 2, E 2 E ∗ 1 + c 2 B 1 B ∗ 2, E 3 E ∗ 1 + c 2 B 1 B ∗ 3] [E 1 cB ∗ 3 − E 3 cB ∗ 1, E 1 E ∗ 2 + c 2 B 2 B ∗ 1, c 2 B 2 B ∗ 2 − E 3 E ∗ 3 − E 1 E ∗ 1, E 3 E ∗ 2 + c 2 B 2 B ∗ 3] [−E 1 cB ∗ 2 + E 2 cB ∗ 1, E 1 E ∗ 3 + c 2 B 3 B ∗ 1, E 2 E ∗ 3 + c 2 B 3 B ∗ 2, c 2 B 3 B ∗ 3 − E 2 E ∗ 2 − E 1 E ∗ 1] (5.11) 3 X µν = [−cB 1 E ∗ 1−cB 2 E ∗ 2−cB 3 E ∗ 3, −c 2 B 2 B ∗ 3+c 2 B 3 B ∗ 2, c 2 B 1 B ∗ 3−c 2 B 3 B ∗ 1, −c 2 B 1 B ∗ 2+c 2 B 2 B ∗ 1] [E 3 E ∗ 2 − E 2 E ∗ 3, cB 1 E ∗ 1 + E 3 cB ∗ 3 + E 2 cB ∗ 2, cB 1 E ∗ 2 − E 2 cB ∗ 1, cB 1 E ∗ 3 − E 3 cB ∗ 1] [−E 3 E ∗ 1 + E 1 E ∗ 3, cB 2 E ∗ 1 − E 1 cB ∗ 2, cB 2 E ∗ 2 + E 3 cB ∗ 3 + E 1 cB ∗ 1, cB 2 E ∗ 3 − E 3 cB ∗ 2] [E 2 E ∗ 1 − E 1 E ∗ 2, cB 3 E ∗ 1 − E 1 cB ∗ 3, cB 3 E ∗ 2 − E 2 cB ∗ 3, cB 3 E ∗ 3 + E 2 cB ∗ 2 + E 1 cB ∗ 1] (5.12) 4 X µν = [−E 1 cB ∗ 1 − E 2 cB ∗ 2 − E 3 cB ∗ 3, −E 3 E ∗ 2 + E 2 E ∗ 3, E 3 E ∗ 1 − E 1 E ∗ 3, −E 2 E ∗ 1 + E 1 E ∗ 2] [c 2 B 2 B ∗ 3 −c 2 B 3 B ∗ 2, E 1 cB ∗ 1 +cB 3 E ∗ 3 +cB 2 E ∗ 2, −cB 2 E ∗ 1 +E 1 cB ∗ 2, −cB 3 E ∗ 1 +E 1 cB ∗ 3] [−c 2 B 1 B ∗ 3+c 2 B 3 B ∗ 1, −cB 1 E ∗ 2+E 2 cB ∗ 1, cB 3 E ∗ 3+E 2 cB ∗ 2+cB 1 E ∗ 1, −cB 3 E ∗ 2+E 2 cB ∗ 3] [c 2 B 1 B ∗ 2−c 2 B 2 B ∗ 1, −cB 1 E ∗ 3+E 3 cB ∗ 1, −cB 2 E ∗ 3+E 3 cB ∗ 2, cB 2 E ∗ 2+E 3 cB ∗ 3+cB 1 E ∗ 1] (5.13)
5.2. THE Y TENSORS 41 5.2 The Y tensors The Y tensors, constructed by the combinations Σ Y µν = 1 2 ( 1 X µν + 2 X µν ) , χ Y µν = 1 ( 3 X µν + 4 X µν ) , 2 ∆ Y µν = 1 ( 1 X µν − 2 X µν ) (5.14) , 2 ψ Y µν = 1 ( 3 X µν − 4 X µν ) , 2 and are then found to be Σ 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+cB 2 E ∗ 3−E 2 cB ∗ 3+E 3 cB ∗ 2, E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, −E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2] [−cB 3 E ∗ 2+cB 2 E ∗ 3−E 2 cB ∗ 3+E 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, E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1] [E 1 cB ∗ 3 − E 3 cB ∗ 1 + cB 3 E ∗ 1 − cB 1 E ∗ 3, E 2 E ∗ 1 + cB 1 cB ∗ 2 + E 1 E ∗ 2 + cB 2 cB ∗ 1, E 2 E ∗ 2−cB 3 cB ∗ 3−cB 1 cB ∗ 1+cB 2 cB ∗ 2−E 3 E ∗ 3−E 1 EC 1 , E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2] [−E 1 cB ∗ 2 + E 2 cB ∗ 1 − cB 2 E ∗ 1 + cB 1 E ∗ 2, E 3 E ∗ 1 + cB 1 cB ∗ 3 + E 1 E ∗ 3 + cB 3 cB ∗ 1, E 3 E ∗ 2+cB 2 cB ∗ 3+E 2 E ∗ 3+cB 3 cB ∗ 2, E 3 E ∗ 3−cB 2 cB ∗ 2−cB 1 cB ∗ 1+cB 3 cB ∗ 3−E 2 E ∗ 2−E 1 E ∗ 1] (5.15) χ 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, −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, 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 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, 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 ∗ 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, cB 3 E ∗ 2−E 2 cB ∗ 3−cB 2 E ∗ 3+E 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] (5.16)
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 48 and 49: 42 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

Towards a covariant formulation of electromagnetic wave polarization

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