Towards a covariant formulation of electromagnetic wave polarization

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26 CHAPTER 2. BACKGROUND Using these tensors to decompose A αβγδ , we obtain A αβγδ = 1 16 Sgαβ g γδ + 1 4! S(p) ɛ αβγδ + 1 2 T αβ g γδ + Ψ αβγδ , (2.123) where Ψ is a reducible rank four tensor. The tensor T αβ is a rank two tensor and thus can be reduced according to Equation (2.116). The irreducible tensors are scalar, pseudo-scalar, symmetric traceless rank two tensor and anti-symmetric rank two tensor. To do a decomposition of higher rank tensors, one makes a representation consisting of these irreducible tensors and products thereof, iterating the symmetry and anti-symmetry between indices. Comparing this with the symmetries of the tensor we want to decompose, it will be easier to construct the proper tensors in the representation. Example: Irreducible tensor representation of the two-dimensional spectral density matrix A simple example of irreducible tensor representation is to take the two dimensional spectral density tensor, Equation (2.53), which is a tensor under the group SO(2), and decompose it into an irreducible representation. In SO(2) we have the invariant metric tensor, δ αβ , and the totally anti-symmetric tensor, ɛ αβ . For a general rank two tensor, A αβ , the irreducible representation will be A αβ = 1 2 Sδ αβ + 1 2 S(p) ɛ αβ + T αβ , (2.124) where we have S as the trace, S = A αβ δ αβ , and the anti-symmetric part, S (p) = A αβ ɛ αβ , which is a pseudo-scalar. Performing this on the two dimensional spectral density matrix, S d (Eq. (2.53)), we have ( ) F1 F1 S d = ∗ F 1 F2 ∗ F 2 F1 ∗ F 2 F2 ∗ = 1 2 (F 1 2 + F2 2 )δ αβ + 1 2 (F 1F † 2 − F 2F † 1 )ɛ αβ+ ( F 2 1 − F2 2 F 1 F2 ∗ + F 2 F1 ∗ 1 2 F 1 F † 2 + F 2F ∗ 1 F 2 2 − F 2 1 ) . (2.125) Comparing this representation with Stokes’ parameters, Equation (2.49), we see that it can be written ( F1 F † S d = 1 F 1 F † ) 2 F 2 F † 1 F 2 F † = 1 2 2 Iδ αβ − i 2 Vɛ αβ + 1 ( ) Q U , (2.126) 2 U −Q where the energy density, I, is the invariant scalar and the degree of circular polarization, V, is the invariant pseudo scalar. The last two Stokes parameters, corresponding to linear polarization along the axis (Q) and linear polarization in 45 ◦ angle to the axis (U), together form a traceless symmetric rank two tensor. This is the irreducible tensor representation of the two-dimensional spectral density tensor under SO(2).
Chapter 3 The covariant spectral density tensor and a generic decomposition scheme In this chapter we introduce the covariant spectral density tensor, τ αβγδ ≡ F αβ (F γδ ) ∗ , which is a manifestly Lorentz covariant rank four tensor, containing all quadratic components of the electromagnetic field. An alternative way of constructing a covariant spectral density tensor is to consider A µ A ν (see Barut [1]), where A µ is the electromagnetic four-potential (see Section 2.1.2), The four-potential is however not a measurable physical quantity and it also depends on the choice of gauge. Since our approach is to consider measurable parameters of polarization (just as Stokes’ parameters) the seemingly more complicated rank four tensor τ αβγδ is preferable. One way to expand the covariant spectral density tensor is to use the Dirac gamma matrices. In this chapter two different definitions of the Dirac gamma matrices is presented and we use the one that comes from the flat metric g µν (see Section 2.1, Equation (2.6)) to extract the 36 parameters needed for a complete representation of the field. The technique here is similar to that in Section 2.2, where a three-dimensional tensor of rank two is decomposed in terms of a basis of matrices. In this Chapter we use the original four Dirac gamma matrices and expand them into a complete 16 dimensional matrix basis. The covariant spectral density tensor is expanded in this basis. 3.1 Covariant spectral density tensor To have all quadratic electromagnetic field components collected into one object, we introduce and define the covariant spectral density tensor, τ αβγδ , by τ αβγδ ≡ F αβ (F γβ ) ∗ . (3.1) The definition for real fields is the same but the complex conjugate is then abundant. This is the deterministic version where we know the pertinent field completely. For a physical field, where one considers a certain bandwidth, this 27
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 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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