outline. skewonless media with no birefringence
TRANSCRIPT
![Page 1: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/1.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Skewonless media with no birefringence.A general constitutive relation.
Alberto Favaro∗ and Luzi Bergamin
∗Department of Physics,Imperial College London, UK.
Department of Radio Science and Engineering,Aalto University, Finland.
March 21, 2011
![Page 2: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/2.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 3: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/3.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 4: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/4.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 5: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/5.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 6: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/6.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 7: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/7.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 8: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/8.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 9: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/9.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 10: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/10.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 11: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/11.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 12: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/12.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Outline.
Basic concepts (Pre-metric Linear-Local Media).
I Constitutive rel: tensor χµναβ and 6×6 χIJ form.I will explain what is the “skewon” in “skewonless”.
I Dispersion rel: a quartic equation in Kµ = (−ω, ki ).“No birefringence” means the equation is bi-quadratic.
Essential Tools.
I Schuller’s classification of χµναβ (caveat: densities).
I Lammerzahl/Hehl/Itin birefringence elimination scheme.
Main result and applications.
I Method: Go numerical, take a hint, repeat analytically.
I Constitutive law for skewonless non-birefringent media.
I Impact: Propels new findings, confirms past ones +Fronts in non-linear media + Inserting skewon.
![Page 13: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/13.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 14: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/14.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 15: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/15.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 16: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/16.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 17: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/17.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 18: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/18.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 19: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/19.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 20: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/20.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek).
Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 21: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/21.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 22: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/22.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Linear local constitutive relation (pre-metric).The EM fields D and H are related to E and B linearly[
DH
]=
[− ε α
− β µ−1
] [− EB
];
[D;H] from [−E;B] at the same local space-time point.
Medium response: 6×6 matrix form (engineering).
Get [W I ] = [D;H] from [FJ ] = [−E;B] via 6×6 matrix χIJ :
W I = χI JFJ ,
very compact. Glues up 3 dimensional objects, not covariant.
Medium response: tensor form (covariant, relativity).
4D indices (Greek). Link: W µν = −W νµ ⇒ 6 components.
W I = χIJFJ ⇒ W µν = χµνJFJ ⇒ W µν =1
2χµναβFαβ .
![Page 23: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/23.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Relate W µν , Fαβ with respective 6 independent entries:
[Fαβ] =
0 −E1 −E2 −E3
E1 0 B3 −B2
E2 −B3 0 B1
E3 B2 −B1 0
,
[W µν ] =
0 D1 D2 D3
−D1 0 H3 −H2
−D2 −H3 0 H1
−D3 H2 −H1 0
,
Important symmetry:
χµναβ = −χνµαβ = −χµνβα,
(E.J. Post, “Formal Structure of Electromagnetics”, 1962).
![Page 24: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/24.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 25: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/25.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + χIJSkew.
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 26: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/26.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 27: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/27.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 28: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/28.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 29: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/29.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 30: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/30.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 31: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/31.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 32: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/32.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0.
In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 33: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/33.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0.
In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 34: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/34.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 35: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/35.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 36: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/36.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 37: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/37.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Further symmetries: principal + skewon + axion.
I From 6×6 matrix split symmetric and skewon parts:χIJ ≡ (χIJ + χJI )/2 + (χIJ − χJI )/2= χIJ
Symm + [2]χIJ .
I Split: χµναβSymm = χµναβPrincipal + χµναβAxion = [1]χµναβ + [3]χµναβ ,axion antisymmetric for any pair of 4D indices swapped.
I Thus: χµναβ = [1]χµναβ + [2]χµναβ + [3]χµναβ .
Skewon [2]χµναβ has not been observed.
Finite skewon violates usual ε = ε T , µ = µ T , α = −β T .
Axion [3]χµναβ is rare but possible.
Axion obeys [3]χµναβ ∝ eµναβ ∈ ±1, 0. In nature:
I Static: de Lange & Raab, J. Opt. A, 2001.
I Waves: Hehl, Obukhov, Rivera & Schmid, PLA, 2008.
Given these considerations the skewon is assumed to vanish.
![Page 38: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/38.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used.
Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 39: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/39.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used.
Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 40: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/40.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used.
Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 41: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/41.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used. Test spacetime ⇒ sharp frontsand causality.
Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 42: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/42.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used. Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 43: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/43.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used. Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 44: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/44.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Dispersion (Fresnel) relation.
I Quartic in waves’ Kµ=(−ω, ki ), cubic in media’s χαβµν :
f (K) =1
4eαβγδ eηθκλχ
αβηθχγµνκχδρσλKµKνKρKσ = 0 ,
Obukhov, Fukui (2000) with Rubilar (2002). Covariant:Lindell (2005) and Itin (2009). Origin: Tamm (1925).
I Geometric optics used. Test spacetime ⇒ sharp frontsand causality. Table-top media ⇒ CW laser light, not aprobe of causality; speed of fronts is c (Milonni, 2002).
I Birefringence is eliminated when f (K) is bi-quadratic:
f (K) ∝ (GαβKαKβ)2 = 0.
Define Gαβ in f (K) as optical metric (Gordon, 1923).
I Example: vacuum, simplest non-birefringent medium
χµναβ0 =√− det(gαβ)(µ0/ε0)−
12
(gµαgνβ − gµβgνα
).
![Page 45: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/45.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.
No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 46: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/46.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.
No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 47: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/47.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.
No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 48: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/48.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.
No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 49: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/49.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.
No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 50: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/50.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.
Same diagonal, χIJC ∼ χIJ
D . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 51: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/51.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD .
But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 52: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/52.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 53: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/53.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 54: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/54.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 55: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/55.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 56: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/56.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 57: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/57.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 58: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/58.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Schuller et al. classify skewonless χαβµν (2010).
I χαβµνA ≈χαβµνB strongly equivalent if linked by change of4D basis. Find typical reps (normal forms) classify all.
I Preview: test normal forms for no-birefringence, test all.
Heuristics of the classification.No skewon, 6×6 matrix χIJ is symmetric: can diagonalise.Same diagonal, χIJ
C ∼ χIJD . But ∼ as 6×6 matrices, weakly!
I Go refined: form κ αβµν = eµνρσχ
ρσαβ/2, valence (2,2).
I Get eigenvalue problem κµναβXαβ = 2ηXµν , and solve.
I Classify ηs: Real-Complex? Multiplicities (geom-algeb)?
⇒ Strong (Schuller) classification: Segre types, Jordan form.
For those like me. . .Schuller provides 23 matrices (6×6 normal forms) thatencode every skewonless medium. Just use them!
![Page 59: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/59.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 60: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/60.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 61: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/61.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 62: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/62.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 63: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/63.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 64: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/64.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
The classification and pre-metric electromag.
I Schuller et al., medium response Ωµναβ transforms as:
Ωµ′ν′α′β′= Lµ′µL
ν′νL
α′αL
β′
β Ωµναβ .
Pre-metrically, medium response χµναβ transforms as:
χµ′ν′α′β′ = | det(Lρ
′ρ)|−1Lµ′µLν
′νL
α′αL
β′
β χµναβ .
I Fix: adapted from Schuller. 4D transform 6×6 dets:
| det(ΩI ′J′)|1/6 = | det(Lρ′ρ)|+1| det(ΩIJ)|1/6 ,
| det(χI ′J′)| = | det(χIJ)| .
I Factors in red can compensate each other. Reconcile:
χµναβ
| det(χIJ)|1/6=
Ωµναβ
| det(ΩIJ)|1/6,
matches pre-metric well; but how about axiomatics?
![Page 65: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/65.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 66: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/66.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 67: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/67.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 68: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/68.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 69: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/69.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 70: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/70.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Tool: constrains medium for no-birefringence.
I Adapt: Lammerzahl, Hehl (PRD ’04) + Itin (PRD ’05).
I Pick component q from Kµ. Dispersion relation w.r.t q,
M0q4 + M1q
3 + M2q2 + M3q + M4 = 0 ,
coeffs dependent on (−ω, ki ), but not on entry Kν = q.
I Compare w/ biquadratic ⇒ no-birefringence scheme:
Quartic Equation
M0=0 N
M1=0 &
M2=M32/(4M4)
M4=(4M0M2-M12)2/(64M0
3)&
M3=M1(4M0M2-M12)/(8M0
2)
N
Y
Y NY
Non birefringent media All other media
M4=0M4=0
![Page 71: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/71.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 72: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/72.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 73: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/73.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 74: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/74.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 75: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/75.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 76: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/76.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 77: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/77.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 78: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/78.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 79: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/79.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 80: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/80.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 81: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/81.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 82: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/82.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Our method: Computer ⇒ “Hint” ⇒ Analytics.
Step 1: computer search of non-birefringent media.
I Input: 23 Schuller matrices (≈ all skewonless media).
I Program: no-birefringence (Lammerzahl/Hehl/Itin).
I Output: 5 matrices (≈ all skewonless non-biref. media).
“Hint”: matrices hint to one intuitive form of χµναβ.
I 5 matrices: no-birefringence solutions; not intuitive.
I Find one “analytic law” χµναβ re-represents 5 matrices.
I Trade-off: the analytic law is too coarse; covers 5matrices, but some birefringent solutions too. Refine!
Step 2: Refine the analytic law, get symbolic result.
I Non-birefringent = optical metric + (bivectors, axion).
![Page 83: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/83.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Mathematica R©: 23 matrices in, 5 matrices out.Lammerzahl/Hehl/Itin birefringence elimination gives 5 χIJ :
−τ 0 0 σ 0 00 −τ 0 0 σ 00 0 −τ 0 0 σσ 0 0 τ 0 00 σ 0 0 τ 00 0 σ 0 0 τ
λ5 0 0 λ6 0 00 λ3 0 0 λ4 00 0 λ1 0 0 λ2λ6 0 0 λ5 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 −τ 0 0 σ 00 0 λ1 0 0 λ2
±λ1 + λ2 0 0 0 0 00 σ 0 0 τ 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 λ3 0 0 λ4 00 0 λ1 0 λ2
±λ1 + λ2 0 0 0 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 λ1 0 00 0 0 0 λ5 00 0 ±(λ3 − λ5) 0 0 λ3λ3 0 0 ε1 0 00 λ5 0 0 0 00 0 λ1 0 0 ±(λ3 − λ5)
Full answer, but ugly: decode into analytic law (“Hint”).
I Magneto-electric diagonals → χµναβ has axion αeµναβ .
I “Rectangles”→ χµναβ has bivector terms AµνAαβ.
![Page 84: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/84.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Mathematica R©: 23 matrices in, 5 matrices out.Lammerzahl/Hehl/Itin birefringence elimination gives 5 χIJ :
−τ 0 0 σ 0 00 −τ 0 0 σ 00 0 −τ 0 0 σσ 0 0 τ 0 00 σ 0 0 τ 00 0 σ 0 0 τ
λ5 0 0 λ6 0 00 λ3 0 0 λ4 00 0 λ1 0 0 λ2λ6 0 0 λ5 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 −τ 0 0 σ 00 0 λ1 0 0 λ2
±λ1 + λ2 0 0 0 0 00 σ 0 0 τ 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 λ3 0 0 λ4 00 0 λ1 0 λ2
±λ1 + λ2 0 0 0 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 λ1 0 00 0 0 0 λ5 00 0 ±(λ3 − λ5) 0 0 λ3λ3 0 0 ε1 0 00 λ5 0 0 0 00 0 λ1 0 0 ±(λ3 − λ5)
Full answer, but ugly: decode into analytic law (“Hint”).
I Magneto-electric diagonals → χµναβ has axion αeµναβ .
I “Rectangles”→ χµναβ has bivector terms AµνAαβ.
![Page 85: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/85.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Mathematica R©: 23 matrices in, 5 matrices out.Lammerzahl/Hehl/Itin birefringence elimination gives 5 χIJ :
−τ 0 0 σ 0 00 −τ 0 0 σ 00 0 −τ 0 0 σσ 0 0 τ 0 00 σ 0 0 τ 00 0 σ 0 0 τ
λ5 0 0 λ6 0 00 λ3 0 0 λ4 00 0 λ1 0 0 λ2λ6 0 0 λ5 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 −τ 0 0 σ 00 0 λ1 0 0 λ2
±λ1 + λ2 0 0 0 0 00 σ 0 0 τ 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 λ3 0 0 λ4 00 0 λ1 0 λ2
±λ1 + λ2 0 0 0 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 λ1 0 00 0 0 0 λ5 00 0 ±(λ3 − λ5) 0 0 λ3λ3 0 0 ε1 0 00 λ5 0 0 0 00 0 λ1 0 0 ±(λ3 − λ5)
Full answer, but ugly: decode into analytic law (“Hint”).
I Magneto-electric diagonals → χµναβ has axion αeµναβ .
I “Rectangles”→ χµναβ has bivector terms AµνAαβ.
![Page 86: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/86.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Mathematica R©: 23 matrices in, 5 matrices out.Lammerzahl/Hehl/Itin birefringence elimination gives 5 χIJ :
−τ 0 0 σ 0 00 −τ 0 0 σ 00 0 −τ 0 0 σσ 0 0 τ 0 00 σ 0 0 τ 00 0 σ 0 0 τ
λ5 0 0 λ6 0 00 λ3 0 0 λ4 00 0 λ1 0 0 λ2λ6 0 0 λ5 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 −τ 0 0 σ 00 0 λ1 0 0 λ2
±λ1 + λ2 0 0 0 0 00 σ 0 0 τ 00 0 λ2 0 0 λ1
0 0 0 ±λ1 + λ2 0 00 λ3 0 0 λ4 00 0 λ1 0 λ2
±λ1 + λ2 0 0 0 0 00 λ4 0 0 λ3 00 0 λ2 0 0 λ1
0 0 0 λ1 0 00 0 0 0 λ5 00 0 ±(λ3 − λ5) 0 0 λ3λ3 0 0 ε1 0 00 λ5 0 0 0 00 0 λ1 0 0 ±(λ3 − λ5)
Full answer, but ugly: decode into analytic law (“Hint”).
I Magneto-electric diagonals → χµναβ has axion αeµναβ .
I “Rectangles”→ χµναβ has bivector terms AµνAαβ.
![Page 87: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/87.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 88: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/88.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 89: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/89.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 90: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/90.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 91: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/91.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 92: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/92.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 93: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/93.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 94: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/94.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
“Hint”: analytic χµναβ suggested by 5 matrices.
χµνρσ= | det(G−1αβ )|
12 M
[(GµρGνσ−GµσGνρ)+sAA
µνAρσ + sAAµν Aρσ
]+αeµνρσ
I Hodge: metric used becomes optical (GαβKαKβ)2 =0.
I Bivector terms: Aαβ, Aαβ antisymmetric; sA, sA signs.
I Preview: Bivector terms vanish if Gαβ signature (3,1).
I Axion term: innocuous, drops from dispersion relation.
I Further stuff: account density and impedance-like M.
Too coarse, skewonless: nonbirefringent + some birefringent.
KILL!
![Page 95: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/95.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.
Substitute analytic “hint” into dispersion relation f (K):[(G − sAA · G−1 · A)µνKµKν
][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν
=sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 96: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/96.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν
=sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 97: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/97.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν
=sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 98: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/98.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν
=sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 99: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/99.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 100: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/100.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 101: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/101.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Refine analytic law, get necessary & sufficient.Substitute analytic “hint” into dispersion relation f (K):[
(G − sAA · G−1 · A)µνKµKν][(G − sAA · G
−1 · A)ρσKρKσ]
−sAsA[(A · G−1 · A)µνKµKν
]2= 0 .
Birefringent. Become non-birefringent (GαβKαKβ)2=0 if:
A · G−1 · A · G−1 = a11 , A · G−1 · A · G−1 = a21 ,
A · G−1 · A · G−1 + A · G−1 · A · G−1 = 2a31 .
Bivect. Aαβand Aαβmust be (anti-)selfdual w.r.t Gαβ-Hodge:
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Aµν =sX2εµναβG (G−1 · A · G−1)αβ := sX (
∗A)µν ,
Density eµναβconverted to tensor εµναβG via optical Gαβ. Also:
∗∗A = A , ∗∗A = A .
![Page 102: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/102.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).
Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 103: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/103.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).
Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 104: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/104.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 105: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/105.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 106: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/106.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.
Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 107: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/107.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Results: all skewonless non-birefringent χµναβ.
χµνρσ= | det(G−1αβ )|
12 M
[GµρGνσ−GµσGνρ + sAA
µνAρσ + sAAµν Aρσ
]+ αeµνρσ
A = sX∗A , A = sX
∗A ,
When the signature of the optical Gαβ is Lorentzian (3,1).Both ∗∗A = A (above) and ∗∗A ≡ −A (Nakahara, ’03), hence:
A ≡ 0 , A ≡ 0 ,
χµνρσ(3,1) = | det(G−1αβ )|
12M[GµρGνσ−GµσGνρ
]+ αeµνρσ
Non-birefringent + Lorentzian ⇔ Hodge dual + Axion part.Other signatures: only metamaterials, hyperlens (Jacob ’06).
![Page 108: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/108.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 109: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/109.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 110: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/110.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 111: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/111.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 112: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/112.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 113: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/113.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 114: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/114.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 115: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/115.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 116: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/116.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 117: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/117.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 118: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/118.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0.
“Note that the vanishing of birefringence equivalent to thevalidity of the reciprocity [closure] relation.”
![Page 119: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/119.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0. CONVERSE?
“Note that the vanishing of birefringence is not equivalent tothe validity of the reciprocity [closure] relation.” (L&H, ’04)
![Page 120: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/120.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Ties together previous literature.
Hehl & Obukhov (2003).
1. EM reciprocity: axionless (tilde) part obeys closure
eIK χKLeLM χ
MJ = −λ2δJI .
2. They further eliminate skewon, setting [2]χIJ = 0.
1. and 2. uniquely give Hodge dual [(3,1)-metric] + axion.
Lammerzahl & Hehl (2004).
1. Enforce that there is a unique light-cone.
2. Hyperbolic propagation: problem is causally posed.
1. and 2. uniquely give optical metric Gαβ signature (3,1).
(3,1): Hodge + axion ⇒ (GαβKαKβ)=0. CONVERSE!
“Note that the vanishing of birefringence IS equivalent tothe validity of the reciprocity [closure] relation.” (F&B, ’11)
![Page 121: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/121.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ . Studygeneral Casimir effect. Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 122: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/122.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ . Studygeneral Casimir effect. Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 123: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/123.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ .
Studygeneral Casimir effect. Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 124: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/124.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ . Studygeneral Casimir effect.
Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 125: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/125.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ . Studygeneral Casimir effect. Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 126: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/126.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
What this work has been cited/used for. . .
Rivera & Schuller, (arXiv:1101.0491).
Quantise skewonless bi-anisotropic W I = χIJFJ . Studygeneral Casimir effect. Usual Casimir = no birefringence.
Matias Dahl (arXiv:1103.3118).
Confirms the result “(3,1): Hodge + axion ⇒(GαβKαKβ)=0. CONVERSE!” using Grobner bases.
![Page 127: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/127.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 128: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/128.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 129: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/129.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 130: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/130.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 131: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/131.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 132: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/132.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 133: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/133.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 134: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/134.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Conclusions.
A. Favaro, L. Bergamin, “The non-birefringent limit of alllinear, skewonless media and its unique light-conestructure.”, Annalen der Physik, 1-19 (2011).
In this talk. . .
I Reviewed the origin of Schuller’s classification.
I Described Lammerzahl/Hehl/Itin no-birefringence.
I Covered all non-birefringent skewonless media.
I For signature (3,1), little more than Hodge dual.
I Haven’t talked of other signatures. Please ask!
I Showed how nicely it all fits in the literature.
![Page 135: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/135.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Beyond: Non-Linear or with Skewon.
Obukhov & Rubilar (PRD, ’02).
I Discuss very general non-linear Lagrangian (Skewon=0):
L = L(I1, I2) , I1 = FµνFµν , I2 = Fµν F
µν ,
I Return to linear optics if propagating very sharp front.
I For this skewonless medium, no-birefringence if only if
χµνρσOR = | det(G−1αβ )|12M[GµρG νσ−GµσG νρ
]+ αeµνρσ
as per this talk. (Made this statement more explicit).
Skewon: divergence from Hodge dual (Bergamin, ’10).
χµναβ ∝ AµνAαβ + [2]χµναβ is non-birefringent.
![Page 136: Outline. Skewonless media with no birefringence](https://reader031.vdocuments.mx/reader031/viewer/2022020620/61e464bca7576b1e87756a78/html5/thumbnails/136.jpg)
Skewonless mediawith no
birefringence.
Outline.
Basics.
Tools.
Method.
Step 1.
Hint.
Step2.
Results.
Impact.
Conclusions.
Beyond.
Thank-you.
Thank-you!