Electromagnetic media with no Fresnel (dispersion) equation and novel jump (boundary) conditions

Transcription

1 Electromagnetic media with no Fresnel (dispersion) equation and novel jump (boundary) Alberto Favaro Inst. Theor. Phys., Univ. of Cologne, Köln, Germany 2 Dept. Radio Science & Engineering, Aalto Univ., Espoo, Finland 3 rd GIF workshop, ZARM, Bremen, June favaro@thp.uni-koeln.de

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3 Part 1. The local and linear electromagnetic response How to represent local in 3D and in 4D. How to decompose electromagnetic response via index symmetries: principal part, skewon part and axion part. Geometrical optics as described by Fresnel (dispersion) equation. What is a medium with no?

4 Part 1. The local and linear electromagnetic response How to represent local in 3D and in 4D. How to decompose electromagnetic response via index symmetries: principal part, skewon part and axion part. Geometrical optics as described by Fresnel (dispersion) equation. What is a medium with no? Part 2. Jump (boundary) useful in engineering In addition to primary electromagnetic jump at interface, have jump useful in engineering. PEMC jump twist polarisers. DB radar invisibility. Linked to media with no Fresnel eq...

5 Part 1. The local and linear electromagnetic response How to represent local in 3D and in 4D. How to decompose electromagnetic response via index symmetries: principal part, skewon part and axion part. Geometrical optics as described by Fresnel (dispersion) equation. What is a medium with no? Part 2. Jump (boundary) useful in engineering In addition to primary electromagnetic jump at interface, have jump useful in engineering. PEMC jump twist polarisers. DB radar invisibility. Linked to media with no Fresnel eq... Part 3. All local with no? We present strong evidence that there exist only three types of materials that give rise to no.

6 & linear electromagnetic response

7 & linear electromagnetic response Field excitation is G αβ = (D i, H j ). Field strength is F αβ = ( E i, B j ). Greek indices range 0 to 3. Latin indices range 1 to 3. We use Einstein summation conv.

8 & linear electromagnetic response Field excitation is G αβ = (D i, H j ). Field strength is F αβ = ( E i, B j ). Greek indices range 0 to 3. Latin indices range 1 to 3. We use Einstein summation conv. Local & linear electromagnetic response (medium): field excitation (D i, H j ) at point p in space and time related linearly to field strength ( E i, B j ) at the same point p.

9 & linear electromagnetic response Field excitation is G αβ = (D i, H j ). Field strength is F αβ = ( E i, B j ). Greek indices range 0 to 3. Latin indices range 1 to 3. We use Einstein summation conv. Local & linear electromagnetic response (medium): field excitation (D i, H j ) at point p in space and time related linearly to field strength ( E i, B j ) at the same point p. Space+time (3D): local & linear material or vacuum is D a = ε 0 ε ab E b + Z0 1 αa b Bb, H a = Z0 1 β a b E b + µ 1 0 µ 1 ab Bb, ε ab called permittivity, µ 1 ab called impermeability, α a b and β a b called magneto-electric terms. But note, we make use of relative quantities. Also, Z 0 =(µ 0 /ε 0 ) 1 2.

10 & linear electromagnetic response Field excitation is G αβ = (D i, H j ). Field strength is F αβ = ( E i, B j ). Greek indices range 0 to 3. Latin indices range 1 to 3. We use Einstein summation conv. Local & linear electromagnetic response (medium): field excitation (D i, H j ) at point p in space and time related linearly to field strength ( E i, B j ) at the same point p. Space+time (3D): local & linear material or vacuum is D a = ε 0 ε ab E b + Z0 1 αa b Bb, H a = Z0 1 β a b E b + µ 1 0 µ 1 ab Bb, ε ab called permittivity, µ 1 ab called impermeability, α a b and β a b called magneto-electric terms. But note, we make use of relative quantities. Also, Z 0 =(µ 0 /ε 0 ) 1 2. Covariant (4D): G αβ = 1 2 χαβµν F µν. Now, decompose...

11 Principal-skewon-axion irreducible decomposition Split medium: principal part, skewon part, axion part. χ αβµν = (1) χ αβµν + (2) χ αβµν + (3) χ αβµν. 36 = (1) χ αβµν is symmetric under [αβ] [µν] and traceless. (2) χ αβµν is antisymmetric under [αβ] [µν]. Moreover, (3) χ αβµν is the trace w.r.t. the Levi-Civita symbol ˆɛ αβµν. Finite axion part observed in nature (Hehl et al. 2008). Finite skewon not yet, but magnetic groups identified (Dmitriev 1998). Know one route to find a violation of ε ab = ε ba, µ 1 ab = µ 1 ba, αa b = β a b.

12 Electromag. waves: Fresnel (dispersion) equation Medium: ε ab =diag(ε 1, ε 2, ε 3 ), µ 1 ab =δ ab and α a b =β b a =0. Taken from: Schaefer (1932).

13 Electromag. waves: Fresnel (dispersion) equation q α = ( ω, k i ) is 4-dimensional wave-covector. Given propagation direction, what is inverse phase velocity k i /ω of electromagnetic waves? Solve : G(q)=ˆɛ αα1 α 2 α 3ˆɛ ββ1 β 2 β 3 χ αα 1ββ 1 χ α 2ρβ 2 σ χ α 3τβ 3 υ q ρ q σ q τ q υ =0. G(q) quartic in q α. Find all response tensors χ αβµν s.t. G(q) = two coinciding quadratics = no-birefringence. Solved for zero skewon: Favaro+Bergamin (2011), Dahl (2012). Using: Lämmerzahl+Hehl (2004), Itin (2005). G(q) = two distinct quadratics = uniaxial generalised. Solved for zero skewon, Lorentz signature: Dahl (2013). Using: Lindell+Wallén (2004), Schuller et al. (2010). G(q) = zero for all possible q α = no. This talk and PIER. Earlier: Hehl+Obukhov (2003). Dahl: find χ αβµν for each possible factorisation of G(q).

14 Electromag. waves: Fresnel (dispersion) equation q α = ( ω, k i ) is 4-dimensional wave-covector. Given propagation direction, what is inverse phase velocity k i /ω of electromagnetic waves? Solve : G(q)=ˆɛ αα1 α 2 α 3ˆɛ ββ1 β 2 β 3 χ αα 1ββ 1 χ α 2ρβ 2 σ χ α 3τβ 3 υ q ρ q σ q τ q υ =0. G(q) quartic in q α. Find all response tensors χ αβµν s.t. G(q) = two coinciding quadratics = no-birefringence. Solved for zero skewon: Favaro+Bergamin (2011), Dahl (2012). Using: Lämmerzahl+Hehl (2004), Itin (2005). G(q) = two distinct quadratics = uniaxial generalised. Solved for zero skewon, Lorentz signature: Dahl (2013). Using: Lindell+Wallén (2004), Schuller et al. (2010). G(q) = zero for all possible q α = no. This talk and PIER. Earlier: Hehl+Obukhov (2003). Dahl: find χ αβµν for each possible factorisation of G(q).

15 Part 2: jump. Fundamental/additional

16 Part 2: jump. Fundamental/additional Interface: have following jump (boundary), ɛ abc n b [E c ] = 0, i.e. n [ E] = 0, ɛ abc n b [H c ] = 0, i.e. n [ H] = 0, n a [D a ] = 0, i.e. n [ D] = 0, n a [B a ] = 0, i.e. n [ B] = 0. [ ]=value in one region minus value in other region as interface approached. n a =surface normal (surf. 1-form).

17 Part 2: jump. Fundamental/additional Interface: have following jump (boundary), ɛ abc n b [E c ] = 0, i.e. n [ E] = 0, ɛ abc n b [H c ] = 0, i.e. n [ H] = 0, n a [D a ] = 0, i.e. n [ D] = 0, n a [B a ] = 0, i.e. n [ B] = 0. [ ]=value in one region minus value in other region as interface approached. n a =surface normal (surf. 1-form). Above, fundamental jump : obtained from Maxwell s equations, thus valid for any two materials. Considered dielectrics and static interface, for simplicity.

18 Part 2: jump. Fundamental/additional Interface: have following jump (boundary), ɛ abc n b [E c ] = 0, i.e. n [ E] = 0, ɛ abc n b [H c ] = 0, i.e. n [ H] = 0, n a [D a ] = 0, i.e. n [ D] = 0, n a [B a ] = 0, i.e. n [ B] = 0. [ ]=value in one region minus value in other region as interface approached. n a =surface normal (surf. 1-form). Above, fundamental jump : obtained from Maxwell s equations, thus valid for any two materials. Considered dielectrics and static interface, for simplicity. There exist additional jump : fulfilled for specific choices of materials. Hence, not fundamental.

19 Part 2: jump. Fundamental/additional Interface: have following jump (boundary), ɛ abc n b [E c ] = 0, i.e. n [ E] = 0, ɛ abc n b [H c ] = 0, i.e. n [ H] = 0, n a [D a ] = 0, i.e. n [ D] = 0, n a [B a ] = 0, i.e. n [ B] = 0. [ ]=value in one region minus value in other region as interface approached. n a =surface normal (surf. 1-form). Above, fundamental jump : obtained from Maxwell s equations, thus valid for any two materials. Considered dielectrics and static interface, for simplicity. There exist additional jump : fulfilled for specific choices of materials. Hence, not fundamental. Examples: perfect electromagnetic conductor (PEMC) and DB jump. Have useful applications...

20 fo an is caused by the external static magnetic field. a ch Perfect electromagnetic conductor (PEMC) jump condition See: Lindell & Sihvola (2005). Further details: ask during Q&A. Fig. 6. Another view of the material described in Fig. 5. can satisfy these requirements: the gyrotropy parameter can be tuned with the external magnetic field and the resonance condition with the physical thickness. It is obvious that there is a sufficient number of parameters for the design of the macroscopic material parameters: 1) permittivity of the dielectric host ; 2) permeability of the magnetic posts ; 3) diameter of the posts ; 4) distance between the posts ; 5) distance between the wires ; 6) thickness of the layer ; 7) external magnetic field amplitude. It is outside the scope of this paper to make any numerical estimations for the parameters required for a practical structure, which would require a considerable amount of extensive computing. A corresponding guiding structure can also be added to the Tellegen medium of Section II, to ensure TEM propagation in At interface with vacuum, given by ɛ abc n b (H c + αe c ) = 0, i.e. n ( H + α E) = 0. The jump operator [ ] is not used. Obtain such PEMC jump condition with a pure-axion medium. In 4D, χ αβµν = αɛ αβµν. ɛ αβµν is Levi-Civita symbol. In 3D, D a = αb a, H a = αe a. Reflected light fully cross-polarised. Can build device: twist polariser. Can realise PEMC condition with metamaterial layer at interface.

21 Perfect electromagnetic conductor (PEMC) jump condition A pure-axion medium has no.

22 The DB jump and radar invisibility Target: a sphere The DB jump at interface with vacuum are: n a D a = 0, n a B a = 0. DB at surface of highly symmetric object give invisibility to the monostatic radar (Lindell et al. 2009).

23 Skewon-axion media and DB jump All skewon-axion media, (1) χ αβµν =0, can be written as: χ αβµν = ɛ αβρ[µ /S ν] ρ ɛ µνρ[α /S β] ρ + αɛ αβµν = 15 1, with /S ρ ρ = 0 & square brackets denoting antisymmetry.

24 Skewon-axion media and DB jump All skewon-axion media, (1) χ αβµν =0, can be written as: χ αβµν = ɛ αβρ[µ /S ν] ρ ɛ µνρ[α /S β] ρ + αɛ αβµν = 15 1, with /S ρ ρ = 0 & square brackets denoting antisymmetry. Nieves and Pal (1989): isotropic skewon-axion medium, D a = ( s + α)b a, H a = ( s α)e a. Link to general 4D law via /S α β =(s/2)diag( 3, 1, 1, 1). See Hehl and Obukhov (2003) for a detailed treatment.

25 Skewon-axion media and DB jump All skewon-axion media, (1) χ αβµν =0, can be written as: χ αβµν = ɛ αβρ[µ /S ν] ρ ɛ µνρ[α /S β] ρ + αɛ αβµν = 15 1, with /S ρ ρ = 0 & square brackets denoting antisymmetry. Nieves and Pal (1989): isotropic skewon-axion medium, D a = ( s + α)b a, H a = ( s α)e a. Link to general 4D law via /S α β =(s/2)diag( 3, 1, 1, 1). See Hehl and Obukhov (2003) for a detailed treatment. Isotropic skewon-axion medium DB jump.

26 Skewon-axion media and DB jump All skewon-axion media, (1) χ αβµν =0, can be written as: χ αβµν = ɛ αβρ[µ /S ν] ρ ɛ µνρ[α /S β] ρ + αɛ αβµν = 15 1, with /S ρ ρ = 0 & square brackets denoting antisymmetry. Nieves and Pal (1989): isotropic skewon-axion medium, D a = ( s + α)b a, H a = ( s α)e a. Link to general 4D law via /S α β =(s/2)diag( 3, 1, 1, 1). See Hehl and Obukhov (2003) for a detailed treatment. Isotropic skewon-axion medium DB jump. Skewon-axion media have no.

27 P-media and DB jump P-medium constructed by straightforward P-roduct, as: χ αβµν = Y ɛ αβρσ P ρ µ Pσ ν. As a specific example, consider uniaxial P-medium. Set: P β α = (ψ P )u α u β + p u α n β + π n α u β + (P P )n α n β + P δ β α. u α is the medium 4-velocity and n α is the preferred axis: u α u α = 1, u α n α = 0, n α n α = +1. (To make example premetric, need extra quantities.) Uniaxial P-medium as formulated in its rest frame, 3D: D a = ε ɛ abc n b E c + α n a n b B b + α (δ a b na n b )B b, H a = µ 1 ˆɛ abcn b B c + β n a n b E b + β (δ b a n a n b )E b.

28 P-media and DB jump (continued) Skewon: permittivity & permeability solenoidal. Aside, ε = Y π P, µ 1 = Yp P, α = YP 2, β = Y ψp Yp π, α = YP P, β = Y ψp.

29 P-media and DB jump (continued) Skewon: permittivity & permeability solenoidal. Aside, ε = Y π P, µ 1 = Yp P, α = YP 2, β = Y ψp Yp π, α = YP P, β = Y ψp. If n a is orthogonal to interface, uniaxial P-medium gives DB. Even if it has a non-zero principal part.

30 P-media and DB jump (continued) Skewon: permittivity & permeability solenoidal. Aside, ε = Y π P, µ 1 = Yp P, α = YP 2, β = Y ψp Yp π, α = YP P, β = Y ψp. If n a is orthogonal to interface, uniaxial P-medium gives DB. Even if it has a non-zero principal part. All P-media have no.

31 P-media and DB jump (continued) Skewon: permittivity & permeability solenoidal. Aside, ε = Y π P, µ 1 = Yp P, α = YP 2, β = Y ψp Yp π, α = YP P, β = Y ψp. If n a is orthogonal to interface, uniaxial P-medium gives DB. Even if it has a non-zero principal part. All P-media have no. P-media in detail: Lindell, Bergamin & Favaro (2011).

32 P-media and DB jump (continued) Skewon: permittivity & permeability solenoidal. Aside, ε = Y π P, µ 1 = Yp P, α = YP 2, β = Y ψp Yp π, α = YP P, β = Y ψp. If n a is orthogonal to interface, uniaxial P-medium gives DB. Even if it has a non-zero principal part. All P-media have no. P-media in detail: Lindell, Bergamin & Favaro (2011). Above, seen that technologically useful PEMC and DB jump are closely related to media with no Fresnel eq. Identify all materials with this property!

33 All local with no? We found strong evidence that there exist three types of local linear materials whose is satisfied for every wave-covector, G(q)=0 for all q α =( ω, k i ). Namely, 1. bivector bivector + bivector bivector + axion part, χ αβµν = A αβ B µν + C αβ D µν + αɛ αβµν ; 2. skewon part + axion part [a.k.a. skewon-axion media], ( ) χ αβµν = ɛ αβρ[µ /S ν] ρ ɛ µνρ[α β] /S ρ + αɛ αβµν ; 3. every P-medium + axion part [a.k.a. P-axion media], ( ) χ αβµν = ɛ αβρσ P ρ µ Pσ ν + αɛ αβµν. Our derivation has some weak points [Lindell and Favaro, Prog. Electromagn. Res. B, vol. 51, pp , 2013].

34 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically.

35 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically. Represent the medium tensor in an alternative way, as κ µν αβ = 1 2ˆɛ αβρσχ ρσµν. Inverse of this linear map is κ 1.

36 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically. Represent the medium tensor in an alternative way, as κ µν αβ = 1 2ˆɛ αβρσχ ρσµν. Inverse of this linear map is κ 1. A response tensor κ of full-rank and its inverse κ 1 have the same, see Dahl (2012).

37 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically. Represent the medium tensor in an alternative way, as κ µν αβ = 1 2ˆɛ αβρσχ ρσµν. Inverse of this linear map is κ 1. A response tensor κ of full-rank and its inverse κ 1 have the same, see Dahl (2012). Hence, if a medium tensor κ of full-rank has no Fresnel equation, its inverse κ 1 has no too.

38 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically. Represent the medium tensor in an alternative way, as κ µν αβ = 1 2ˆɛ αβρσχ ρσµν. Inverse of this linear map is κ 1. A response tensor κ of full-rank and its inverse κ 1 have the same, see Dahl (2012). Hence, if a medium tensor κ of full-rank has no Fresnel equation, its inverse κ 1 has no too. But, taking inverse yields no new classes of materials: material κ bivector pairs + axion skewon-axion specific P-axion general P-axion inverse κ 1 bivector pairs + axion specific P-axion skewon-axion general P-axion

39 All media with no Fresnel? It is likely, because... Argument below suggests we have found all local linear materials whose is satisfied identically. Represent the medium tensor in an alternative way, as κ µν αβ = 1 2ˆɛ αβρσχ ρσµν. Inverse of this linear map is κ 1. A response tensor κ of full-rank and its inverse κ 1 have the same, see Dahl (2012). Hence, if a medium tensor κ of full-rank has no Fresnel equation, its inverse κ 1 has no too. But, taking inverse yields no new classes of materials: material κ bivector pairs + axion skewon-axion specific P-axion general P-axion inverse κ 1 bivector pairs + axion specific P-axion skewon-axion general P-axion

40 a) At an interface with vacuum... PEMC jump condition: build a twist polariser. DB: achieve invisibility to radar. b) PEMC jump condition: from pure-axion medium. DB: from e.g. isotropic skewon-axion or uniaxial P-medium c) These are all media with no. Study local linear materials s.t. G(q)=0 is satisfied for all q α. d) Likely that all materials with no Fresnel are of 3 types: bivector pairs plus axion, skewon-axion and P-axion. e) In support of this finding, taking the inverse does not produce new classes of local completeness.

41 Thank-you!