Constitutive Relations in Chiral Media

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2 Constitutive Relations in Chiral Media Covariance and Chirality Coefficients in Biisotropic Materials Roger Scott Montana State University, Department of Physics March 2 nd, 2010

3 Optical Activity Polarization Rotation - Observed early 19 th century - Independent of wave-vector orientation - Independent of linear polarization Resolved though Biisotropic Constitutive Relations - Consistent with treatment of sub-wavelength chiral objects - Constrained by Covariance Requirements

4 Example of Chiral Object

5 Induced Dipole Moments Direct Dependencies p = 1 2 l ldlλ and m = l r 2 dl I

6 Induced Dipole Moments Direct Dependencies p = 1 2 Specific Case - Solenoid l ldlλ and m = l r 2 dl I p 1 2ẑ h hdh λ h = ẑnπr h hdh λ l m ı πr r I ı = (±)ẑnπr 2 h dh I l

7 Dipole Interdependence Inspection of Magnetic Dipole m z = (±)nπr 2 h dh I l

8 Dipole Interdependence Inspection of Magnetic Dipole m z = (±)nπr 2 h dh I l = (±)nπr 2 h [d(hi l) h I l h dh] = ( )nπr 2 l h h h I l l dh = (±)nπr 2 2nπr h hdh λ l t = (±)nπr 2 2 t (nπr h hdhλ l)

9 Dipole Interdependence Inspection of Magnetic Dipole m z = (±)nπr 2 h dh I l = (±)nπr 2 h [d(hi l) h I l h dh] = ( )nπr 2 l h h h I l l dh = (±)nπr 2 2nπr h hdh λ l t = (±)nπr 2 2 t (nπr h hdhλ l) Dipole Coupling m = (±)2nπr 2 t p harmonic case m = (±)2nπr 2 ıω p

10 Constitutive Relations Polarization Vectors p =γ pe E ẑ +γ pb B ẑ m =γ mb Ḃ ẑ +γ me Ė ẑ

11 Constitutive Relations Polarization Vectors p =γ pe E ẑ +γ pb B ẑ m =γ mb Ḃ ẑ +γ me Ė ẑ P =ǫ o {χ e E +χ eb B} M = 1 µ o {χ b B +χ be E}

12 Constitutive Relations Polarization Vectors p =γ pe E ẑ +γ pb B ẑ m =γ mb Ḃ ẑ +γ me Ė ẑ P =ǫ o {χ e E +χ eb B} M = 1 µ o {χ b B +χ be E} Example Cases D =ǫe +ξ db B H = 1 µ B +ξ hee } with ξ db =ξ he =ξ

13 Constitutive Relations Polarization Vectors p =γ pe E ẑ +γ pb B ẑ m =γ mb Ḃ ẑ +γ me Ė ẑ P =ǫ o {χ e E +χ eb B} M = 1 µ o {χ b B +χ be E} Example Cases D =ǫe +ξ db B H = 1 µ B +ξ hee } with ξ db =ξ he =ξ General Linear Form D =ǫe +αb H = 1 µ B +βe } with {α, β} unrelated

14 Maxwell s Wave Equation Source Free, Harmonic Maxwell Equations B = 0 E ıωb = 0 D = 0 H+ ıωd = 0

15 Maxwell s Wave Equation Source Free, Harmonic Maxwell Equations B = 0 E ıωb = 0 D = 0 H+ ıωd = 0 Use of Constitutive Equations ( 1 B +βe) = ıω(ǫe +αb) µ

16 Maxwell s Wave Equation Source Free, Harmonic Maxwell Equations B = 0 E ıωb = 0 D = 0 H+ ıωd = 0 Use of Constitutive Equations ( 1 B +βe) = ıω(ǫe +αb) µ Curl Wave Equation E = ıω B

17 Maxwell s Wave Equation Source Free, Harmonic Maxwell Equations B = 0 E ıωb = 0 D = 0 H+ ıωd = 0 Use of Constitutive Equations ( 1 B +βe) = ıω(ǫe +αb) µ Curl Wave Equation E = ıω B 2 E +κ 2 E +δ E = 0 κ 2 = ω2 c2, δ = ıωµ(α +β)

18 Maxwell Revisited Divergeance of D D = (ǫe +αb) E = 0

19 Maxwell Revisited Divergeance of D D = (ǫe +αb) E = 0 Curl of H H+ ıωd = ( 1 B +βe) + ıω(ǫe +αb) µ = 1 B+ ıωǫe + [β E+ ıωαb] µ B+ ıωµǫe = µ[α +β] E

20 Maxwell Revisited Divergeance of D Curl of H D = (ǫe +αb) E = 0 H+ ıωd = ( 1 B +βe) + ıω(ǫe +αb) µ = 1 B+ ıωǫe + [β E+ ıωαb] µ B+ ıωµǫe = µ[α +β] E Ambiguous Representations D =ǫe +αb D =ǫe H = 1 µ B αe H = 1 µ B for α = β

21 Four-Vector and Tensor Notation Invariance of Charge s := { ρ, J} A := { ϕ, A }

22 Four-Vector and Tensor Notation Invariance of Charge s := { ρ, J} A := { ϕ, A } Vacuum Field Tensor F µν = µ A ν ν A µ A ν = g νσ A σ

23 Four-Vector and Tensor Notation Invariance of Charge s := { ρ, J} A := { ϕ, A } Vacuum Field Tensor F µν = µ A ν ν A µ A ν = g νσ A σ Covariant Maxwell s Equations [σ F µν] = 0 and ν G µν = s µ

24 Field Tensor Elements Vacuum Field Tensor [F µν ] = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 Material Field Tensor [G µν ] = 0 D x D y D z D x 0 H z H y D y H z 0 H x D z H y H x 0

25 Field Tensor Elements Vacuum Field Tensor [F µν ] = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 Material Field Tensor [G µν ] = 0 D x D y D z D x 0 H z H y D y H z 0 H x D z H y H x 0 Covariant Constitutive Relation G σκ =χ σκµν F µν

26 Constitutive Tensor Relation General Linear Medium χ σκµν F 01 F 02 F 03 F 23 F 31 F 12 E x E y E z B x B y B z G 01 D x ǫ 11 ǫ 12 ǫ 13 α 11 α 12 α 13 G 02 D y ǫ 21 ǫ 22 ǫ 23 α 21 α 22 α 23 G 03 D z ǫ 31 ǫ 32 ǫ 33 α 31 α 32 α 33 G 23 H x β 11 β 12 β 13 ζ 11 ζ 12 ζ 13 G 31 H y β 21 β 22 β 23 ζ 21 ζ 22 ζ 23 G 12 H z β 31 β 32 β 33 ζ 31 ζ 32 ζ 33 Linear Biisotropic Medium χ σκµν F 01 F 02 F 03 F 23 F 31 F 12 E x E y E z B x B y B z G 01 D x ǫ 0 0 α 0 0 G 02 D y 0 ǫ 0 0 α 0 G 03 D z 0 0 ǫ 0 0 α G 23 H x β 0 0 ζ 0 0 G 31 H y 0 β 0 0 ζ 0 G 12 H z 0 0 β 0 0 ζ

27 Immediate Antisymmetry and the Lagrangian First Antisymmetry G σκ =χ σκµν F µν

28 Immediate Antisymmetry and the Lagrangian First Antisymmetry G σκ =χ σκµν F µν χ σκµν = χ κσµν = χ σκνµ

29 Immediate Antisymmetry and the Lagrangian First Antisymmetry G σκ =χ σκµν F µν χ σκµν = χ κσµν = χ σκνµ Lagrangian L = 1 8 χµνσκ F µν F σκ

30 Immediate Antisymmetry and the Lagrangian First Antisymmetry G σκ =χ σκµν F µν χ σκµν = χ κσµν = χ σκνµ Lagrangian L = 1 8 χµνσκ F µν F σκ Euler-Lagrange Derivitive uniform media L x λ ( A η / x λ ) = ( L ),λ = 0 A η,λ

31 Consequence of Lagrange Derivitive Computing the Lagrange Derivitive 4 L =χ µνσκ (F µνf σκ) A η,λ (A η,λ )

32 Consequence of Lagrange Derivitive Computing the Lagrange Derivitive 4 L =χ µνσκ (F µνf σκ) A η,λ (A η,λ ) =A [µ,ν] (χ µνηλ χ µνλη ) +A [σ,κ] (χ ηλσκ χ λησκ ) = F µν (χ µνηλ +χ ηλµν ) = F µν χ µνηλ + G ηλ

33 Consequence of Lagrange Derivitive Computing the Lagrange Derivitive 4 L =χ µνσκ (F µνf σκ) A η,λ (A η,λ ) =A [µ,ν] (χ µνηλ χ µνλη ) +A [σ,κ] (χ ηλσκ χ λησκ ) = F µν (χ µνηλ +χ ηλµν ) = F µν χ µνηλ + G ηλ ( L A η,λ ),λ = 0 F µν,λ χ µνηλ + G ηλ,λ = 0

34 General Symmetry Second Antisymmetry F µν,λ χ µνηλ = 0 χ ηλµν =±χ µνηλ

35 General Symmetry Second Antisymmetry F µν,λ χ µνηλ = 0 χ ηλµν =±χ µνηλ Sub-Matrix Symmetries ǫ ĳ =ǫ ji ζ kl =ζ lk α mn =±β nm

36 General Symmetry Second Antisymmetry F µν,λ χ µνηλ = 0 χ ηλµν =±χ µνηλ Sub-Matrix Symmetries ǫ ĳ =ǫ ji ζ kl =ζ lk α mn =±β nm Uniform Biisotropic Linear Media α =β = ıγ

37 General Symmetry Second Antisymmetry F µν,λ χ µνηλ = 0 χ ηλµν =±χ µνηλ Sub-Matrix Symmetries ǫ ĳ =ǫ ji ζ kl =ζ lk α mn =±β nm Uniform Biisotropic Linear Media α =β = ıγ This is the punch-line!

38 Concluding Remarks Chiral Coupling D =ǫe H = 1 µ B Coupling Coefficients D =ǫe +αb H = 1 µ B +βe α =β required for covariant theory Antisymmetric Biisotropic Media is A BooJum, You See!

39 Sources Texts 1 Jackson, J.D. : Classical Electrodynamics, Third Edition, Kritikos and Jaggard : Recent Advances in Electromagnetic Theory, Lakhtakia et al : Time-Harmonic Electromagnetic Fields in Chiral Media, Post, E. J. : Formal Structure of Electromagnetics, Shelkunoff, I.S. : Antennas: Theory and Practice, 1952 Papers 1 Jaggard et al : On Electromagnetic Waves in Chiral Media, Lakhtakia and Weiglhofer : Are Linear, Nonreciprocal, Biisotropic Media Forbidden?, Lakhtakia, A. : The Tellegen Medium is a A BooJum, You See, Tellegen, B. D. H., The gyrator, A New Electric Network Element, 1948

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