Constitutive Relations in Chiral Media Covariance and Chirality - PowerPoint PPT Presentation

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

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

Example of Chiral Object

Induced Dipole Moments Direct Dependencies p = 1 � r � � ℓ � � d ℓλ m = � 2 d ℓ × I and 2 ℓ ℓ

Induced Dipole Moments Direct Dependencies p = 1 � r � � ℓ � � d ℓλ m = � 2 d ℓ × I and 2 ℓ ℓ Specific Case - Solenoid ≃ 1 r × � p � 2 ˆ z � h h d h · λ h � ≃ � ı π r � m I ı | | zn π r 2 � = ˆ zn π r � h h d h · λ ℓ = ( ± )ˆ h d h · I ℓ

Dipole Interdependence Inspection of Magnetic Dipole = ( ± ) n π r 2 � m z h d h · I ℓ

Dipole Interdependence Inspection of Magnetic Dipole = ( ± ) n π r 2 � m z h d h · I ℓ | = ( ± ) n π r 2 � h [ d ( h I ℓ ) − h ∂ I ℓ ∂ h d h ] | h h ∂ I ℓ = ( ∓ ) n π r 2 ∂ℓ � ∂ℓ d h ∂ h | h h d h ∂λ ℓ = ( ± ) n π r 2 2 n π r � ∂ t | = ( ± ) n π r 2 2 · ∂ t ( n π r � h h d h λ ℓ )

Dipole Interdependence Inspection of Magnetic Dipole = ( ± ) n π r 2 � m z h d h · I ℓ | = ( ± ) n π r 2 � h [ d ( h I ℓ ) − h ∂ I ℓ ∂ h d h ] | h h ∂ I ℓ = ( ∓ ) n π r 2 ∂ℓ � ∂ℓ d h ∂ h | h h d h ∂λ ℓ = ( ± ) n π r 2 2 n π r � ∂ t | = ( ± ) n π r 2 2 · ∂ t ( n π r � h h d h λ ℓ ) Dipole Coupling − → m = ( ± ) 2 n π r 2 ∂ t � m = ( ± ) 2 n π r 2 ˙ � � ıω� p p harmonic case

Constitutive Relations Polarization Vectors � p = γ pe E · ˆ z + γ pb B · ˆ z m = γ mb ˙ z + γ me ˙ � B · ˆ E · ˆ z

Constitutive Relations Polarization Vectors � � p = γ pe E · ˆ z + γ pb B · ˆ z P = ǫ o { χ e E + χ eb B } ⇒ m = γ mb ˙ z + γ me ˙ � M = − 1 � B · ˆ E · ˆ z µ o { χ b B + χ be E }

Constitutive Relations Polarization Vectors � � p = γ pe E · ˆ z + γ pb B · ˆ z P = ǫ o { χ e E + χ eb B } ⇒ m = γ mb ˙ z + γ me ˙ � M = − 1 � B · ˆ E · ˆ z µ o { χ b B + χ be E } Example Cases � = ǫ E + ξ db B D with ξ db = ξ he = ξ = 1 µ B + ξ he E H

Constitutive Relations Polarization Vectors � � p = γ pe E · ˆ z + γ pb B · ˆ z P = ǫ o { χ e E + χ eb B } ⇒ m = γ mb ˙ z + γ me ˙ � M = − 1 � B · ˆ E · ˆ z µ o { χ b B + χ be E } Example Cases � = ǫ E + ξ db B D with ξ db = ξ he = ξ = 1 µ B + ξ he E H General Linear Form � D = ǫ E + α B with { α, β } unrelated = 1 µ B + β E H

Maxwell’s Wave Equation Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıω B = 0 ∇ · D = 0 ∇ × H + ˙ ıω D = 0

Maxwell’s Wave Equation Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıω B = 0 ∇ · D = 0 ∇ × H + ˙ ıω D = 0 Use of Constitutive Equations ∇ × ( 1 µ B + β E ) = − ˙ ıω ( ǫ E + α B )

Maxwell’s Wave Equation Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıω B = 0 ∇ · D = 0 ∇ × H + ˙ ıω D = 0 Use of Constitutive Equations ∇ × ( 1 µ B + β E ) = − ˙ ıω ( ǫ E + α B ) Curl Wave Equation ∇ × ∇ × E =˙ ıω ∇ × B

Maxwell’s Wave Equation Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıω B = 0 ∇ · D = 0 ∇ × H + ˙ ıω D = 0 Use of Constitutive Equations ∇ × ( 1 µ B + β E ) = − ˙ ıω ( ǫ E + α B ) Curl Wave Equation ∇ × ∇ × E =˙ ıω ∇ × B ⇓ κ 2 = ω 2 ∇ 2 E + κ 2 E + δ ∇ × E = 0 → c 2 , δ = − ˙ ıωµ ( α + β )

Maxwell Revisited Divergeance of D ∇ · D = ∇ · ( ǫ E + α B ) ⇓ ∇ · E = 0

Maxwell Revisited Divergeance of D ∇ · D = ∇ · ( ǫ E + α B ) ⇓ ∇ · E = 0 Curl of H ıω D = ∇ × ( 1 ∇ × H + ˙ µ B + β E ) + ˙ ıω ( ǫ E + α B ) = 1 µ ∇ × B + ˙ ıωǫ E + [ β ∇ × E + ˙ ıωα B ] ⇓ ∇ × B + ˙ ıωµǫ E = − µ [ α + β ] ∇ × E

Maxwell Revisited Divergeance of D ∇ · D = ∇ · ( ǫ E + α B ) ⇓ ∇ · E = 0 Curl of H ıω D = ∇ × ( 1 ∇ × H + ˙ µ B + β E ) + ˙ ıω ( ǫ E + α B ) = 1 µ ∇ × B + ˙ ıωǫ E + [ β ∇ × E + ˙ ıωα B ] ⇓ ∇ × B + ˙ ıωµǫ E = − µ [ α + β ] ∇ × E Ambiguous Representations D = ǫ E + α B µ B − α E ⇔ D = ǫ E α = − β for = 1 = 1 H H µ B

Four-Vector and Tensor Notation Invariance of Charge � ρ, J } � ϕ, A � s := ← → A :=

Four-Vector and Tensor Notation Invariance of Charge � ρ, J } � ϕ, A � s := ← → A := Vacuum Field Tensor F µν = ∂ µ A ν − ∂ ν A µ ← → A ν = g νσ A σ

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

Field Tensor Elements Vacuum Field Tensor   0 − E x − E y − E z − B y E x 0 B z   [ F µν ] =   E y − B z 0 B x     E z B y − B x 0 Material Field Tensor   0 D x D y D z − D x 0 H z − H y   [ G µν ] =   − D y − H z 0 H x     − D z − H x H y 0

Field Tensor Elements Vacuum Field Tensor   0 − E x − E y − E z − B y E x 0 B z   [ F µν ] =   E y − B z 0 B x     E z B y − B x 0 Material Field Tensor   0 D x D y D z − D x 0 H z − H y   [ G µν ] =   − D y − H z 0 H x     − D z − H x H y 0 Covariant Constitutive Relation G σκ = χ σκµν F µν

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 − β 31 − β 32 − β 33 ζ 31 ζ 32 ζ 33 G 12 H z 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

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

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

Immediate Antisymmetry and the Lagrangian First Antisymmetry G σκ = χ σκµν F µν χ σκµν = − χ κσµν = − χ σκνµ ⇒ Lagrangian L = 1 8 χ µνσκ F µν F σκ

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

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

Consequence of Lagrange Derivitive Computing the Lagrange Derivitive 4 ∂ L = χ µνσκ ∂ ( F µν F 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

Consequence of Lagrange Derivitive Computing the Lagrange Derivitive 4 ∂ L = χ µνσκ ∂ ( F µν F 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 µνηλ + G ) ,λ = 0 ⇒ F µν,λ χ ηλ ,λ = 0 ∂ A η,λ

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

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

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

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! ← −