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Constitutive Relations in Chiral Media

Covariance and Chirality Coefficients in Biisotropic Materials

Roger Scott

Montana State University, Department of Physics

March 2nd, 2010

Optical Activity

Polarization Rotation

• Observed early 19th 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

2

• ℓ

ℓ dℓλ and

• m =
• ℓ
• r

2dℓ × I

Induced Dipole Moments

Direct Dependencies

• p = 1

2

• ℓ

ℓ dℓλ and

• m =
• ℓ
• r

2dℓ × I Specific Case - Solenoid

• p

≃ 1

2ˆ

z

• h hdh · λh

| = ˆ znπr

• h hdh · λℓ
• m

≃

ı πr

r × Iı | = (±)ˆ znπr2

h dh · Iℓ

Dipole Interdependence

Inspection of Magnetic Dipole mz = (±)nπr2

h dh · Iℓ

Dipole Interdependence

Inspection of Magnetic Dipole mz = (±)nπr2

h dh · Iℓ

| = (±)nπr2

h[d(hIℓ) − h ∂Iℓ ∂h dh]

| = (∓)nπr2 ∂ℓ

∂h

• h h ∂Iℓ

∂ℓ dh

| = (±)nπr22nπr

• h hdh ∂λℓ

∂t

| = (±)nπr22 · ∂t(nπr

• h hdhλℓ)

Dipole Interdependence

Inspection of Magnetic Dipole mz = (±)nπr2

h dh · Iℓ

| = (±)nπr2

h[d(hIℓ) − h ∂Iℓ ∂h dh]

| = (∓)nπr2 ∂ℓ

∂h

• h h ∂Iℓ

∂ℓ dh

| = (±)nπr22nπr

• h hdh ∂λℓ

∂t

| = (±)nπr22 · ∂t(nπr

• h hdhλℓ)

Dipole Coupling

• m = (±)2nπr2∂t

p − →

harmonic case

• m = (±)2nπr2˙

ıω p

Constitutive Relations

Polarization Vectors

• p = γpeE · ˆ

z + γpbB · ˆ z

• m = γmb ˙

B · ˆ z + γme ˙ E · ˆ z

Constitutive Relations

Polarization Vectors

• p = γpeE · ˆ

z + γpbB · ˆ z

• m = γmb ˙

B · ˆ z + γme ˙ E · ˆ z ⇒

• P = ǫo{χeE + χebB}
• M = − 1

µo {χbB + χbeE}

Constitutive Relations

Polarization Vectors

• p = γpeE · ˆ

z + γpbB · ˆ z

• m = γmb ˙

B · ˆ z + γme ˙ E · ˆ z ⇒

• P = ǫo{χeE + χebB}
• M = − 1

µo {χbB + χbeE}

Example Cases D = ǫE + ξdbB H = 1

µB + ξheE

• with

ξdb = ξhe = ξ

Constitutive Relations

Polarization Vectors

• p = γpeE · ˆ

z + γpbB · ˆ z

• m = γmb ˙

B · ˆ z + γme ˙ E · ˆ z ⇒

• P = ǫo{χeE + χebB}
• M = − 1

µo {χbB + χbeE}

Example Cases D = ǫE + ξdbB H = 1

µB + ξheE

• with

ξdb = ξhe = ξ General Linear Form D = ǫE + αB H = 1

µB + βE

• with

{α, β} unrelated

Maxwell’s Wave Equation

Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıωB = 0 ∇ · D = 0 ∇ × H + ˙ ıωD = 0

Maxwell’s Wave Equation

Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıωB = 0 ∇ · D = 0 ∇ × H + ˙ ıωD = 0 Use of Constitutive Equations ∇ × ( 1 µB + βE) = −˙ ıω(ǫE + αB)

Maxwell’s Wave Equation

Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıωB = 0 ∇ · D = 0 ∇ × H + ˙ ıωD = 0 Use of Constitutive Equations ∇ × ( 1 µB + βE) = −˙ ıω(ǫE + αB) Curl Wave Equation ∇ × ∇ × E =˙ ıω∇ × B

Maxwell’s Wave Equation

Source Free, Harmonic Maxwell Equations ∇ · B = 0 ∇ × E − ˙ ıωB = 0 ∇ · D = 0 ∇ × H + ˙ ıωD = 0 Use of Constitutive Equations ∇ × ( 1 µB + βE) = −˙ ıω(ǫE + αB) Curl Wave Equation ∇ × ∇ × E =˙ ıω∇ × B ⇓ ∇2E + κ2E + δ∇ × E = 0 → κ2 = ω2 c2 , δ = −˙ ıωµ(α + β)

Maxwell Revisited

Divergeance of D ∇ · D = ∇ · (ǫE + αB) ⇓ ∇ · E = 0

Maxwell Revisited

Divergeance of D ∇ · D = ∇ · (ǫE + αB) ⇓ ∇ · E = 0 Curl of H ∇ × H + ˙ ıωD = ∇ × ( 1 µB + βE) + ˙ ıω(ǫE + αB) = 1 µ∇ × B + ˙ ıωǫE + [β∇ × E + ˙ ıωαB] ⇓ ∇ × B + ˙ ıωµǫE = −µ[α + β]∇ × E

Maxwell Revisited

Divergeance of D ∇ · D = ∇ · (ǫE + αB) ⇓ ∇ · E = 0 Curl of H ∇ × H + ˙ ıωD = ∇ × ( 1 µB + βE) + ˙ ıω(ǫE + αB) = 1 µ∇ × B + ˙ ıωǫE + [β∇ × E + ˙ ıωαB] ⇓ ∇ × B + ˙ ıωµǫE = −µ[α + β]∇ × E Ambiguous Representations D = ǫE + αB H = 1

µB − αE ⇔ D

= ǫE H = 1

µB

for α = −β

Four-Vector and Tensor Notation

Invariance of Charge s :=

ρ, J}

← → A :=

ϕ, A

Four-Vector and Tensor Notation

Invariance of Charge s :=

ρ, J}

← → A :=

ϕ, A

• Vacuum Field Tensor

Fµν = ∂µAν − ∂νAµ ← → Aν = gνσAσ

Four-Vector and Tensor Notation

Invariance of Charge s :=

ρ, J}

← → A :=

ϕ, A

• Vacuum Field Tensor

Fµν = ∂µAν − ∂νAµ ← → Aν = gνσAσ Covariant Maxwell’s Equations ∂[σFµν] = 0 and ∂νGµν = sµ

Field Tensor Elements

Vacuum Field Tensor [Fµν] =

    

−Ex −Ey −Ez Ex Bz −By Ey −Bz Bx Ez By −Bx

    

Material Field Tensor [Gµν] =

    

Dx Dy Dz −Dx Hz −Hy −Dy −Hz Hx −Dz Hy −Hx

    

Field Tensor Elements

Vacuum Field Tensor [Fµν] =

    

−Ex −Ey −Ez Ex Bz −By Ey −Bz Bx Ez By −Bx

    

Material Field Tensor [Gµν] =

    

Dx Dy Dz −Dx Hz −Hy −Dy −Hz Hx −Dz Hy −Hx

    

Covariant Constitutive Relation Gσκ = χσκµνFµν

Constitutive Tensor Relation

General Linear Medium

χσκµν F01 F02 F03 F23 F31 F12 −Ex −Ey −Ez Bx By Bz G01 Dx −ǫ11 −ǫ12 −ǫ13 α11 α12 α13 G02 Dy −ǫ21 −ǫ22 −ǫ23 α21 α22 α23 G03 Dz −ǫ31 −ǫ32 −ǫ33 α31 α32 α33 G23 Hx −β11 −β12 −β13 ζ11 ζ12 ζ13 G31 Hy −β21 −β22 −β23 ζ21 ζ22 ζ23 G12 Hz −β31 −β32 −β33 ζ31 ζ32 ζ33

Linear Biisotropic Medium

χσκµν F01 F02 F03 F23 F31 F12 −Ex −Ey −Ez Bx By Bz G01 Dx −ǫ α G02 Dy −ǫ α G03 Dz −ǫ α G23 Hx −β ζ G31 Hy −β ζ G12 Hz −β ζ

Immediate Antisymmetry and the Lagrangian

First Antisymmetry Gσκ = χσκµνFµν

Immediate Antisymmetry and the Lagrangian

First Antisymmetry Gσκ = χσκµνFµν ⇒ χσκµν = −χκσµν = −χσκνµ

Immediate Antisymmetry and the Lagrangian

First Antisymmetry Gσκ = χσκµνFµν ⇒ χσκµν = −χκσµν = −χσκνµ Lagrangian L = 1 8χµνσκFµνFσκ

Immediate Antisymmetry and the Lagrangian

First Antisymmetry Gσκ = χσκµνFµν ⇒ χσκµν = −χκσµν = −χσκνµ Lagrangian L = 1 8χµνσκFµνFσκ Euler-Lagrange Derivitive

uniform media

→ ∂ ∂x λ ∂L ∂(∂Aη/∂x λ) = ( ∂L ∂Aη,λ ),λ = 0

Consequence of Lagrange Derivitive

Computing the Lagrange Derivitive 4 ∂L ∂Aη,λ = χµνσκ ∂(FµνFσκ) ∂(Aη,λ)

Consequence of Lagrange Derivitive

Computing the Lagrange Derivitive 4 ∂L ∂Aη,λ = χµνσκ ∂(FµνFσκ) ∂(Aη,λ) | = A[µ,ν](χµνηλ − χµνλη) + A[σ,κ](χηλσκ − χλησκ) | = Fµν(χµνηλ + χηλµν) | = Fµνχµνηλ + Gηλ

Consequence of Lagrange Derivitive

Computing the Lagrange Derivitive 4 ∂L ∂Aη,λ = χµνσκ ∂(FµνFσκ) ∂(Aη,λ) | = A[µ,ν](χµνηλ − χµνλη) + A[σ,κ](χηλσκ − χλησκ) | = Fµν(χµνηλ + χηλµν) | = Fµνχµνηλ + Gηλ

( ∂L ∂Aη,λ ),λ = 0 ⇒ Fµν,λχ

µνηλ + G ηλ ,λ = 0

General Symmetry

Second Antisymmetry Fµν,λχµνηλ = 0 ⇒ χηλµν = ±χµνηλ

General Symmetry

Second Antisymmetry Fµν,λχµνηλ = 0 ⇒ χηλµν = ±χµνηλ Sub-Matrix Symmetries ǫij = ǫji ζkl = ζlk αmn = ±βnm

General Symmetry

Second Antisymmetry Fµν,λχµνηλ = 0 ⇒ χηλµν = ±χµνηλ Sub-Matrix Symmetries ǫij = ǫji ζkl = ζlk αmn = ±βnm Uniform Biisotropic Linear Media

α = β = ˙ ıγ

General Symmetry

Second Antisymmetry Fµν,λχµνηλ = 0 ⇒ χηλµν = ±χµνηλ Sub-Matrix Symmetries ǫij = ǫji ζkl = ζlk αmn = ±βnm Uniform Biisotropic Linear Media

α = β = ˙ ıγ ← − ← − ← −

This is the punch-line!

Concluding Remarks

Chiral Coupling D = ǫE H = 1

µB

− → D = ǫE + αB H = 1

µB + βE

Coupling Coefficients α = β

required for covariant theory

Antisymmetric Biisotropic Media is “A BooJum, You See!”

Sources

Texts

1 Jackson, J.D. : Classical Electrodynamics, Third Edition, 1999 2 Kritikos and Jaggard : Recent Advances in Electromagnetic

Theory, 1990

3 Lakhtakia et al : Time-Harmonic Electromagnetic Fields in

Chiral Media, 1989

4 Post, E. J. : Formal Structure of Electromagnetics, 1962 5 Shelkunoff, I.S. : Antennas: Theory and Practice, 1952

Papers

1 Jaggard et al : “On Electromagnetic Waves in Chiral Media”,

1978

2 Lakhtakia and Weiglhofer : “Are Linear, Nonreciprocal,

Biisotropic Media Forbidden?”, 1994

3 Lakhtakia, A. : “The Tellegen Medium is a ‘A BooJum, You

See”’, 1994

4 Tellegen, B. D. H., “The gyrator, A New Electric Network

Element”, 1948