Appendix 8A: Wave Propagation in Chiral Media

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It can be shown that the R and L waves are the normal modes of propagation in a chiral medium.

A mode is said to be a normal mode of propagation if the state of polarization of the wave is unaltered as it propagates. In a sourceless chiral medium, Maxwell’s equations are

(8A.1)

\nabla \times E = - \frac{\partial B}{\partial t},

$\nabla \times E = - \frac{\partial B}{\partial t},$

(8A.2)

\nabla \times H = \frac{\partial D}{\partial t},

$\nabla \times H = \frac{\partial D}{\partial t},$

(8A.3)

\nabla \cdot B = 0,

$\nabla \cdot B = 0,$

(8A.4)

\nabla \cdot D = 0.

$\nabla \cdot D = 0.$

Let us investigate the propagation of an R wave in such a medium. Let

(8A.5)

E = (\hat{x} - j \hat{y}) E_{0} e^{j (ω t - k_{c} z)},

$E = (\hat{x} - j \hat{y}) E_{0} e^{j (ω t - k_{c} z)},$

(8A.6)

D = (\hat{x} - j \hat{y}) D_{0} e^{j (ω t - k_{c} z)},

$D = (\hat{x} - j \hat{y}) D_{0} e^{j (ω t - k_{c} z)},$

(8A.7)

B = (j \hat{x} + \hat{y}) B_{0} e^{j (ω t - k_{c} z)},

$B = (j \hat{x} + \hat{y}) B_{0} e^{j (ω t - k_{c} z)},$

(8A.8)

H = (j \hat{x} + \hat{y}) H_{0} e^{j (ω t - k_{c} z)},

$H = (j \hat{x} + \hat{y}) H_{0} e^{j (ω t - k_{c} z)},$

where k_c is the wave number in the chiral medium. From Equation 8A.1, we obtain

|\begin{matrix} \hat{x} & \hat{y} & \hat{z} \\ 0 & 0 & - j k_{c} \\ E_{0} & - j E_{0} & 0 \end{matrix}| = - j ω B_{0},

$|\begin{matrix} \hat{x} & \hat{y} & \hat{z} \\ 0 & 0 & - j k_{c} \\ E_{0} & - j E_{0} & 0 \end{matrix}| = - j ω B_{0},$

which leads to

(8A.9)

k_{c} E_{0} = ω B_{0} .

$k_{c} E_{0} = ω B_{0} .$

From Equation 8A.2, similarly we get

(8A.10)

k_{c} H_{0} = ω D_{0} .

$k_{c} H_{0} = ω D_{0} .$

From the constitutive relations for the chiral medium given by Equations 8.174 and 8.175, on substitution of Equation 8A.5 into Equation 8A.8, we obtain

(8A.11)

D_{0} = ε E_{0} + ξ_{c} B_{0},

$D_{0} = ε E_{0} + ξ_{c} B_{0},$

(8A.12)

H_{0} = - ξ_{c} E_{0} + \frac{B_{0}}{μ} .

$H_{0} = - ξ_{c} E_{0} + \frac{B_{0}}{μ} .$

From Equation 8A.10, Equation 8A.11 and Equation 8A.12, we can obtain a relation between E₀ and B₀:

(8A.13)

(- k_{c} ξ_{c} - ε ω) E_{0} + (\frac{k_{c}}{μ} - ω ξ_{c}) B_{0} = 0.

$(- k_{c} ξ_{c} - ε ω) E_{0} + (\frac{k_{c}}{μ} - ω ξ_{c}) B_{0} = 0.$

Equations 8A.13 and 8A.9 may be arranged in the matrix form as

(8A.14)

[\begin{matrix} - k_{c} ξ_{c} - ω ε & \frac{k_{c}}{μ} - ω ξ_{c} \\ k_{c} & - ω \end{matrix}] [\begin{matrix} E_{0} \\ B_{0} \end{matrix}] = 0.

$[\begin{matrix} - k_{c} ξ_{c} - ω ε & \frac{k_{c}}{μ} - ω ξ_{c} \\ k_{c} & - ω \end{matrix}] [\begin{matrix} E_{0} \\ B_{0} \end{matrix}] = 0.$

This set of homogeneous equations has a nontrivial solution only when the determinant of the matrix is zero, giving rise to the following equation:

(8A.15)

k_{c}^{2} - 2 ω μ ξ_{c} k_{c} - k^{2} = 0.

$k_{c}^{2} - 2 ω μ ξ_{c} k_{c} - k^{2} = 0.$

The wave number k_cR of this R wave is thus given by

(8A.16)

k_{c R} = ω μ ξ_{c} + {[k^{2} + {(ω μ ξ_{c})}^{2}]}^{1 / 2} .

$k_{c R} = ω μ ξ_{c} + {[k^{2} + {(ω μ ξ_{c})}^{2}]}^{1 / 2} .$

It can be further shown that the wave impedance is

(8A.17)

η_{c} = \frac{E_{0}}{H_{0}} = {[\frac{μ}{ε + μ ξ_{c}^{2}}]}^{1 / 2} .

$η_{c} = \frac{E_{0}}{H_{0}} = {[\frac{μ}{ε + μ ξ_{c}^{2}}]}^{1 / 2} .$

For an L wave, the wave number is given by

(8A.18)

k_{c L} = - ω μ ξ_{c} + {[k^{2} + {(ω μ ξ_{c})}^{2}]}^{1 / 2}

$k_{c L} = - ω μ ξ_{c} + {[k^{2} + {(ω μ ξ_{c})}^{2}]}^{1 / 2}$

and the wave (characteristic) impedance is still given by Equation 8A.17.

Appendix 11A discusses Faraday rotation in a magnetoplasma and compares it with the natural rotation in a chiral medium.

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Table of Contents for Appendix 8A: Wave Propagation in Chiral Media

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Table of Contents for
Appendix 8A: Wave Propagation in Chiral Media