Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 8A: Wave Propagation in Chiral Media

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

(8A.2)$$\nabla \times H=\frac{\partial D}{\partial t},$$

(8A.3)$$\nabla \cdot B=0,$$

(8A.4)$$\nabla \cdot D=0.$$

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

(8A.5)$$E=\left(\widehat{x}-\text{j}\widehat{y}\right)E_0{\text{e}}^{\text{j}\left(\mathrm{\omega }t-k_{\text{c}}z\right)},$$

(8A.6)$$D=\left(\widehat{x}-\mathrm{j}\widehat{y}\right)D_0{\text{e}}^{\text{j}\left(\mathrm{\omega }t-k_{\text{c}}z\right)},$$

(8A.7)$$B=\left(\text{j}\widehat{x}+\widehat{y}\right)B_0{\text{e}}^{\text{j}\left(\mathrm{\omega }t-k_{\text{c}}z\right)},$$

(8A.8)$$H=\left(\text{j}\widehat{x}+\widehat{y}\right)H_0{\text{e}}^{\text{j}\left(\mathrm{\omega }t-k_{\text{c}}z\right)},$$

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

$$\left|\begin{array}{ccc}\hfill \widehat{x}\hfill & \hfill \widehat{y}\hfill & \hfill \widehat{z}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\text{j}{k}_{\text{c}}\hfill \\ \hfill E_0\hfill & \hfill -\text{j}{E}_0\hfill & \hfill 0\hfill \end{array}\right|=-\text{j}\mathrm{\omega }{B}_0,$$

which leads to

(8A.9)$$k_{\text{c}}{E}_0=\mathrm{\omega }{B}_0.$$

From Equation 8A.2, similarly we get

(8A.10)$$k_{\text{c}}{H}_0$$