Speed and Wavelength of Light in a Meta-Material

For light propagating in any optical medium, including an engineered meta-material where the refractive index $n$ can be negative, the physical speed of propagation $v$ is a positive scalar quantity.

By definition, the refractive index $n$ relates the speed of light in vacuum $c$ to the phase velocity of light in the medium via: $|n| = \frac{c}{v}$

Rearranging this expression for the speed of light $v$ in the meta-material gives: $v = \frac{c}{|n|}$

Furthermore, since the frequency of light $\nu$ remains invariant when light transitions from air into the meta-material, the wavelength of light in the meta-material $\lambda_m$ is related to its speed by: $\lambda_m = \frac{v}{\nu} = \frac{c}{|n|\nu} = \frac{\lambda_{\text{air}}}{|n|}$ where $\lambda_{\text{air}} = \frac{c}{\nu}$ is the wavelength of light in air (or vacuum).

Analyzing the given choices:

• Option (A) is incorrect because $v = \frac{c}{|n|}$, not $c|n|$.
• Option (B) is correct because $v = \frac{c}{|n|}$.
• Option (C) is incorrect because light changes speed in a medium where $|n| \neq 1$.
• Option (D) is incorrect because $\lambda_m = \frac{\lambda_{\text{air}}}{|n|}$, not $\lambda_{\text{air}}|n|$.

Therefore, the correct statement is given by option B.