Do EM waves have negative frequency inside negative-index materials?

[jeast]

The speed of light in a vacuum, $c$, is defined as positive.

The refractive index of a material, $n$, can be positive or negative.

The dispersion relation for light inside the material is given by

$$\omega=\frac{c}{n}|\mathbf{k}|.$$
The magnitude of the wavevector, $|\mathbf{k}|$, must be positive by definition therefore the sign of the wave frequency $\omega$ is determined solely by the refractive index $n$.

Thus if the refractive index $n$ is negative then the frequency $\omega$ is negative.

Is this reasoning correct?

 

[Dale]

The speed of light in a vacuum, $c$, is defined as positive.

The refractive index of a material, $n$, can be positive or negative.

The dispersion relation for light inside the material is given by

$$\omega=\frac{c}{n}|\mathbf{k}|.$$
The magnitude of the wavevector, $|\mathbf{k}|$, must be positive by definition therefore the sign of the wave frequency $\omega$ is determined solely by the refractive index $n$.

Thus if the refractive index $n$ is negative then the frequency $\omega$ is negative.

Is this reasoning correct?

You very often get negative frequencies even in vacuum, so sure you can have them in materials too. For example, when you take a signal at baseband and then you modulate it, with e.g. a $\cos$ carrier, the result has both positive frequency and negative frequency components.
And with a quadrature transmitter and detector you can even have a purely negative frequency signal.

 

[vanhees71]

Any local relativistic field has always wavemmodes with both positive and negative frequencies. The decomposition of free em. waves in plane-wave modes reads (in radiation gauge)
$$\vec{A}(t,\vec{x})=\sum_{\lambda \in \{1,-1\}} \int_{\mathbb{R}^3} \frac{\mathrm{d}^3 k}{(2 \pi)^3} \left [A_{\lambda}(\vec{k}) \vec{\epsilon}_{\lambda}(\vec{k}) \exp(-\mathrm{i} \omega_k t + \mathrm{i} \vec{k} \cdot \vec{x}) + A^*_{\lambda}(\vec{k}) \vec{\epsilon}_{\lambda}^*(\vec{k}) \exp[+\mathrm{i} \omega_k t -\mathrm{i} \vec{k} \cdot \vec{x} \right].$$
Here $\vec{\epsilon}_{\lambda}(\vec{k})$ are orthogonal to $\vec{k}$ and helicitity eigenstates (referring to left- and right-circular polarization) and $\omega_k=c |\vec{k}|$.

 

[DrDu]

Classically, the distinction between positive and negative frequencies makes no sense as an electromagnetic wave is always a superposition $a\exp(i(\mathbf{kr}-\omega t))+a^*\exp(-i(\mathbf{kr}-\omega t))$.

 

[jedishrfu]

It seems this is a good time to close this thread and put the discussion of negative indices and frequencies to rest.

Thank you all for contributing here.

Jedi