Please, let’s agree on this. Complex number \(z\) has real part \(a\) and imaginary part \(b\) so

\(z=a+ib\).

Complex dielectric function value \(\varepsilon\) has real part \(\varepsilon_1\), which can be positive or negative, and imaginary part \(\varepsilon_2\), which is non-negative in any transparent or absorbing material,

\(\varepsilon=\varepsilon_1+i\varepsilon_2\).

Complex refractive index \(N\) has real part \(n\), which is the real refractive index, and non-negative imaginary part \(k\), known as the absorption coefficient,

\(N=n+ik\).

In a dielectric (non-magnetic) material, \(N\) is the positive square root of \(\varepsilon\), which means that while \(\varepsilon\) covers the upper half of the \(\mathbb{C}\) plane, then refractive index \(N\) acquires values only from the first quadrant of the \(\mathbb{C}\) plane. There is, of course, the case of negative refractive index materials, which requires simultaneously negative dielectric permittivity \(\varepsilon\) and magnetic permeability \(\mu\). In such case the real refractive index \(n\) turns to be negative, but absorption coefficient \(k\) remains non-negative.
Negative values of \(k\) or \(\varepsilon_2\) would describe linear response of material amplifying the fields, which is somewhat conceivable as stimulated emission, but not realistic.
This sign convention requires the negative time harmonic term \(-i\omega t\), that is, plane wave propagation in vacuum is given by

\(\mathbf{E}=\mathbf{E_0}e^{i(\mathbf{kr}-\omega t)}\),

where \(\mathbf{k}\) is the wave-vector pointing in the direction of propagation in real space coordinates \(\mathbf{r}\) – when talking about sanity of the definitions, let’s also say that we shall work with vectors written in right-handed \(\mathbf{x,y,z}\) system. Magnitude of \(\mathbf{k}\) is given by \(|\mathbf{k}| = \omega/c\) in vacuum. With positive real angular frequency \(\omega\) and increasing time \(t\), the wave fronts indeed propagate in the direction of \(\mathbf{k}\), as desired. Here we consider the amplitude \(\mathbf{E_0}\) to be real vector and only the real part of \(\mathbf{E}\) has physical meaning. Maxwell’s equations require that \(\mathbf{E}\) and \(\mathbf{E_0}\) are perpendicular to \(\mathbf{k}\).
In conventional isotropic material with refractive index \(N=n+ik\) we shall multiply the real vacuum wave-vector \(\mathbf{k_0}\), \(|\mathbf{k_0}| = \omega/c\), by the complex index \(N=n+ik\), i.e. \(\mathbf{k} = N\mathbf{k_0}\). At this point we could be bit alarmed about appearance of complex-valued real-space vector, but I will leave the discussion for some later post and just say here that we will split the real and imaginary part of the product \(\mathbf{kr}\) in the exponent of the propagation formula written above. Then the real part produces the harmonic wavy behavior, while the imaginary part is responsible for the wave amplitude attenuation along the direction of propagation.

Why am I writing this? There are some optics textbooks that employ the positive time-harmonic convention \(i\omega t\). With that, one should change the sign of the imaginary part of the complex refractive index to keep the attenuation along propagation, but since it is conventional to have non-negative absorption coefficient, one has to define refractive index as \(N=n-ik\) in such case. Then it no longer looks like usual complex number \(z=a+ib\) and one has to keep it in mind when doing further operations, e.g. \(\varepsilon=N^2\). This is of course no disaster, but to me it seems that using the convention presented here leads to smoother theory and one less elusive minus to track later down the road.

Finally, let us define another complex quantity, the optical conductivity \(\sigma\), which relates the dimensionless dielectric function \(\varepsilon\) to physical quantity that can be (on low frequencies) directly determined with AC transport experiment,

\(\sigma=\sigma_1+i\sigma_2=-i\omega\varepsilon_0 (\varepsilon-1)\).

Here the real part \(\sigma_1\) resembles the \(\varepsilon_2\), so it is always non-negative, while imaginary \(\sigma_2\) can have any value and appears inverted with respect to the \(\varepsilon_1\).

Ok, with that being sorted out, we can proceed with the search for the Elusive Minus.