How does $\sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu}$ having either a positive or negative sign determine $\alpha_0$ and $\beta_0$?

Question

I am told that Maxwell's equations take the form

$$\text{curl} \ \mathbf{E} = - \mu j \omega \mathbf{H}, \ \ \ \ \ \text{curl} \ \mathbf{H} = (\sigma + \epsilon j \omega) \mathbf{E},$$

where $\sigma$ is the conductivity, the dielectric constant is $\kappa_e = \epsilon / \epsilon_0$, the magnetic permeability is $\kappa_m = \mu / \mu_0$, $j = \sqrt{-1}$, and $\omega$ is the angular frequency.

I am then told that these equations have a simple solution in the form of a plane wave, propagated along a given direction, such as the $z$-axis, so that each component of $\mathbf{E}$ and $\mathbf{H}$ varies with $t$ and $z$ with the exponential $e^{j \omega t - \gamma_0 z}$, where $\gamma_0$, the propagation constant, is given by

$$\gamma_0 = \alpha_0 + j \beta_0 = \sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu} \tag{1.1}$$

For a non-conducting medium, we see in (1.1) that $\alpha_0$ is zero, the propagation constant is imaginary, and we have propagation without attenuation, with a velocity of $\omega/\beta_0 = 1/\sqrt{\epsilon \mu}$, and wavelength $2\pi/\beta_0$:

$$\sigma = 0: \ \ \ \ \ \gamma_0 = \sqrt{-\omega^2 \epsilon \mu} = j \sqrt{\omega^2 \epsilon \mu} = j \beta_0$$

For a conducting medium, on the other hand, there is attenuation, $\alpha_0$ being the attenuation constant. I am told here that we always choose the sign of the square root in (1.1) that makes $\alpha_0$ and $\beta_0$ positive, and that they will either both be positive or both be negative, and if we wish the other square root, we indicate it by $-\gamma_0$.

I don't understand the mathematical reasoning behind this last claim. $\alpha_0 + j \beta_0$ is a complex number, and the square root $\sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu}$ can either have a positive or negative sign. But how does $\sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu}$ having either a positive or negative sign determine that $\alpha_0$ and $\beta_0$ will be either both positive or both negative? What is the mathematical reasoning behind this?

Answer 1

Squaring $\,\alpha_0 + j \beta_0 = \sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu}\,$ and identifying the real and imaginary parts:

$$ \alpha_0^2-\beta_0^2 + 2j \alpha_0\beta_0 = -\omega^2\epsilon\mu + j\,\omega\sigma\mu \;\;\implies\;\; \begin{cases} \begin{align} \alpha_0^2-\beta_0^2 &= -\omega^2\epsilon\mu \\ 2\alpha_0\beta_0 &= \omega\sigma\mu \end{align} \end{cases} $$

Since $\,\alpha_0\beta_0=\frac{1}{2} \omega\sigma\mu \gt 0\,$ it follows that $\,\alpha_0,\beta_0\,$ must have the same sign.