I am currently studying the textbook Physics of Photonics Devices, Second Edition, by Shun Lien Chuang. In a section discussing The Invention of Semiconductor Lasers, the author says the following:
However, if we are able to inject enough electrons and holes into the semiconductor to reach the so-called population inversion condition, which means that there are more downward than upward stimulated transitions, there will be a net gain of the photon number or optical intensity. Gain is not the only requirement for a laser. It requires a resonator, which can be a one-, two-, or three-dimensional structure. The most common one is the Fabry-Perot resonator formed by two parallel mirrors with a cavity length $L$. The light is reflected back and forth between the two mirrors, thus a standing wave pattern can be formed for certain resonant wavelengths (Fig. 1.5a). When the round-trip gain of the optical intensity is large enough to balance the loss due to waveguide absorption and mirror transmission, a threshold condition can be reached. It means that the optical field after the round-trip propagation reaches a resonance condition with a constructive phase and an amplitude of $1$,
$$r_1 r_2 e^{i2kL + (G - \alpha)L} = 1 \tag{1.2.1}$$
where $r_1$ and $r_2$ are the reflection coefficients of the optical fields from the two end facets, $k$ is the propagation constant,
$$k = 2 \pi n / \lambda = 2 \pi v n / c, \tag{1.2.2}$$
and $n$ is the refractive index of the semiconductor. $G$ is the modal gain coefficient of the guided optical mode in the semiconductor waveguide, and $\alpha$ is the absorption coefficient. Equation (1.2.1) leads to the phase and magnitude conditions for lasing,
$$2kL = 2m \pi \tag{1.2.3}$$
$$G = \alpha + \dfrac{1}{2L} \ln\left( \dfrac{1}{R_1 R_2} \right) \tag{1.2.4}$$
where $R_1 = \vert r_1 \vert^2$ and $R_2 = \vert r_2 \vert$ are the power reflectivities. The phase condition (1.2.3) leads to the Fabry-Perot resonance spectrum
$$v_m = \dfrac{mc}{2nL} \ \ \ m = \text{integer.} \tag{1.2.5}$$
I'm trying to understand (1.2.1). The equation takes the form
$$\begin{align} r_1 r_2 e^{i2kL + (G - \alpha)L} &= 1 \\ \Rightarrow r_1 r_2 e^{i2kL} e^{(G - \alpha)L} &= 1 \end{align}$$
If I'm not mistaken, this is the phasor/analytic representation of the (harmonic) wave. Is my understanding here correct?
Assuming this is the phasor/analytic representation of a (harmonic) wave, I now want to understand the structure of this wave, in terms of its equation. According to (2.37) of Optics, fifth edition, by Hecht, the phasor/analytic representation of a harmonic wave is $\psi(x, t) = Ae^{i(\omega t - kx + \epsilon)} = Ae^{i \varphi}$, where $\varphi$ is the phase. We are told that (1.2.3) is the phase. But then what is the $e^{(G - \alpha)L}$? Given what what I've just stated about the equation of a simple harmonic wave, I'm struggling to reconcile these two equations.
And where is the "constructive phase" part of this equation? I wonder if that's what the $i2kL + (G - \alpha)L$ in $r_1 r_2 e^{i2kL + (G - \alpha)L} = r_1 r_2 e^{i2kL} e^{(G - \alpha)L}$ is, where $e^{i2kL}$ and $e^{(G - \alpha)L}$ are two waves? But there is no $i$ multiplying the $(G - \alpha)L$, so I don't see how this is a wave? So how does $e^{(G - \alpha)L}$ fit into this wave equation?
I also wonder if there are any errors here on the author's part, since that's always a possibility.
I would greatly appreciate it if people would please take the time to clarify this.
