# Understanding the equation $r_1 r_2 e^{i2kL + (G - \alpha)L} = 1$

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.

The first part of the exponent, $$ikL$$ is the oscillatory part of the expression. Basically, the requirement that $$2kL = 2M\pi$$ says that an integral number of waves need to fit into the laser cavity.
The other part of the exponent contributes to the amplitude or absolute value of the expression. The gain is equal to that amplitude: $$r_1 r_2 e^{(G-\alpha) L}$$. If the amplitude is less than 1, then the power of the light decreases with each round trip between the mirrors and there is no net gain and therefore no lasing. If the amplitude is greater than 1, then there is net gain and the system lases.
• Thanks for the answer. Can you please elaborate on how $2kL = 2M\pi$ implies that an integral number of waves need to fit into the laser cavity? – The Pointer Feb 14 '20 at 5:17
• Sure. The left side of Equation (1.2.1) can only be real-valued if the imaginary part of the exponent is equal to $2M\pi$. If the exponent has any other value, the left side will have a complex value. – S. McGrew Feb 14 '20 at 5:59
• Ahh, yes, because having the imaginary part of the exponent equal to $2M\pi$ means that the $i\sin(2M\pi)$ term equals $0$, right? – The Pointer Feb 14 '20 at 6:11