Laser transparency and gain saturation

Question

I learned that a laser system, like a conventional solid-state material inside a Fabry-Perot cavity, achieves lasing once the threshold condition is met, in this case through population inversion. Then, I learned that at (or after?) the lasing threshold, transparency occurs where gain is equal to zero (g = 0).

(1) My question is: What is the difference between transparency and saturation of a laser?

My understanding is that after steady-state operation is achieved, the laser media achieves transparency, where the net gain (gain = loss) is zero and the gain coefficient is pinned to the threshold value.

Would it be correct to think of transparency as the same as saturation then? My idea of saturation is that no more gain can be achieved, even if you have external factors such as increased electrical or optical pumping.

I am confused about the distinction in parameters such as transparency and transparency carrier density with other parameters such as saturation intensity or saturation photon density. Would it be correct to also think of saturation intensity as "transparency intensity" and transparency carrier density as "saturation carrier density"?

I feel like there may be a difference, since in the homogeneous broadening of laser operation, the gain saturation does not equal zero.

Thank you!

Answer 1

In the beginning, let's agree on conventions.

First, lasing threshold occurs when power losses for light during its round trip in your cavity are balanced by the gain introduced with a gain medium. That means $g l = 1$, where $g$ is a total gain coefficient that can only be bigger or equal to $1$ and $l$ is a total loss coefficient between $1$ and $0$.

Second, one can say that a medium is transparent when it does not introduce any change to the intensity of your light. Generally speaking, $g_{\text{medium}}$ can be any positive number because your medium can introduce not only gain but also losses due to light absorption or scattering. The term transparency is often used in the case of a gain medium for laser amplifiers. Sometimes those can be operated in an "idle" mode, when the output power equals to the seeding power, so the introduced losses are compensated. In that case $g_{\text{medium}} = 1$. However, there is a second case when that coefficient can be equal to $1$. It is the case of saturation.

Gain saturation comes from the fact that a gain medium stores a finite amount of energy. You can imagine it like this: let's say that you have available 10 atoms that can participate in the act of stimulated emission. If at that moment through the medium are passing hundreds of photons, the resulting gain is negligible.

A formula used to describe the relationship between gain $g$ and input power $P$ is:

$$ g = \frac{g_\text{ss}}{1 + \frac{P}{P_\text{sat}}}, $$

where $g_\text{ss}$ is a small signal gain and the $P_\text{sat}$ is the saturation power. You can see here that the $P_\text{sat}$ is defined as a power for which gain is reduced by half.

And finally: the role of gain saturation in laser operation. When you reached lasing threshold for the initial small intra-cavity power, the condition $g_\text{ss}l = 1$ is met. Then if you crank up your gain to $g'_\text{ss} > g_\text{ss}$ (for example with a more intense pumping) the power will start to build up inside the cavity, but because of the aforementioned formula, the gain will simultaneously start to decrease. The power will increase until the equilibrium $g(P, g'_\text{ss})l = 1$ is reached again.