Fiber lasers can be operated over a wide variety of wavelengths, from 0.45 to 3.5 $\mu{m}$, with the lasing wavelength determined primarily by the dopant ion. However, the glass host medium where the doping is located plays an important role as well. The most common host medium is silica, but in cases where silica cannot be used, ZBLAN fibers are used. ZBLAN fibers may also be called fluorozirconate, or fluoride, fibers when the glass composition is dominated by zirconium. The type of host medium affects the fluorescence lifetime $T_1$, the dipole relaxation time $T_2$, the absorption and emission cross sections, and the shape and bandwidth of the gain spectrum.

**Table of Contents:**

**PUMPING AND OPTICAL GAIN**

There are two main types of pumping schemes: three- and four-level pumping. There is also a third type of pumping process, known as up-conversion, or up-conversion lasers. A schematic for the pumping scheme of up-conversion is shown in the figure below.

As can be seen in the schematic, a photon corresponding to 1140 nm in energy is absorbed from a pumping source. Further absorption then causes “up-conversion” of the photon to the highest energy level, where it then emits as a single photon at 480 nm. This process of multiple absorptions can occur in the presence of a high-intensity laser. Notice also in this figure that for the case of Thulium in ZBLAN, one pump laser can be used for all three transitions, but in other cases multiple pumping sources may need to be used.

As a sneak peak into the next section for Yb-doped lasers, the following image shows the pumping scheme for Yb:Silica fiber lasers.

As can be seen in the figure, we have a quasi-three level pumping scheme for Yb:Silica gain media. What this means is, is that at thermal equilibrium, there is a significant portion of elections in the lower laser level because it is very close to the ground state. As such, there can be re-absorption of laser photons, which means that there needs to be higher pumping intensity in order to reach pump transparency. Transparency means that the photons are not reabsorbed. There will be more discussion of this in the next section on “Laser Amplifiers”!

**LASER THRESHOLD**

The two defining characteristics of a laser are the threshold pump power, or the power required to achieve lasing, and the efficiency of pump power conversion into laser output power. This second relation is oftentimes called the slope efficiency because on a plot of the output power vs pump, a slope efficiency of, say, 80% means that the output power of the laser is 80% of the input pump power. Different factors may lower the efficiency, but we will be talking about that more in depth in the proceeding section of this page. First we will discuss the laser threshold condition. The threshold condition is reached when the total gain in a cavity equals the cavity loss. The total cavity loss is influenced by the types of mirrors used in the resonator (how “leaky” they are) as well as any internal cavity losses due to absorption, thermal effects, etc. If we consider our cavity to be a Fabry-Perot (described in the “Classical Optics” section under “Interferometry”), then we can consider two mirrors of reflectivities “$R_1$” and “$R_2$” that are placed at either end of a fiber of length “L”. The threshold condition corresponding to this case is then:

\begin{equation} G^2R_1R_2 = e^{2\alpha_{int}L} \label{thres}\end{equation}

where “G” is the single-pass gain and “$\alpha_{int}$” is the internal cavity losses mentioned above. We can derive from the fact that the gain is reduced by a factor of $R_1$ and $R_2$ each time the beam of light is reflected by the mirrors, and the exponential term is effectively a loss term that accounts for the total loss through the round-trip length (2L) that would modulate and decrease the total amplitude of the wave traveling through the cavity. The single-pass gain “G” can be found as:

\begin{equation} G = e^{\int_0^Lg(z)dz} \label{g_z}\end{equation}

where the local gain coefficient “g(z)” is given as:

\begin{equation} g(z) = \sigma_s[N_2(z)-N_1(z)] \end{equation}

where “$\sigma_s$” is the transition cross section and “$N_1$” and “$N_2$” are the ion densities for the upper and lower energy states that are being used for the laser transition. By substituting Equation \eqref{g_z} into Equation \eqref{thres}, we have:

\begin{equation} \frac{1}{L}\int_0^Lg(z)dz = \alpha_{mir} + \alpha_{int} = \alpha_{cav} \label{thres_new}\end{equation}

where $\alpha_{mir} = -ln(R_1R_2)/2L$ is the effective mirror loss and $\alpha_{cav}$ is the total cavity loss. Due to the fact that the inversion $N_2(z)-N_1(z)$ and thus the gain changes at each value of “z”, Equation \eqref{thres_new} generally needs to be solved numerically such that each step in “z” can be calculated separately. However, we can use an approximate method of solving here in order to demonstrate generally how the gain would evolve.

As we know from the “Lasers” section, the inversion depends on the pumping strength (i.e. what power, wavelengths, cross sections, etc., are used for the pump). In the interest of space, we won’t go through the derivation again of the rate equations for the three-level laser system, but we will use the rate equation for the upper energy level for a four-level system. In this case for four-level systems, we can consider the lower energy level $N_1 \approx 0$, since the ground state would be approximately empty in the case of population inversion. We can also consider the lower energy level $N_2 << N_t$, since for the four-level system, the lower energy level necessarily needs to empty quickly in order to maintain population inversion. Note that this notation of the ground state being labeled as “$N_1$” is different from Siegman’s text, where the ground state is labeled as “$N_0$”. Now, the transition rate for the second energy level is given as:

\begin{equation} \frac{\partial{N_2}}{\partial{t}} = W_pN_t – W_sN_2 – \frac{N_2}{T_1} \label{N_2}\end{equation}

where the transition rates $W_p$ and $W_s$ are related to the pump (p=pump) and laser (s=signal) powers via the relations:

\begin{equation} W_p = \frac{\sigma_pP_p}{a_ph\nu_p} \end{equation}

\begin{equation} W_s = \frac{\sigma_sP_s}{a_sh\nu_s} \end{equation}

Then, by setting the rate of change $\frac{\partial N_2}{\partial{t}} = 0$ for the steady state case, the solution to \eqref{N_2} is given as:

\begin{equation} N_2 = \frac{(P_p/P_p^{sat})N_t}{1+P_s/P_s^{sat}} \end{equation}

where the saturation powers (derived in the “Lasers” section) are given as:

\begin{equation} P_p^{sat} = \frac{a_ph\nu_p}{\sigma_pT_1} \end{equation}

\begin{equation} P_s^{sat} = \frac{a_sh\nu_s}{\sigma_sT_1} \end{equation}

FINISH THIS SECTION

**Output Power and Slope Efficiency**

The total amount of output power can be obtained from the threshold condition since the saturated gain remains clamped to the threshold value once the pump power exceeds the threshold. We then have:

\begin{equation} \frac{\alpha_s}{L}\int^L_0\frac{P_p/P_p^{sat}}{1+P_s/P_s^{sat}}dz = \alpha_{cav} \end{equation}

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The output power from the mirror of reflectivity $R_1$ is written as:

\begin{equation} P_{out} = (1-R_1)P_s = \eta_s(P_{abs}-P_{th}) \end{equation}