# Rare-earth ion-doped fiber amplifiers

This is a continuation from the previous tutorial - **laser amplifiers**.

Among the many different types of optical amplifiers, those that are guided-wave devices have many advantages over bulk devices.

There are two important, but distinctly different, groups of guided-wave optical amplifiers: fiber devices and semiconductor devices.

Fiber devices can be further subdivided into two categories: those based on active rare-earth ion-doped fibers and those based on the nonlinear optical processes in fibers.

Therefore, there are three types of established guided-wave optical amplifiers:

- Rare-earth ion-doped fiber amplifiers
- Nonlinear Raman or Brillouin fiber amplifiers
- Semiconductor optical amplifiers

Each type can be made into lasers by arranging some proper optical feedback to the amplifiers.

Optical amplifiers and lasers based on polymer waveguides are also of great interest, but they are not well established yet.

Fiber amplifiers utilize the waveguiding effect of optical fibers, which are nonconductive dielectric glass materials but can be made very long.

Semiconductor optical amplifiers use semiconductor waveguides, which are conductive but are of limited length.

Fiber devices require optical pumping. Due to the fiber geometry, the only practical and efficient pumping arrangement is ** longitudinal optical pumping**. The pump beam is launched into the fiber waveguide either through a fiber coupler or through the end of the fiber.

As illustrated in Figure 10-13 [refer to the laser amplifiers tutorial], the longitudinal optical pumping arrangement can be unidirectional forward, with the pump and signal waves propagating codirectionally, as shown in Figure 10-13(d), unidirectional backward, with the pump and signal waves propagating contradirectionally, as shown in Figure 10-13(e), or bidirectional, as shown in Figure 10-13(f).

Semiconductor devices are normally pumped with electric current injection though they can be optically pumped as well.

In this section, we consider only the rare-earth ion-doped fiber amplifiers. The basic principles of Raman and Brillouin amplifiers are discussed in the Raman and Brillouin devices tutorial. Semiconductor optical amplifiers are discussed in later tutorials.

In comparison to bulk optical amplifiers, fiber amplifiers have several advantages. A few are unique to the fiber geometry, but most are common features of waveguide devices and are shared by semiconductor optical amplifiers as well. Some of the advantages are listed below.

1. **Low pump power**.

With longitudinal optical pumping, the waveguiding nature of an optical fiber keeps the pump power confined and concentrated in the active core region, allowing the pump power to be completely absorbed and utilized.

In comparison, the pump beam for a longitudinally pumped bulk amplifier cannot be kept focused over a long distance because it is subject to the diffraction limit.

In a fiber amplifier, the pump spot size is solely determined by the fiber core diameter, while the effective pumping length is determined only by the absorption coefficient of the fiber gain medium, which in turn is determined by the doping concentration of the rare-earth ions in the fiber core.

Decoupling of the pump spot size from consideration of the effective pumping length makes the design of long fiber amplifiers with low rare-earth ion-doping concentrations and low pump powers possible, which are not possible for bulk optical amplifiers.

2. **Good overlap of pump and signal waves**.

In a fiber amplifier, the pump and signal waves, though of different wavelengths, overlap over the entire length of the device due to the waveguiding effect of the fiber. This feature improves the efficiency and reduces the required pump power of a fiber amplifier.

3. **Easy control of transverse mode characteristics**.

The transverse spatial characteristics of the output beam from a fiber amplifier can be easily and precisely controlled by choosing the fiber to have the desired mode property. To have a diffraction-limited single-mode output beam, even at a very high pumping level, it is only necessary to use a single-mode fiber for the signal wavelength because a single-mode waveguide is the most effective spatial filter that automatically produces a diffraction-limited beam.

4. **Reduced thermal effects**.

The fiber geometry, with its small cross-sectional area and large length, naturally has a good efficiency for heat dissipation, thus eliminating the thermal lensing and stress problems often encountered in bulk devices at high pumping levels.

5. **Compatibility with fiber transmission systems**.

This physical compatibility leads to efficient integration of fiber amplifiers in an optical fiber transmission system, greatly reducing coupling losses. It also allows large flexibility in the design and handling of the system, including the flexibility of using different pumping arrangements.

**Rare-Earth Ion-Doped Fibers**

Fiber laser amplifiers are based on rare-earth ion-doped fibers. The majority of rare-earth ion-doped fibers are low-loss silica or fluorozirconate glass fibers doped with active rare-earth ions, such as Pr^{3+}, Nd^{3+}, Sm^{3+}, Ho^{3+}, Er^{3+}, Tm^{3+}, and Yb^{3+}.

A rare-earth ion-doped fiber can be either a three-level or a four-level gain system. In some special instances, such as Er^{3+} pumped at 1.48 μm, a rare-earth ion can even operate as a quasi-two-level system.

As example, the energy levels of praseodymium, neodymium, and erbium ions are shown in Figure 10-16. Some important optical transitions of these ions are summarized in Table 10-2.

One important property of rare-earth ion-doped fibers is that their transition characteristics, including the spectral broadening mechanism, the spectral shape and width, the spectral peak wavelength, and the fluorescence lifetime, are influenced by the molecular environment of the rare-earth ions.

The most important factor is the structure and composition of the host material. Also important is the operating temperature.

The absorption and emission spectral widths of a given ion doped in a glass fiber are generally very broad, much broader than those of the same ion doped in a crystalline material.

The spectral characteristics can be significantly varied by using a completely different glass material for the fiber or by adjusting the composition of the glass.

As an example, the absorption cross-section spectrum, \(\sigma_\text{a}(\lambda)\), and the emission cross-section spectrum, \(\sigma_\text{e}(\lambda)\), measured at room temperature in the spectral region around 1.53 μm for Er^{3+} doped in a silica fiber doped with Al_{2}O_{3} and P_{2}O_{5} are shown in Figure 10-17(a), which can be compared with those shown in Figure 10-17(b) for Er^{3+} doped in an Al_{2}O_{3}/GeO_{2}-silica fiber.

Other factors that affect the spectral characteristics and the fluorescence lifetime include the doping concentration of the active ion and the codopants. The laser emission wavelength corresponding to each laser transition shown in Table 10-2 can be varied and tuned within a rather broad range.

Because of their glass hosts, rare-earth ion-doped fibers have mixed homogeneous and inhomogeneous broadening characteristics. The relative significance between the two varies with the fiber host material, the dopant, the doping concentration, and temperature.

For many rare-earth ion-doped fibers of interest, the homogeneous line broadening at room temperature is about the same as the inhomogeneous broadening. Experimental results on fiber amplifiers seem to be adequately explained by simple models that assume pure homogeneous broadening.

**Fiber Amplifiers**

The development of rare-earth ion-doped fiber amplifiers was driven primarily by their applications in fiber-optic communication systems for amplifying weak optical signals.

For this reason, the major effort has been on the development of erbium-doped fiber amplifiers (EDFAs) for amplifying optical signals in the spectral region around 1.55 μm, where silica fiber transmission lines have minimum attenuation loss.

Also of interest are praseodymium-doped fiber amplifiers (PDFAs) and neodymium-doped fiber amplifiers (NDFAs) for the 1.3-μm spectral region, where many of the existing optical communication systems operate because of the minimum dispersion and low attenuation loss of silica fibers in this spectral window.

These fiber amplifiers can be used as power amplifiers (postamplifiers), optical repeaters (inline amplifiers), or optical preamplifiers, shown in Figure 10-18, in optical communication systems.

**Figure 10-18**. Use of a fiber amplifier as (a) a power amplifier (b) an optical repeater, and (c) an optical preamplifier in a fiber-optic communication system.

Fiber amplifiers have to be optically pumped. Practical considerations such as reliability, package size, cost, and power efficiency dictate that semiconductor lasers be used to pump fiber amplifiers in most applications.

Efficient semiconductor laser sources are available for the pump bands of Nd^{3+} and Er^{3+} listed in Table 10-2: AlGaAs/GaAs lasers for the 800-nm pump band, InGaAs/GaAs strained quantum-well lasers for the 980-nm pump band, and InGaAsP/InP lasers for the 1.48-μm pump band.

The 1.017-μm pump band of Pr^{3+} is not so conveniently located for pumping with a semiconductor laser. However, by codoping with Yb^{3+}, a praseodymium-doped fiber can be pumped at 980 nm.

An EDFA can be pumped at 800 or 980 nm, or 1.48 μm, but pumping in the 800-nm pump band is not very efficient due to a phenomenon known as ** excited-state absorption** (ESA) at 800 nm. Therefore, practical EDFAs are pumped at either 980 nm or 1.48 μm.

A silica-based NDFA operating in the 1.34-μm signal wavelength region also suffers from ESA, but due to absorption of signal photons rather than pump photons. This problem can be avoided in an NDFA that is based on a fluorozirconate glass fiber instead of a silica fiber.

It is important to recognize whether a particular fiber amplifier in a certain operating condition functions as a quasi-two-level system, a three-level system, or a four-level system because the characteristics of laser amplifiers vary significantly among the three systems.

As indicated in Table 10-2, this depends on the combination of the active ion in the fiber, the pump transition used, and the laser wavelength of interest. For example, an EDFA for the signal laser wavelength of 1.53 μm functions as a three-level system when it is pumped at 800 or 980 nm and as a quasi-two-level system when pumped at 1.48 μm, but it is never a four-level system.

Rare-earth ion-doped fiber amplifiers have the general characteristics of laser amplifiers discussed in the laser amplifiers tutorial. However, because of the waveguide structure of optical fibers, evaluation of certain parameters has to be modified when the general formulation in the laser amplifiers tutorial are applied to fiber amplifiers.

The required modification depends largely on the concentration profile of the active rare-earth ions in the fiber. In general, these ions are doped only in the core of the fiber, but they may distribute throughout the core area or reside only in a fraction of the core area.

In the following discussions, we assume that the entire fiber core is uniformly doped with active ions at a concentration of \(N_\text{t}\) but the cladding contains no active ions.

In this situation, only a fraction, quantified by the fiber mode confinement factor defined in (64) [refer to the weakly guiding fibers tutorial], of the power of an optical beam guided in the fiber interacts with the active ions.

Because the pump and the signal of a given fiber amplifier have different wavelengths, their confinement factors, \(\Gamma_\text{p}\) and \(\Gamma_\text{s}\), respectively, have different values.

With this understanding, we can find the following required modifications when applying the formulations in the laser amplifiers tutorial to a fiber amplifier:

\[\tag{10-119}\alpha_\text{p}=\Gamma_\text{p}\sigma_\text{a}^\text{p}N_\text{t}\]

\[\tag{10-120}P_\text{p}^\text{sat}=\pi{w}_\text{p}^2\frac{h\nu_\text{p}}{\Gamma_\text{p}\eta_\text{p}\tau_2\sigma_\text{a}^\text{p}}\]

\[\tag{10-121}P_\text{p}^\text{tr}=\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}P_\text{p}^\text{sat}\]

for the pump, where \(w_\text{p}\) is the effective mode radius of the pump beam, and

\[\tag{10-122}\begin{align}g_0(z)&=\frac{\Gamma_\text{s}(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}}{1+(1+p)P_\text{p}(z)/P_\text{p}^\text{sat}}\left[\frac{P_\text{p}(z)}{P_\text{p}^\text{sat}}-\frac{P_\text{p}^\text{tr}}{P_\text{p}^\text{sat}}\right]\\&=\frac{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{1+(1+p)P_\text{p}(z)/P_\text{p}^\text{sat}}\frac{P_\text{p}(z)}{P_\text{p}^\text{sat}}-\Gamma_\text{s}\sigma_\text{a}N_\text{t}\end{align}\]

\[\tag{10-123}P_\text{sat}=\pi{w}_\text{s}^2\frac{h\nu_\text{s}}{\Gamma_\text{s}\tau_\text{s}\sigma_\text{e}}\]

for the signal, where \(w_\text{s}\) is the effective mode radius of the signal beam.

For a fiber amplifier of a length \(l\), the integral of the unsaturated gain coefficient for both cases of \(p\ne0\) and \(p=0\) has the following closed-form solution:

\[\tag{10-124}\begin{align}\displaystyle\int\limits_0^lg_0(z)\text{d}z&=\Gamma_\text{s}\sigma_\text{e}N_\text{t}l+\frac{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\ln\frac{P_\text{p}^\text{out}}{P_\text{p}^\text{in}}\\&=\Gamma_\text{s}\sigma_\text{e}N_\text{t}l+\frac{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\ln(1-\xi_\text{p})\end{align}\]

Similar to the relation between (10-106) and (10-108) [refer to the laser amplifiers tutorial], this expression can be transformed into

\[\tag{10-125}\displaystyle\int\limits_0^lg_0(z)\text{d}z=\left\{\begin{array}{l}\frac{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{p\alpha_\text{p}}\ln\frac{P_\text{p}^\text{sat}+pP_\text{p}^\text{in}}{P_\text{p}^\text{sat}+pP_\text{p}^\text{out}}-\Gamma_\text{s}\sigma_\text{a}N_\text{t}l,\qquad\text{for }p\ne0\\\frac{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\left(\frac{P_\text{p}^\text{in}-P_\text{p}^\text{out}}{P_\text{p}^\text{sat}}\right)-\Gamma_\text{s}\sigma_\text{a}N_\text{t}l,\qquad\text{ for }p=0\end{array}\right.\]

**Example 10-13**

An EDFA uses a step-index Al_{2}O_{3}/GeO_{2}-silica fiber doped with an Er concentration of \(N_\text{t}=2.2\times10^{24}\text{ m}^{-3}\) in its core of \(a=4.5\text{ μm}\) radius. It is pumped in the forward direction at \(\lambda_\text{p}=1.48\text{ μm}\) to amplify a signal at \(\lambda_\text{s}=1.53\text{ μm}\). At both wavelengths, the fiber is single moded supporting only the fundamental HE_{11} mode with effective mode radii of \(w_\text{p}=4.0\text{ μm}\) and \(w_\text{s}=4.1\text{ μm}\) and confinement factors of \(\Gamma_\text{p}=0.72\) and \(\Gamma_\text{s}=0.70\) for the pump and signal, respectively. The fluorescence lifetime is \(\tau_2=10\text{ ms}\). At the pump wavelength of 1.48 μm, \(\sigma_\text{a}^\text{p}=2.2\times10^{-25}\text{ m}^2\) and \(\sigma_\text{e}^\text{p}=1.2\times10^{-26}\text{ m}^2\). At the signal wavelength of 1.53 μm, \(\sigma_\text{e}=7.9\times10^{-25}\text{ m}^2\) and \(\sigma_\text{a}=5.75\times10^{-25}\text{ m}^2\). The background absorption of the fiber at both pump and signal wavelengths are negligible. The fiber length is chosen to be \(l=20\text{ m}\), and the input pump power is \(P_\text{p}^\text{in}=20\text{ mW}\). The pumping efficiency is \(\eta_\text{p}=1\).

(a) Find \(P_\text{p}^\text{sat}\), \(P_\text{p}^\text{tr}\), and \(P_\text{sat}\).

(b) Find the unsaturated power gain \(G_0\).

(c) If the power of the input signal is \(P_\text{s}^\text{in}=1\text{ μm}\), what is the output signal power?

**(a)**

The pump photon energy is \(h\nu_\text{p}=(1.2398/1.48)\text{ eV}=0.838\text{ eV}\). The signal photon energy is \(h\nu_\text{s}=(1.2398/1.53)\text{ eV}=0.810\text{ eV}\). When pumped at \(\lambda_\text{p}=1.48\text{ μm}\), this EDFA is a quasi-two-level system with \(p=\sigma_\text{e}^\text{p}/\sigma_\text{a}^\text{p}=1.22/22=0.055\). We find, by using (10-120) and (10-121), respectively, that

\[P_\text{p}^\text{sat}=\pi{w}_\text{p}^2\frac{h\nu_\text{p}}{\Gamma_\text{p}\eta_\text{p}\tau_2\sigma_\text{a}^\text{p}}=\frac{\pi\times(4.0\times10^{-6})^2\times0.838\times1.6\times10^{-19}}{0.72\times1\times10\times10^{-3}\times2.2\times10^{-25}}\text{ W}=4.25\text{ mW}\]

and

\[P_\text{p}^\text{tr}=\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}P_\text{p}^\text{sat}=\frac{5.75\times10^{-25}}{7.9\times10^{-25}-0.055\times5.75\times10^{-25}}\times4.25\text{ mW}=3.22\text{ mW}\]

Without pumping, \(W_\text{p}\tau_2=0\), and

\[\tau_\text{s}=\tau_2\left(1+\frac{\sigma_\text{a}}{\sigma_\text{e}}\right)=\left(1+\frac{5.75}{7.9}\right)\times10\text{ ms}=17.3\text{ ms}\]

from (10-76) [refer to the population inversion and optical gain tutorial]. Therefore, from (10-123), the intrinsic saturation power is

\[P_\text{sat}=\pi{w}_\text{s}^2\frac{h\nu_\text{s}}{\Gamma_\text{s}\tau_\text{s}\sigma_\text{e}}=\frac{\pi\times(4.1\times10^{-6})^2\times0.810\times1.6\times10^{-19}}{0.70\times17.3\times10^{-3}\times7.9\times10^{-25}}\text{ W}=716\text{ μW}\]

For \(P_\text{p}^\text{in}=20\text{ mW}\), the pumping ratio is \(s=P_\text{p}^\text{in}/P_\text{p}^\text{sat}=20/4.25=4.71\). Therefore, \(W_\text{p}\tau_2=s=4.71\), and from (10-76) [refer to the population inversion and optical gain tutorial], we get

\[\tau_\text{s}=\tau_2\frac{1+\sigma_\text{a}/\sigma_\text{e}}{1+(1+p)W_\text{p}\tau_2}=\frac{1+5.75/7.9}{1+1.055\times4.71}\times10\text{ ms}=2.9\text{ ms}\]

From (10123), the saturation power for the signal at this pumping level is

\[P_\text{sat}=\pi{w}_\text{s}^2\frac{h\nu_\text{s}}{\Gamma_\text{s}\tau_\text{s}\sigma_\text{e}}=\frac{\pi\times(4.1\times10^{-6})^2\times0.810\times1.6\times10^{-19}}{0.70\times2.9\times10^{-3}\times7.9\times10^{-25}}\text{ W}=4.27\text{ mW}\]

This is the signal saturation power at the input pump power of \(P_\text{p}^\text{in}=20\text{ mW}\). As the pump power decays along the fiber due to pump absorption, \(\tau_\text{s}\) increases toward the value of 17.3 ms and, as a consequence, \(P_\text{sat}\) decreases toward its intrinsic value of 716 μW.

**(b)**

To find \(G_0\), we have to find the integral of \(g_0(z)\) over the entire length of the fiber. This integral can be evaluated by using (10-124) or, equivalently, by using (10-125).

For this fiber, from (10-119), we have

\[\alpha_\text{p}=\Gamma_\text{p}\sigma_\text{a}^\text{p}N_\text{t}=0.72\times2.2\times10^{-25}\times2.2\times10^{24}\text{ m}^{-1}=0.348\text{ m}^{-1}\]

With \(P_\text{p}^\text{sat}=4.25\text{ mW}\) found above and \(P_\text{p}(0)=P_\text{p}^\text{in}=20\text{ mW}\), we can solve (10-102) [refer to the laser amplifiers tutorial] for \(z=l=20\text{ m}\) to find that \(P_\text{p}^\text{out}=P_\text{p}(l)=984\text{ μW}\). All other parameters needed for calculating the integral in (10-124) are known. We find from (10-124) that

\[\begin{align}\displaystyle\int\limits_0^lg_0(z)\text{d}z&=0.7\times7.9\times10^{-25}\times2.2\times10^{24}\times20\\&+\frac{0.7\times(7.9+5.75)\times10^{-25}\times2.2\times10^{24}}{0.348}\times\ln\frac{0.984}{20}\\&=6.14\end{align}\]

Then, we have

\[G_0=\text{e}^\text{6.14}=464\]

which is 26.7 dB.

Although we have found \(G_0\) by evaluating the integral of \(g_0(z)\) through (10-124), it is instructive to find the distributions of the pump power \(P_\text{p}(z)\) and the unsaturated gain coefficient \(g_0(z)\) as a function of distance along the EDFA.

It can then by shown that the value of the integral of \(g_0(z)\) obtained by directly integrating over its distribution is the same as that evaluated above using (10-124) to confirm the validity of (10-124).

To find \(P_\text{p}(z)\) and \(g_0(z)\) as a function of distance along the EDFA, we first find \(P_\text{p}(z)\) through (10-102) [refer to the laser amplifiers tutorial]. We then find \(g_0(z)\) by using (10-122) and \(G_0\) as a function of \(z\) by using (10-99) [refer to the laser amplifiers tutorial].

The results are plotted in Figure 10-19.

**Figure 10-19**. (a) Pump power evolution and (b) gain variation in an EDFA. Plotted as a function of distance \(z\) along the fiber from the input of the EDFA are (a) pump power normalized to the input pump power, \(P_\text{p}(z)/P_\text{p}^\text{in}\), and exponentially decaying function \(\text{e}^{-\alpha_\text{p}z}\) for comparison with the pump power evolution and (b) unsaturated gain coefficient \(g_0(z)\) per meter and unsaturated power gain \(G_0(z)\) in decibels.

Also plotted in Figure 10-19(a) is the function \(\exp(-\alpha_\text{p}z)\) for comparison with \(P_\text{p}(z)/P_\text{p}^\text{in}\) to show that the pump power decays much more slowly than the exponential function because of the absorption saturation of the pump at the high pumping ratio of \(s=4.71\). We find from the curve in Figure 10-19(b) for \(G_0(z)\) that \(G_0=26.7\text{ dB}\), or \(G_0=464\) at \(z=l=20\text{ m}\), confirming the result obtained above.

**(c)**

From (a), we know that the signal saturation power varies along the EDFA in the range of 716 μW < \(P_\text{sat}\) < 4.27 mW.

If we take \(P_\text{sat}\) = 4.27 mW, we find that \(P_\text{s}^\text{in}/P_\text{sat}=2.34\times10^{-4}\) for \(P_\text{s}^\text{in}\) = 1 μW. Then, with \(G_0\) = 464 obtained above, we find from (10-98) [refer to the laser amplifiers tutorial] that \(G\) = 420.

If we take \(P_\text{sat}=716\text{ μW}\), we find \(G\) = 308.

The actual gain is somewhere between these two limits, about \(G\) = 364. Therefore, the output signal power is about \(P_\text{s}^\text{out}=364\text{ μW}\).

Here, we have not considered the effect of the ASE. As the ASE can be significant, thus depleting a significant portion of the population inversion, the realistic gain available for the signal could be much smaller than that estimated here.

For an accurate solution, we need to solve (10-95) [refer to the laser amplifiers tutorial] together with (10-101) numerically while including the effect of ASE in the process.

The length of a fiber amplifier can easily be made large. However, at a given pump power for a longitudinally pumped fiber amplifier, if the fiber gain medium is too long, part of the fiber gain medium will not be pumped because the pump power will be totally absorbed before any of it reaches the far end of the fiber. If the fiber gain medium is too short, part of the pump power will not be absorbed. Therefore, ** when considering the optimum length of a fiber amplifier at a given pump power level, the difference between a four-level system and a three-level or quasi-two-level system is significant**.

In a four-level system, such as a PDFA or an NDFA operating in the 1.3-μm region, unexcited active ions are transparent to the signal photons. Therefore, in order to obtain the maximum gain at a given pump power level for a longitudinally pumped four-level fiber amplifier, it is only necessary to make sure that the fiber gain medium is long enough so that all of the pump power is absorbed. Further increasing the length of the fiber amplifier does not have much effect on the overall gain of the amplifier, provided that the background attenuation coefficient of the host fiber is low.

In a three-level or a quasi-two-level system, such as an EDFA operating in the 1.53-μm region, the situation is very different because unexcited active ions that remain in the ground state are in resonance with the signal frequency to absorb the signal photons. Therefore, at a given pump power level, there is an optimum length for a three-level or a quasi-two-level fiber amplifier to have the maximum overall gain. A three-level fiber amplifier that is shorter than the optimum length does not use the pump power efficiently, whereas one that is longer than the optimum length suffers from absorption of the signal by the gain medium in the unpumped section of the fiber.

The optimum length of a fiber amplifier can be found by considering the fact that the maximum value of the integral of \(g_0(z)\) over the length of the fiber occurs when \(g_0(l)=0\) at the end of the fiber. Physically, this condition is equivalent to requiring that \(P_\text{p}(l)=P_\text{p}^\text{tr}\) so that \(G_0\) is maximized, as can be seen by considering \(g_0(l)=0\) for (10-122). By applying this condition, we find from (10-102) [refer to the laser amplifiers tutorial] that the optimum length of a quasi-two-level fiber amplifier, for which \(p\ne0\), is

\[\tag{10-126}l_\text{opt}=-\frac{1}{\alpha_\text{p}}\left(\ln\frac{P_\text{p}^\text{tr}}{P_\text{p}^\text{in}}+\frac{1}{p}\ln\frac{P_\text{p}^\text{sat}+pP_\text{p}^\text{tr}}{P_\text{p}+pP_\text{p}^\text{in}}\right)\]

and from (10-103) that the optimum length of a three-level fiber amplifier, for which \(p=0\), is

\[\tag{10-127}l_\text{opt}=-\frac{1}{\alpha_\text{p}}\left(\ln\frac{P_\text{p}^\text{tr}}{P_\text{p}^\text{in}}+\frac{P_\text{p}^\text{tr}-P_\text{p}^\text{in}}{P_\text{p}^\text{sat}}\right)\]

The optimum length of a fiber amplifier clearly depends on the input pump power.

**Example 10-14**

Find the optimum length of the EDFA described in Example 10-13 for an input pump power of \(P_\text{p}^\text{in}=20\text{ mW}\). What is the unsaturated power gain \(G_0\) when the optimum length is used for such an EDFA?

This EDFA is a quasi-two-level system with \(p=0.055\). From Example 10-13, we find that \(\alpha_\text{p}=0.348\text{ m}^{-1}\), \(P_\text{p}^\text{sat}=4.25\text{ mW}\), and \(P_\text{p}^\text{tr}=3.22\text{ mW}\). Using (10-126), we find that the optimum length of this amplifier at an input power of \(P_\text{p}^\text{in}=20\text{ mW}\) is

\[l_\text{opt}=-\frac{1}{0.348}\times\left(\ln\frac{3.22}{20}+\frac{1}{0.055}\times\ln\frac{4.25+0.055\times3.22}{4.25+0.055\times20}\right)\text{ m}=15.14\text{ m}\]

We find from Figure 10-19(b) that \(g_0(z)=0\) and \(G_0\) has a maximum value of 32 dB at \(z=15.14\text{ m}\), in agreement with what is found here. Therefore, the unsaturated power gain is \(G_0=32\text{ dB}\) for the EDFA with an optimum length of \(15.14\text{ m}\) at \(P_\text{p}^\text{in}=20\text{ mW}\).

In comparison to other laser gain media, a rare-earth ion-doped fiber typically has a broad gain bandwidth, a relatively small emission cross section, and a long fluorescence lifetime, as can easily be seen from Table 10-2 and Figure 10-17.

These characteristics have very important implications for the practical applications of fiber amplifiers.

The broad gain bandwidth allows tunability and tolerance on the wavelength of the input signal.

The small emission cross section implies that the amplifier gain is not easily perturbed by variations of the signal power because variations in the population inversion caused by stimulated emission are small. Therefore, nonlinear effects such as distortion of the signal waveform and cross interference between different signal channels are minimized. Together with the broad gain bandwidth, this characteristic allows a single rare-earth ion-doped fiber amplifier to be used for amplifying multiple optical channels in a wavelength division multiplexing system.

The long fluorescence lifetime is no less important. In particular, the fluorescence lifetime of the \(^4\text{I}_{13/2}\) level of an erbium-doped fiber is on the order of \(\tau_2=10\text{ ms}\), though it varies somewhat among different host glasses. Indeed, the success of EDFAs is due in large part to this long fluorescence lifetime because it allows an EDFA to maintain a high gain at a modest pump power under constant CW pumping.

In addition, a long fluorescence lifetime for the upper laser level means that population inversion and, therefore, gain do not respond to any pump fluctuations or noise that varies on a time scale less than the saturation lifetime \(\tau_\text{s}\). Though \(\tau_\text{s}\) is smaller than \(\tau_2\) at a pumping level of \(s=W_\text{p}\tau_2=P_\text{p}^\text{in}/P_\text{p}^\text{sat}\gt1\) required for an EDFA, it is still on the order of \(1\text{ ms}\) at a high pumping level of \(s=10\). For this reason, an EDFA is not susceptible to noise or intermodulation distortions at frequencies higher than 1 kHz even at such a high pumping level.

Another significant characteristic of a rare-earth ion-doped glass fiber is that the orientation of the active rare-earth ions doped in the fiber is randomized because of the amorphous structure of the host glass. This effect results in polarization-independent absorption and emission cross sections. Consequently, the optical gain in a rare-earth ion-doped glass fiber amplifier is insensitive to the polarization state of the optical signal.

The next tutorial covers **resonant optical cavities**.