# Laser Amplifiers

This is a continuation from the previous tutorial - **population inversion and optical gain**.

Any medium that has an optical gain can be used to amplify an optical signal. Depending on the physical mechanism responsible for the optical gain, there are two different categories of optical amplifiers: the ** nonlinear optical amplifiers** and the

**.**

*laser amplifiers*The optical gain of a nonlinear optical amplifier is associated with a nonlinear optical process in a nonlinear medium, whereas the gain of a laser amplifier results for the population inversion in a medium.

Important nonlinear optical amplifiers include the OPAs, discussed in the optical frequency converters tutorial, and the Raman and Brillouin amplifiers, discussed in the Raman and Brillouin devices tutorial.

In this tutorial, the general characteristics of laser amplifiers are addressed. We consider only continuously pumped laser amplifiers operating in the steady state. Not considered here are pulsed laser amplifiers that require transient dynamical analysis, including those for regenerative amplification of ultrashort laser pulses and those using transient pumping for high-power amplification of giant laser pulses.

We consider single-pass, traveling-wave laser amplifiers, as shown in Figure 10-13 below.

Such a laser amplifier does not form a resonant optical cavity; therefore, the optical signal being amplified passes through it only once as a traveling wave.

A laser amplifier can be pumped in many different ways, but the most commonly employed techniques are ** electrical pumping** and

**.**

*optical pumping*For electrical pumping, a ** transverse pumping** arrangement is more convenient and is most often used though a

**arrangement is also possible.**

*longitudinal pumping*For optical pumping, both transverse and longitudinal pumping arrangements can be easily implemented. However, for an optically pumped amplifier that has a long length but a relatively small absorption coefficient at the pump frequency, such as the fiber amplifier discussed in the following section, the longitudinal pumping arrangement is much more efficient than the transverse pumping arrangement.

Longitudinal optical pumping can be arranged as unidirectional forward, unidirectional backward, or bidirectional. The concepts of these different pumping schemes are also illustrated in Figure 10-13.

The most important characteristics of a laser amplifier are ** gain**,

**,**

*efficiency***, and**

*bandwidth***. These four characteristics are addressed in the following discussions.**

*noise*

**Amplifier Gain**

Ignoring the contribution of noise, the amplification of the intensity, \(I_\text{s}\), of an optical signal propagating through a laser amplifier can be described by [refer to (10-73) of the population inversion and optical gain tutorial]

\[\tag{10-94}\frac{\text{d}I_\text{s}}{\text{d}z}=gI_\text{s}=\frac{g_0(z)}{1+I_\text{s}/I_\text{sat}}I_\text{s}\]

where \(g_0(z)\) is the unsaturated gain coefficient, which can be spatially varying in the longitudinal direction, and \(I_\text{sat}\) is the saturation intensity of the gain medium, both defined in the population inversion and optical gain tutorial.

Here we assume transverse uniformity but consider the possibility of longitudinal nonuniformity by taking the unsaturated gain coefficient \(g_0(z)\) to be a function of \(z\).

Such a longitudinal nonuniform gain distribution is a common scenario in an amplifier under longitudinal optical pumping because of pump absorption by the gain medium.

In the following discussions, we assume for simplicity that the signal beam is collimated throughout the length of the amplifier such that divergence of the beam is negligible. This assumption allows us to consider the power, \(P_\text{s}\), of the optical signal and to convert (10-94) into

\[\tag{10-95}\frac{\text{d}P_\text{s}}{\text{d}z}=gP_\text{s}=\frac{g_0(z)}{1+P_\text{s}/P_\text{sat}}P_\text{s}\]

where \(P_\text{sat}\) is the ** saturation power** obtained by integrating \(I_\text{sat}\) over the cross-sectional area of the signal beam.

By integrating (10-95), the following relation is obtained:

\[\tag{10-96}\frac{P_\text{s}(z)}{P_\text{s}(0)}\exp\left[\frac{P_\text{s}(z)-P_\text{s}(0)}{P_\text{sat}}\right]=\exp\displaystyle\int\limits_0^zg_0(z)\text{d}z\]

where \(P_\text{s}(0)\) is the power of the signal beam at \(z=0\).

When \(P_\text{s}\ll{P}_\text{sat}\), the power of the optical signal grows exponentially with distance. As \(P_\text{s}\) approaches the value of \(P_\text{sat}\), the growth slows down. Eventually, the signal grows only linearly with distance when \(P_\text{s}\gg{P}_\text{sat}\).

The ** power gain** of a signal amplified by a laser amplifier is defined as

\[\tag{10-97}G=\frac{P_\text{s}^\text{out}}{P_\text{s}^\text{in}}\]

where \(P_\text{s}^\text{in}\) and \(P_\text{s}^\text{out}\) are the input and output powers of the signal, respectively.

By using the relation in (10-96) while identifying \(P_\text{s}^\text{out}\) and \(P_\text{s}^\text{in}\) with \(P_\text{s}(l)\) and \(P_\text{s}(0)\), respectively, for an amplifier of a length \(l\), the following relation for the power gain of the signal is found:

\[\tag{10-98}G=G_0\text{e}^{(1-G)P_\text{s}^\text{in}/P_\text{sat}}\]

where \(G_0\) is the ** unsaturated power gain**, or the

**.**

*small-signal power gain*For a single pass through the amplifier, \(G_0\) is given by

\[\tag{10-99}G_0=\exp\displaystyle\int\limits_0^lg_0(z)\text{d}z\]

Note that, according to (10-98), \(G_0\ge{G}\gt1\) because \(g_0\gt0\) for an amplifier.

For a weak optical signal such that \(P_\text{s}^\text{in}\lt{P}_\text{s}^\text{out}\ll{P}_\text{sat}\), the power gain is simply the small-signal power gain, \(G=G_0\).

If the signal power approaches or even exceeds the saturation power of the amplifier, the relation in (10-98) clearly indicates that \(G\lt{G}_0\) because of gain saturation. In this situation, the overall gain, \(G\), can be found by solving (10-98) when the values of \(P_\text{s}^\text{in}\) and \(P_\text{sat}\), as well as that of \(G_0\), are given. Figure 10-14 shows the amplifier gain as a function of input signal power for a few different values of the unsaturated power gain.

The unsaturated gain coefficient \(g_0\) of an optically pumped laser amplifier depends on the pump intensity according to (10-88) [refer to the population inversion and optical gain tutorial].

For both longitudinal and transverse pumping, the pump intensity normally varies in space because of absorption and diffraction of the pump beam. In the case of longitudinal optical pumping, the pump intensity is still a function of distance from the input end even when transverse uniformity is assumed, as is done in the above discussion.

In general, \(I_\text{p}(z)\) and \(g_0(z)\) depend on many geometric parameters of the amplifier and have to be found numerically for each particular case.

A special situation of interest is when transverse divergence of the pump beam is nonexistent, such as in the case of an optical fiber amplifier, or can be ignored, such as in a short bulk laser amplifier with a highly collimated pump beam.

In this situation, we can express \(g_0(z)\) in terms of the pump power \(P_\text{p}(z)\) instead of the pump intensity by integrating \(I_\text{p}(z)\) over the transverse cross section. A saturation pump power, \(P_\text{p}^\text{sat}\), can be defined by integrating \(I_\text{p}^\text{sat}\) over the transverse cross section.

Then, by replacing \(I_\text{p}/I_\text{p}^\text{sat}\) with \(P_\text{p}/P_\text{p}^\text{sat}\) in (10-91) [refer to the population inversion and optical gain tutorial] for \(N_2\), we find that the absorption coefficient of the pump beam as a function of distance in the amplifier can be expressed as

\[\tag{10-100}\alpha_\text{p}(z)=\sigma_\text{a}^\text{p}[N_\text{t}-N_2(z)]=\alpha_\text{p}\frac{1+pP_\text{p}(z)/P_\text{p}^\text{sat}}{1+(1+p)P_\text{p}(z)/P_\text{p}^\text{sat}}\]

where \(\alpha_\text{p}=\sigma_\text{a}^\text{p}N_\text{t}\) is the intrinsic absorption coefficient at the pump wavelength in the absence of a strong pump beam so that pump depletion of the ground-state population is negligible.

Here we have assumed a low-loss medium where the pumping mechanism fully accounts for absorption of the pump beam.

With this spatially varying pump absorption coefficient, we can write the following equation for the spatial evolution of the pump power:

\[\tag{10-101}\frac{\text{d}P_\text{p}}{\text{d}z}=-\alpha_\text{p}(z)P_\text{p}=-\alpha_\text{p}\frac{1+pP_\text{p}/P_\text{p}^\text{sat}}{1+(1+p)P_\text{p}/P_\text{p}^\text{sat}}P_\text{p}\]

This equation can be integrated to find the following solutions:

\[\tag{10-102}\frac{P_\text{p}(z)}{P_\text{p}(0)}\left[\frac{P_\text{p}^\text{sat}+pP_\text{p}(z)}{P_\text{p}^\text{sat}+pP_\text{p}(0)}\right]^{1/p}=\text{e}^{-\alpha_\text{p}z},\qquad\text{ for }p\ne0\]

and

\[\tag{10-103}\frac{P_\text{p}(z)}{P_\text{p}(0)}\exp\left[\frac{P_\text{p}(z)-P_\text{p}(0)}{P_\text{p}^\text{sat}}\right]=\text{e}^{-\alpha_\text{p}z},\qquad\text{ for }p=0\]

It can be seen from the relations in (10-102) and (10-103) that besides the absorption coefficient \(\alpha_\text{a}\), longitudinal variations of pump power in the amplifier strongly depend on the ** pumping ratio** defined as

\[\tag{10-104}s=\frac{P_\text{p}(0)}{P_\text{p}^\text{sat}}=\frac{P_\text{p}^\text{in}}{P_\text{p}^\text{sat}}\]

where \(P_\text{p}^\text{in}=P_\text{p}(0)\) is the input pump power.

Once the pump power distribution, \(P_\text{p}(z)\), is found from the implicit solutions given in (10-102) and (10-103), the distribution of the small-signal gain coefficient \(g_0(z)\) can be found from (10-88) [refer to the population inversion and optical gain tutorial] as

\[\tag{10-105}\begin{align}g_0(z)&=\frac{(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}}{1+(1+p)P_\text{p}(z)/P_\text{p}^\text{sat}}\left[\frac{P_\text{p}(z)}{P_\text{p}^\text{sat}}-\frac{P_\text{p}^\text{tr}}{P_\text{p}^\text{sat}} \right]\\&=\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{1+(1+p)P_\text{p}(z)/P_\text{p}^\text{sat}}\frac{P_\text{p}(z)}{P_\text{p}^\text{sat}}-\sigma_\text{a}N_\text{t}\end{align}\]

In general, numerical solution is required to find \(P_\text{p}(z)\) from the implicit solutions given in (10-102) and (10-103) in order to find \(g_0(z)\).

However, what really matters for an amplifier is the integral of \(g_0(z)\) over the entire length of the amplifier, which gives the value of \(G_0\) in (10-99). Closed form solutions for both cases of \(p\ne0\) and \(p=0\) can be found by using (10-101) to integrate \(g_0(z)\) in (10-105).

For an amplifier of a length \(l\), the integral can be expressed conveniently in terms of the input pomp power \(P_\text{p}^\text{in}=P_\text{p}(0)\) launched at the input end and the remaining pump power \(P_\text{p}^\text{out}=P_\text{p}(l)\) at the output end of the amplifier as

\[\tag{10-106}\begin{align}\displaystyle\int\limits_0^lg_0(z)\text{d}z&=\sigma_\text{e}N_\text{t}l+\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\ln\frac{P_\text{p}^\text{out}}{P_\text{p}^\text{in}}\\&=\sigma_\text{e}N_\text{t}l+\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\ln(1-\zeta_\text{p})\end{align}\]

where \(\zeta_\text{p}\) is the ** pump power utilization factor** that accounts for the pump power absorbed by the gain medium and is defined as

\[\tag{10-107}\zeta_\text{p}=\frac{P_\text{p}^\text{in}-P_\text{p}^\text{out}}{P_\text{p}^\text{in}}\]

The relation given in (10-106) is valid for both \(p\ne0\) and \(p=0\). It is a convenient form because all that is needed to evaluate the value of the gain integral is the value of \(\zeta_\text{p}\) besides the basic parameters of the amplifier.

In theory, \(P_\text{p}^\text{out}\) never completely vanishes and \(\zeta_\text{p}\) never reaches the value of unity no matter how long the amplifier is because the pump power can only continue to decay. Therefore, there is no problem in utilizing (10-106) in principle.

In practice, however, great uncertainty arises in using (10-106) when \(P_\text{p}^\text{out}\) becomes very small. In an experimental setting, (10-106) yields no meaningful result when \(P_\text{p}^\text{out}\) approaches the detection limit.

To avoid such a limitation, (10-102) for \(p\ne0\) and (10-103) for \(p=0\) can be used to transform (10-106) into

\[\tag{10-108}\displaystyle\int\limits_0^lg_0(z)\text{d}z=\left\{\begin{array}{l}\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{p\alpha_\text{p}}\ln\frac{P_\text{p}^\text{sat}+pP_\text{p}^\text{in}}{P_\text{p}^\text{sat}+pP_\text{p}^\text{out}}-\sigma_\text{a}N_\text{t}l,\qquad\text{ for }p\ne0\\\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{\alpha_\text{p}}\left(\frac{P_\text{p}^\text{in}-P_\text{p}^\text{out}}{P_\text{p}^\text{sat}}\right)-\sigma_\text{a}N_\text{t}l,\qquad\text{ for }p=0\end{array}\right.\]

Given an input pump power of \(P_\text{p}^\text{in}=P_\text{p}(0)\) at \(z=0\), the remaining pump power, \(P_\text{p}^\text{out}=P_\text{p}(l)\), at the output end \(z=l\) of the amplifier can be found from (10-102) for \(p\ne0\) or from (10-103) for \(p=0\).

In an experimental setting, both \(P_\text{p}^\text{in}\) and \(P_\text{p}^\text{out}\) can be measured directly. Once the integral of \(g_0(z)\) is found, \(G_0\) can be found through (10-99).

**Example 10-9**

A CW Nd : YAG laser amplifier for an optical signal at \(\lambda_\text{s}=1.064\text{ μm}\) is pumped with the output of a high-power semiconductor laser at \(\lambda_\text{p}=808\text{ nm}\). The Nd : YAG crystal, which is doped with 1.1 at. \(\%\) Nd for a concentration of \(N_\text{t}=1.52\times10^{26}\text{ m}^{-3}\), has a length of 5 mm and a cross-sectional diameter of 5 mm.

The pump optical beam is delivered through a multimode optical fiber of a 200-μm core diameter and is collimated to define a circular pumping spot of \(w=100\text{ μm}\) radius throughout the length of the crystal. The signal beam is collimated to a spot size that matches the pumping area exactly.

As shown in Figure 10-15 below, the crystal surfaces are coated for a single pass of the pump beam through the crystal, but for double passes of the signal beam.

At the 808 nm pump wavelength, the peak absorption cross section is \(5.6\times10^{-24}\text{ m}^2\). However, because the emission of the pump semiconductor laser has a broad spectral width of \(\Delta\lambda_\text{p}=3.5\text{ nm}\), the effective absorption cross section for the pump beam is reduced to \(\sigma_\text{a}^\text{p}=3.0\times10^{-24}\text{ m}^2\).

The emission cross section accounting for all effects including the population ratio in the upper laser level is \(\sigma_\text{e}=3.1\times10^{-23}\text{ m}^2\) at the signal wavelength.

The fluorescence lifetime is \(\tau_2=240\text{ μs}\). The pump quantum efficiency is found to be \(\eta_\text{p}=0.8\). The amplifier is pumped with a pump power of \(P_\text{p}=2\text{ W}\).

Assume that there is no additional attenuation to the pump beam besides the absorption for the pumping transition with the cross section \(\sigma_\text{a}^\text{p}\).

(a) Find the double-pass unsaturated power gain of the amplifier.

(b) Find the gain and the output power for a signal with an input power of \(P_\text{s}^\text{in}=20\text{ mW}\).

**(a)**

The pump photon energy is \(h\nu_\text{p}=(1.2398/0.808)\text{ eV}=1.534\text{ eV}\). The signal photon energy is \(h\nu_\text{s}=(1.2398/1.064)\text{ eV}=1.165\text{ eV}\).

Because the pump and signal beams are all well collimated, we consider the pump and signal powers directly instead of the pump and signal intensities.

We first find that [refer to (10-85) in the population inversion and optical gain tutorial]

\[P_\text{p}^\text{sat}=\pi{w}^2I_\text{p}^\text{sat}=\pi{w}^2\frac{h\nu_\text{p}}{\eta_\text{p}\tau_2\sigma_\text{a}^\text{p}}=\frac{\pi\times(100\times10^{-6})^2\times1.534\times1.6\times10^{-19}}{0.8\times240\times10^{-6}\times3.0\times10^{-24}}=13.4\text{ W}\]

Then, with an input pump power of \(P_\text{p}^\text{in}=2\text{ W}\), the pumping ratio defined in (10-104) is

\[s=\frac{P_\text{p}(0)}{P_\text{p}^\text{sat}}=\frac{2}{13.4}=0.149\]

The pump absorption coefficient is [refer to (10-100)] \(\alpha_\text{p}=\sigma_\text{a}^\text{p}N_\text{t}=3.0\times10^{-24}\times1.52\times10^{26}\text{ m}^{-1}=456\text{ m}^{-1}\). With an amplifier length of \(l=5\text{ mm}\), we find that \(\alpha_\text{p}l=2.28\).

Because Nd : YAG is a four-level system for the signal at 1.064 μm wavelength, we have \(p=0\) and \(\sigma_\text{a}=0\). We then find \(P_\text{p}^\text{out}=P_\text{p}(l)\) from the implicit solution in (10-103). By taking \(z=l\) and defining the variable \(x=P_\text{p}(l)/P_\text{p}^\text{sat}\), (10-103) can be expressed as

\[x\text{e}^x=s\text{e}^s\text{e}^{-\alpha_\text{p}l}=0.149\times\text{e}^{0.149-2.28}\]

which has a solution of \(x=1.74\times10^{-2}\). Thus

\[x=\frac{P_\text{p}(l)}{P_\text{p}^\text{sat}}=1.74\times10^{-2}\]

and \(P_\text{p}^\text{out}=P_\text{p}(l)=1.74\times10^{-2}\times{P}_\text{p}^\text{sat}=1.74\times10^{-2}\times13.4\text{ W}=0.233\text{ W}\).

With \(p=0\) and \(\sigma_\text{a}=0\), we then obtain from (10-108) the following integral:

\[\displaystyle\int\limits_0^lg_0(z)\text{d}z=\frac{\sigma_\text{e}N_\text{t}}{\alpha_\text{p}}\left(\frac{P_\text{p}^\text{in}-P_\text{p}^\text{out}}{P_\text{p}^\text{sat}}\right)=\frac{3.1\times10^{-23}\times1.52\times10^{26}}{456}\times\frac{2-0.233}{13.4}=1.36\]

The double-pass unsaturated power gain can now be found as

\[G_0=\exp\left[2\displaystyle\int\limits_0^lg_0(z)\text{d}z\right]=\exp(2\times1.36)=\text{e}^{2.72}=15.2\]

This unsaturated gain is about 11.8 dB.

**(b)**

To find the power gain for the signal, we need to consider the gain saturation effect by first finding the saturation power of the amplifier. At the given pumping level, \(W_\text{p}\tau_2=s=0.149\) at \(z=0\) [refer to (10-104), and (10-84), (10-85) in the population inversion and optical gain tutorial to get this relation]. Taking this value as an approximation throughout the gain medium, we have [refer to (10-80) in the population inversion and optical gain tutorial]

\[\tau_\text{s}=\frac{\tau_2}{1+W_\text{p}\tau_2}=\frac{240}{1.149}\text{ μs}=209\text{ μs}\]

Then, the saturation intensity is [refer to (10-74) in the population inversion and optical gain tutorial]

\[I_\text{sat}=\frac{h\nu_\text{s}}{\tau_\text{s}\sigma_\text{e}}=\frac{1.165\times1.6\times10^{-19}}{209\times10^{-6}\times3.1\times10^{-23}}\text{ W m}^{-2}=28.8\text{ MW m}^{-2}\]

and the saturation power is

\[P_\text{sat}=\pi{w}^2I_\text{sat}=\pi\times(100\times10^{-6})^2\times28.8\times10^6\text{ W}=905\text{ mW}\]

For an input signal power of \(P_\text{s}^\text{in}=20\text{ mW}\), \(P_\text{s}^\text{in}/P_\text{sat}=0.022\). Therefore, according to (10-98), the gain can be found by solving

\[G=G_0\text{e}^{(1-G)P_\text{s}^\text{in}/P_\text{sat}}=\text{e}^{2.72+0.022(1-G)}\]

By solving this relation iteratively, we find that the signal power gain is \(G=11.9\), which is 10.8 dB. The output signal power is

\[P_\text{s}^\text{out}=GP_\text{s}^\text{in}=11.9\times20\text{ mW}=238\text{ mW}\]

**Amplifier Efficiency**

The efficiency of a laser amplifier can be measured either as power efficiency or as quantum efficiency.

The ** power conversion efficiency**, \(\eta_\text{c}\), of a laser amplifier is defined as

\[\tag{10-109}\eta_\text{c}=\frac{P_\text{s}^\text{out}-P_\text{s}^\text{in}}{P_\text{p}}\]

Another useful concept is the ** differential power conversion efficiency**, also known as the

**, \(\eta_\text{s}\), of an amplifier, which is defined as**

*slope efficiency*\[\tag{10-110}\eta_\text{s}=\frac{\text{d}P_\text{s}^\text{out}}{\text{d}P_\text{p}}\]

The differential power conversion efficiency measures the increase of the output signal power as the pump power increases. It is generally somewhat larger than the total power conversion efficiency measured by \(\eta_\text{c}\).

The ** quantum efficiency**, \(\eta_\text{q}\), of a laser amplifier is defined as the number of signal photons generated per pump photon, in the case of optical pumping, or per pump electron, in the case of electrical pumping, that is absorbed by the gain medium. It can be expressed as

\[\tag{10-111}\eta_\text{q}=\frac{\Phi_\text{s}^\text{out}-\Phi_\text{s}^\text{in}}{\xi_\text{p}\Phi_\text{p}}\]

where \(\Phi_\text{s}^\text{in}\) and \(\Phi_\text{s}^\text{out}\) are the input and output photon fluxes, respectively, and \(\Phi_\text{p}\) is the pump photon or electron flux. The maximum possible value of \(\eta_\text{q}\) is unity.

The power conversion efficiency is always less than the quantum efficiency. For the case of optical pumping, they have the following relationship:

\[\tag{10-112}\eta_\text{q}=\frac{\nu_\text{p}}{\nu_\text{s}}\frac{\eta_\text{c}}{\xi_\text{p}}=\frac{\lambda_\text{s}}{\lambda_\text{p}}\frac{\eta_\text{c}}{\xi_\text{p}}\]

where \(\nu_\text{s}\) and \(\nu_\text{p}\) are the signal and pump frequencies, respectively, and \(\lambda_\text{s}\) and \(\lambda_\text{p}\) are the free-space signal and pump wavelengths, respectively.

Because the maximum value of \(\eta_\text{q}\) is unity, the maximum possible power conversion efficiency of an optically pumped laser amplifier is \(\lambda_\text{p}/\lambda_\text{s}\).

**Example 10-10**

Find the power conversion efficiency and the quantum efficiency of the Nd : YAG laser amplifier described in Example 10-9 operated with a pump power of 2 W and an input signal power of 20 mW.

From Example 10-9, the output signal power is \(P_\text{s}^\text{out}=238\text{ mW}\) when the amplifier is operated with \(P_\text{p}^\text{in}=2\text{ W}\) and \(P_\text{s}^\text{in}=20\text{ mW}\). Therefore, according to (10-109), the power conversion efficiency is

\[\eta_\text{c}=\frac{238\times10^{-3}-20\times10^{-3}}{2}=10.9\%\]

For this amplifier, we have, from Example 10-9, \(P_\text{p}^\text{out}=0.233\text{ W}\). The pump power utilization factor is

\[\xi_\text{p}=\frac{P_\text{p}^\text{in}-P_\text{p}^\text{out}}{P_\text{p}^\text{in}}=\frac{2-0.233}{2}=0.884\]

Using (10-112), we find that the quantum efficiency is

\[\eta_\text{q}=\frac{\lambda_\text{s}}{\lambda_\text{p}}\frac{\eta_\text{c}}{\xi_\text{p}}=\frac{1.064\times10^{-6}}{808\times10^{-9}}\times\frac{10.9\%}{0.884}=16.2\%\]

**Amplifier Bandwidth**

The optical bandwidth, \(B_\text{o}\), of a laser amplifier is determined by the spectral width, \(\Delta\nu_\text{g}\), of the gain coefficient \(g(\nu)\) and any optical filter that might be incorporated into the device. In the case when there is no additional optical filter, \(B_\text{o}=\Delta\nu_\text{g}\).

On the other hand, if a narrow-band optical filter with a bandwidth much smaller than \(\Delta\nu_\text{g}\) is used at the output of the amplifier, then \(B_\text{o}\) is simply that of the filter.

From the results obtained in the preceding section regarding regarding the gain coefficient, \(g(\nu)\) is a function of \(\sigma_\text{e}(\nu)\), \(\sigma_\text{a}(\nu)\), and the pumping rate \(W_\text{p}\).

For a gain medium whose laser transition levels are narrow enough so that \(\sigma_\text{e}(\nu)\) and \(\sigma_\text{a}(\nu)\) have the same spectral distribution, or for a four-level system whose lower laser level is empty so that \(g(\nu)\) is independent of \(\sigma_\text{a}(\nu)\), the spectral distribution of \(g(\nu)\) is simply that of \(\sigma_\text{e}(\nu)\).

However, as discussed in the optical absorption and amplification tutorial, the spectral distribution of \(\sigma_\text{e}(\nu)\) can be very different from that of \(\sigma_\text{a}(\nu)\) for many practical laser materials.

For a quasi-two-level or a three-level system whose \(\sigma_\text{e}(\nu)\) and \(\sigma_\text{a}(\nu)\) have different spectral distributions, the spectral distribution of \(g(\nu)\) not only depends on both \(\sigma_\text{e}(\nu)\) and \(\sigma_\text{a}(\nu)\) but also varies as the pumping rate \(W_\text{p}\) is varied, as can easily be observed by examining (10-75) and (10-77) [refer to the population inversion and optical gain tutorial].

Consequently, the optical bandwidth of a laser amplifier that consists of a quasi-two-level or a three-level gain medium is generally a function of the pumping rate, but that of a four-level amplifier is less sensitive to the pumping rate.

Because of the resonant nature of the laser transition that is responsible for the gain of a laser amplifier, the optical bandwidth of a laser amplifier is generally quite small in the sense that \(B_\text{o}\ll\nu_\text{s}\), where \(\nu_\text{s}\) is the frequency of an optical signal being amplified.

**Example 10-11**

Find the optical bandwidth of the Nd : YAG laser amplifier described in Example 10-9.

The bandwidth of a laser amplifier is determined by the spectral width of its optical gain, which in turn is largely determined by the spontaneous emission linewidth of the gain medium. The linewidth of Nd : YAG varies with temperature, doping concentration, and crystal quality, but it typically falls between 120 and 180 GHz at room temperature.

According to Table 10-1 [refer to the optical transitions for laser amplifiers tutorial], we find that the typical spontaneous linewidth of Nd : YAG is \(\Delta\nu=150\text{ GHz}\). Therefore, we can expect that \(B_\text{o}=\Delta\nu_\text{g}\approx\Delta\nu=150\text{ GHz}\).

**Amplifier Noise**

There are two intrinsic noise sources in a laser amplifier: quantum noise due to spontaneous emission and thermal noise associated with blackbody radiation.

At room temperature, these two noise sources have the same magnitude at an electromagnetic wavelength of \(\lambda=44\text{ μm}\). Thermal noise dominates at long wavelengths, whereas quantum noise dominates at short wavelengths.

Therefore, thermal noise in a laser amplifier that operates in the optical region at room temperature is negligible in the presence of quantum noise caused by spontaneous emission.

The spontaneous emission noise power at the output of a laser amplifier is the result of the ** amplified spontaneous emission** (ASE) in the amplifier. It is a function of the gain and bandwidth of the amplifier and is given by

\[\tag{10-113}P_\text{sp}=N_\text{sp}h\nu_\text{s}B_\text{o}(G-1)\]

where

\[\tag{10-114}N_\text{sp}=\frac{\sigma_\text{e}N_2}{\sigma_\text{e}N_2-\sigma_\text{a}N_1}\]

is the amplifier ** spontaneous emission factor** that measures the degree of population inversion in the amplifier.

In a given amplifier, the value of \(N_\text{sp}\) varies with pumping rate and signal wavelength. It can be seen from (10-114) that \(N_\text{sp}\ge1\) for an amplifier with \(\sigma_\text{e}N_2\gt\sigma_\text{a}N_1\) so that \(G\gt1\). The minimum value, \(N_\text{sp}=1\), corresponds to complete population inversion with \(N_1=0\).

In the case of insufficient pumping with \(\sigma_\text{e}N_2\lt\sigma_\text{a}N_1\) so that \(G\lt1\), the amplifier actually attenuates the optical signal rather than amplifying it. Then, \(N_\text{sp}\) has a negative value, but the noise power \(P_\text{sp}\) is still positive because \(G\lt1\).

** Therefore, an ideal amplifier has the minimum noise factor when the amplifying medium has complete population inversion so that \(N_\text{sp}=1\)**.

By ignoring the effect of gain saturation, the spontaneous emission factor defined in (10-114) can be expressed in the following form in terms of pump intensity for an optically pumped system:

\[\tag{10-115}N_\text{sp}=\left(1-p\frac{\sigma_\text{a}}{\sigma_\text{e}}-\frac{I_\text{p}^\text{sat}}{I_\text{p}}\frac{\sigma_\text{a}}{\sigma_\text{e}}\right)^{-1}=\left(1-p\frac{\sigma_\text{a}}{\sigma_\text{e}}-\frac{P_\text{p}^\text{sat}}{P_\text{p}}\frac{\sigma_\text{a}}{\sigma_\text{e}}\right)^{-1}\]

where \(p\ne0\) and \(\sigma_\text{a}\ne0\) for a quasi-two-level system, \(p=0\) but \(\sigma_\text{a}\ne0\) for a three-level system, and \(p=0\) and \(\sigma_\text{a}=0\) for a four-level system.

It can be seen from (10-115) that a four-level amplifier is normally less noisy than a quasi-two-level or a three-level amplifier because the lower laser level of a four-level system is normally not populated, so that \(N_\text{sp}=1\).

Everything else being equal, a quasi-two-level system is expected to be noisier than a three-level system.

We also see from (10-115) that the spontaneous emission factor is reduced toward its minimum value of unity when the pump intensity is increased.

For a three-level erbium-doped fiber amplifier, an \(N_\text{sp}\) approaching the ideal minimum value of 1 can be obtained near the peak of the emission spectrum at a high pumping level.

For a semiconductor laser amplifier, the value of \(N_\text{sp}\) typically ranges from 1.4 to more than 4, depending on the operating condition.

From (10-113), we see that the ASE of a laser amplifier is directly proportional to the optical bandwidth \(B_\text{o}\) of the amplifier. To increase the ** signal-to-noise ratio** (SNR) at the amplifier output, the total noise power can be reduced to a minimum by placing at the output end of the amplifier an optical filter that has a narrow bandwidth matching the bandwidth of the optical signal.

Because of the spontaneous emission noise, the SNR of an optical signal always degrades after the optical signal passes through an amplifier. The degradation of the SNR of the optical signal passing through an amplifier is measured by the ** optical noise figure** of the amplifier defined as

\[\tag{10-116}F_\text{o}=\frac{\text{SNR}_\text{in}}{\text{SNR}_\text{out}}\]

where \(\text{SNR}_\text{in}\) and \(\text{SNR}_\text{out}\) represent the values of the optical SNR at the input and output ends of the amplifier, respectively.

The optical noise figure of an amplifier is a function of the gain and the spontaneous emission factor of the amplifier. It also depends on the photon statistics of the optical signal.

For an optical signal that is characterized by a classical electromagnetic field with a large number of photons, the photon statistics can be described by the Poisson statistics. Then, the optical noise figure can be approximated by

\[\tag{10-117}F_\text{o}\approx\frac{1+2N_\text{sp}(G-1)}{G}=2N_\text{sp}+\frac{1-2N_\text{sp}}{G}\]

If the amplifying medium has complete population inversion so that \(N_\text{sp}=1\), then \(F_\text{o}=2-1/G\) and \(2\gt{F}_\text{o}\gt1\) for \(G\gt1\). For a high-gain amplifier, \(G\gg1\). Then (10-117) can be further approximated by

\[\tag{10-118}F_\text{o}(G\gg1)\approx2N_\text{sp}\ge2\]

because \(N_\text{sp}\ge1\) for an amplifier of \(G\gt1\).

Therefore, unless complete population inversion is achieved in the amplifying medium, the optical noise figure of a high-gain laser amplifier is subject to the ** quantum limit** of \(F_\text{o}\ge2\), or \(F_\text{o}\ge3\text{ dB}\).

For a low-gain amplifier with sufficient population inversion, it is possible to have a noise figure less than 2.

However, the relation in (10-117) does not imply that the value of \(F_\text{o}\) can be less than unity if \(G\lt1\) because \(N_\text{sp}\) has a negative value when \(G\lt1\).

Therefore, an optical amplifier can never improve the SNR of an optical signal. No matter how a laser amplifier is pumped or operated and no matter whether the optical signal is amplified or attenuated, the optical noise figure is always larger than unity, \(F_\text{o}\gt1\).

**Example 10-12**

Find the ASE noise power and the optical noise figure of the Nd : YAG laser amplifier described in Example 10-9 operated with a pump power of 2 W and an input signal power of 20 mW.

From Example 10-9, we find that the power gain is \(G=11.9\) when the amplifier is operated with \(P_\text{p}=2\text{ W}\) and \(P_\text{s}^\text{in}=20\text{ mW}\). From Example 10-11, we have \(B_\text{o}=150\text{ GHz}\). We also have \(h\nu_\text{s}=1.165\text{ eV}\) for the signal photon energy. Because Nd : YAG operating at the signal wavelength of 1.064 μm is a four-level system with \(N_1=0\), we have \(N_\text{sp}=1\). Therefore, according to the relation in (10-113), the ASE noise power for this amplifier under the specified operating condition is

\[P_\text{sp}=1\times1.165\times1.6\times10^{-19}\times150\times10^{9}\times(11.9-1)\text{ W}=305\text{ nW}\]

This is the power of the ASE noise at the output of the amplifier. It is only \(1.28\times10^{-6}\) of the output signal power of 238 mW. This noise power is small primarily because of the narrow bandwidth of the Nd : YAG laser amplifier.

The optical noise figure of the amplifier can be found by using (10-117) to be

\[F_\text{o}=\frac{1+2\times1\times(11.9-1)}{11.9}=1.92\]

This amplifier has a noise figure of \(F_\text{o}\lt2\) because it is a four-level system with \(N_\text{sp}=1\).

Laser amplifiers and nonlinear optical amplifiers find their primary applications as ** power amplifiers** for laser beams, as

**in long-distance optical transmission systems, and as**

*optical repeaters***to optical receivers.**

*optical preamplifiers*In addition, some laser amplifiers, particularly the semiconductor laser amplifiers, can be used as ** optical switches** and nonlinear optical processing devices.

Basically all laser amplifiers and nonlinear optical amplifiers can be used as power amplifiers. In fact, because of their large noise figures and narrow bandwidths, many laser amplifiers and nonlinear optical amplifiers are unsuitable for other applications.

As power amplifiers, they are used to amplify high-quality but low-power laser beams for the generation of high-power laser beams of good spatial and temporal qualities that cannot be easily generated directly from laser oscillators.

The primary consideration in this application is the power conversion efficiency. Therefore, a laser amplifier is normally operated at saturation level.

Not all laser amplifiers and nonlinear optical amplifiers are suitable for the other applications mentioned above. The suitability for each application varies form one type of amplifier to another.

Some laser amplifiers, however, are suitable for multiple applications. For example, the erbium-doped fiber amplifiers, which are discussed in the next tutorial, have found very important applications as power amplifiers, optical repeaters, and preamplifiers in optical communication systems.

The next tutorial covers **rare-earth ion-doped fiber amplifiers**.