Optical Fiber Lasers
This is a continuation from the previous tutorial - pulsed lasers.
A fiber laser can be constructed by simply creating some form of optical feedback to a fiber amplifier. Nevertheless, while most interest in fiber amplifiers has concentrated on the 1.3- and 1.5-μm spectral regions for optical communication systems, the development of fiber lasers has covered a broad spectral range, from a holmium-doped fiber laser at 550 nm and a praseodymium-doped fiber laser at 610 nm in the visible spectral region to an erbium-doped fiber laser at 2.7 μm and a holmium-doped fiber laser at 2.9 μm in the infrared region. Besides, the active ions used for fiber lasers include almost all rare-earth ions doped in either silica or fluoride glass fibers.
The operation of a fiber laser follows the general laser principles discussed in earlier tutorials. Besides CW oscillation, fiber lasers can also be \(Q\) switched or mode locked to deliver very short and intense laser pulses. The geometry and the waveguiding nature of the fiber gain medium, however, lead to many unique configurations, along with some special characteristics, for fiber lasers.
Several different cavity configurations for fiber lasers are shown in Figure 11-17.
The most straightforward configuration, shown in Figure 11-17(a), is a Fabry-Perot cavity created by attaching a dichroic mirror to each end of a fiber that contains a section of rare-earth ion-doped fiber. The dichroic mirrors are selected to have high reflectivities at the laser wavelength but have nearly \(100\%\) transmittance at the pump wavelength. The pump beam is launched through one end of the fiber, while the laser output exits from the other end.
An alternative configuration, which has two different arrangements shown in Figure 11-17(b) and (c), is a transversely coupled fiber Fabry-Perot cavity in which a fiber directional coupler is used to couple the pump power into, and part of the resonating laser power out of, the cavity. In this configuration, both the pump and the laser beams never leave the fiber, avoiding the coupling of these beams in and out of the fiber through mirrors and lens.
An all-fiber Fabry-Perot laser, shown in Figure 11-17(d), can be constructed using two fiber loop reflectors in place of mirrors. Similarly to the dichroic mirrors used in the Fabry-Perot cavity of Figure 11-17(a), the fiber loop reflectors are chosen to have \(100\%\) transmittance for the pump wave and high reflectance for the resonating laser wave.
Another all-fiber configuration is the fiber ring cavities shown in Figure 11-17(e) and (f). Note the significant difference between a fiber ring cavity and a fiber loop reflector. A ring cavity is an optical resonator, which stores energy, but a loop reflector is a nonresonant optical interferometer, which does not store energy.
Because the host materials are glasses, most rare-earth ion-doped fibers are at least partially inhomogeneously broadened at room temperature. This property, coupled with the broad gain bandwidth and the usually long cavity length of a fiber laser, leads to the fact that fiber laser normally oscillates in multiple longitudinal modes.
There are a few approaches to forcing a fiber laser to oscillate in a single longitudinal mode, thus delivering a narrow-linewidth, single-frequency laser output.
An all-fiber approach is to use frequency-selective fiber Bragg gratings. The frequency of the single-frequency laser output can be tuned if tunable fiber gratings are used.
One possible arrangement, shown in Figure 11-18(a), is a kind of fiber DBR laser, in which a fiber grating is used as a frequency-selective distributed Bragg reflector to replace the output coupling loop reflector of Figure 11-17(d).
Another possibility, shown in Figure 11-18(b), is a fiber DFB laser, in which no localized reflector is used but laser oscillation is accomplished by frequency-selective distributed feedback of a fiber grating throughout the entire section of the rare-earth ion-doped fiber gain medium.
Because of the waveguiding nature of a fiber, the transverse-mode characteristics of a fiber laser are not a function of the cavity configuration but are solely determined by the mode characteristics of the fiber waveguide. By using a single-mode fiber, single-transverse-mode oscillation of a fiber laser can be guaranteed, irrespective of other parameters of the fiber laser resonator.
Because a fiber laser generally has a longitudinal optical pumping arrangement, the general characteristics of solid-state lasers with longitudinal optical pumping discussed in the earlier tutorials apply equally well.
As in the case of the fiber amplifiers discussed in the rare-earth ion-doped fiber amplifiers tutorial, some quantitative modifications on the formulations of certain relations are needed to account for the waveguiding nature of the fiber.
Specifically, it is necessary to incorporate the confinement factors \(\Gamma_\text{p}\) for the pump beam and \(\Gamma_\text{s}\) for the signal beam into the formulation.
Except for the transparency pump power, \(P_\text{p}^\text{tr}\), given in (10-121), [refer to the rare-earth ion-doped fiber amplifiers tutorial], all of the formulations listed in (10-119) to (10-125) for a fiber amplifier are valid for a fiber laser.
The transparency pump power of a laser is defined differently from that of an amplifier. The transparency pump power of a fiber laser is still that given in (11-62) [refer to the laser oscillation tutorial] without modification.
The only additional formulations that have to modified for a fiber laser are those for the pump power utilization factor at threshold and the threshold pump power:
\[\tag{11-109}\zeta_\text{p}^\text{th}=1-\exp\left[-\frac{\Gamma_\text{s}\sigma_\text{e}N_\text{t}-g_\text{th}}{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]\]
and
\[\tag{11-110}P_\text{p}^\text{th}=\begin{cases}\frac{1}{p}\frac{\exp\left[p\frac{\Gamma_\text{s}\sigma_\text{a}N_\text{t}+g_\text{th}}{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]-1}{1-\exp\left[-\frac{\Gamma_\text{s}(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}-(1+p)g_\text{th}}{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]}P_\text{p}^\text{sat},\qquad\text{for }p\ne0\\\frac{\Gamma_\text{s}\sigma_\text{a}N_\text{t}+g_\text{th}}{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\frac{\alpha_\text{p}l_\text{g}}{1-\exp\left[-\frac{\Gamma_\text{s}\sigma_\text{e}N_\text{t}-g_\text{th}}{\Gamma_\text{s}(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}\alpha_\text{p}l_\text{g}\right]}P_\text{p}^\text{sat},\qquad\text{for }p=0\end{cases}\]
By setting \(g_\text{th}\) to zero in (11-109) and (11-110), \(\zeta_\text{p}^\text{tr}\) and \(P_\text{p}^\text{tr}\) can be found.
With these modifications, all of the relations found in the resonant optical cavities tutorial, the laser oscillation tutorial, the laser power tutorial, and the pulsed lasers tutorial are applicable to fiber lasers.
However, because at least one or two of the conditions for the applicability of (11-87) [refer to the laser power tutorial] are often violated in a fiber laser, the output power of a fiber laser should be found by using (11-88) [refer to the laser power tutorial].
Example 11-9
Because an erbium-doped fiber is a high-gain medium that can have a long length, an EDFA can be made into an erbium-doped fiber laser with a relatively small amount of optical feedback. The simplest approach is to cleave the two ends of the EDFA and leave them uncoated for a \(4\%\) reflectivity each, thus forming a Fabry-Perot cavity of \(R_1=R_2=4\%\) and a gain-medium length of \(l_\text{g}=l\) with a unity filling factor of \(\Gamma=1\). The EDFA of \(l=20\text{ m}\) pumped at \(\lambda_\text{p}=1.48\text{ μm}\) with a gain peak at \(\lambda=1.53\text{ μm}\) described in Example 10-13 [refer to the rare-earth ion-doped fiber amplifiers tutorial] is made into a fiber laser in this manner. Aside from the absorption associated with the laser transition levels, this fiber has a background distributed loss of \(\bar{\alpha}=2\text{ dB km}^{-1}\)
(a) Find the threshold pump power \(P_\text{p}^\text{th}\) of this fiber laser. What is the pump power utilization factor \(\zeta_\text{p}^\text{th}\) at the laser threshold?
(b) Find the transparency pump power \(P_\text{p}^\text{tr}\) of this fiber laser. What is the pump power utilization factor \(\zeta_\text{p}^\text{tr}\) at transparency?
(c) What is the output power of the laser if it is pumped with an input pump power of \(P_\text{p}=20\text{ mW}\)?
We find from Example 10-13 [refer to the rare-earth ion-doped fiber amplifiers tutorial] the following parameters for this fiber: \(\eta_\text{p}=1\), \(p=0.055\), \(\alpha_\text{p}=0.3485\text{ m}^{-1}\), and \(P_\text{p}^\text{sat}=4.25\text{ mW}\) at the pump wavelength of \(\lambda_\text{p}=1.48\text{ μm}\); \(\sigma_\text{a}=5.75\times10^{-25}\text{ m}^2\), \(\sigma_\text{e}=7.9\times10^{-25}\text{ m}^2\), and \(\Gamma_\text{s}=0.70\) at the signal wavelength of \(\lambda=1.53\text{ μm}\); \(N_\text{t}=2.2\times10^{24}\text{ m}^{-3}\).
Therefore, \(\alpha_\text{p}l_\text{g}=6.97\), \(\sigma_\text{a}N_\text{t}=1.265\), \(\sigma_\text{e}N_\text{t}=1.738\), \((\sigma_\text{e}+\sigma_\text{a})N_\text{t}=3.003\), and \((\sigma_\text{e}-p\sigma_\text{a})N_\text{t}=1.668\). The distributed loss is \(\bar{\alpha}=2\text{ dB km}^{-1}=0.46\text{ km}^{-1}=4.6\times10^{-4}\text{ m}^{-1}\). Because \(l_\text{g}=l=20\text{ m}\) and \(R_1=R_2=0.04\), the threshold gain coefficient of the laser is
\[g_\text{th}=\bar{\alpha}-\frac{\ln\sqrt{R_1R_2}}{l_\text{g}}=\left(4.6\times10^{-4}-\frac{\ln0.04}{2}\right)\text{ m}^{-1}=0.1614\text{ m}^{-1}\]
(a)
The threshold pump power can be found from (11-110) for \(p\ne0\):
\[\begin{align}P_\text{p}^\text{th}&=\frac{1}{0.055}\frac{\exp\left(0.055\times\frac{0.7\times1.265+0.1614}{0.7\times3.003}\times6.97\right)-1}{1-\exp\left(-\frac{0.7\times1.668-1.055\times0.1614}{0.7\times3.003}\times6.97\right)}\times4.25\text{ mW}\\&=16.87\text{ mW}\end{align}\]
The pump power utilization factor at threshold can be found from (11-109):
\[\zeta_\text{p}^\text{th}=1-\exp\left(-\frac{0.7\times1.738-0.1614}{0.7\times3.003}\times6.97\right)=0.970\]
(b)
The transparency pump power can be found from (11-110) for \(p\ne0\) by setting \(g_\text{th}\) to zero:
\[\begin{align}P_\text{p}^\text{tr}&=\frac{1}{0.055}\frac{\exp\left(0.055\times\frac{0.7\times1.265}{0.7\times3.003}\times6.97\right)-1}{1-\exp\left(-\frac{0.7\times1.668}{0.7\times3.003}\times6.97\right)}\times4.25\text{ mW}\\&=13.83\text{ mW}\end{align}\]
The pump power utilization factor at transparency can be found from (11-109) by setting \(g_\text{th}\) to zero:
\[\zeta_\text{p}^\text{tr}=1-\exp\left(-\frac{0.7\times1.738}{0.7\times3.003}\times6.97\right)=0.982\]
(c)
The output power of the laser can be found using (11-88) [refer to the laser power tutorial]. We find from Example 10-13 [refer to the rare-earth ion-doped fiber amplifiers tutorial] that \(P_\text{p}^\text{out}=0.984\text{ mW}\) when \(P_\text{p}^\text{in}=20\text{ mW}\). Thus, \(\zeta_\text{p}=(20-0.984)/20=0.951\). We also find that
\[\frac{\gamma_\text{out}}{\gamma_\text{c}}=\frac{-\ln\sqrt{R_1R_2}}{g_\text{th}l}=\frac{-\ln0.04}{0.1614\times20}=0.997\]
We can then find the output laser power from (11-88) [refer to the laser power tutorial]:
\[\begin{align}P_\text{out}&=\eta_\text{p}\frac{\gamma_\text{out}}{\gamma_\text{c}}\frac{\lambda_\text{p}}{\lambda}(\zeta_\text{p}P_\text{p}-\zeta_\text{p}^\text{th}P_\text{p}^\text{th})\\&=1\times0.997\times\frac{1.48}{1.53}\times(0.951\times20-0.970\times16.87)\text{ mW}\\&=2.56\text{ mW}\end{align}\]
We see from this example that \(P_\text{p}\gt{P}_\text{p}^\text{th}\gt{P}_\text{p}^\text{tr}\) and \(\zeta_\text{p}\lt\zeta_\text{p}^\text{th}\lt\zeta_\text{p}^\text{tr}\), as expected from the discussion following (11-86) in the laser power tutorial.
We also see that this laser has a relatively high transparency pump power compared to its threshold pump power, with \(P_\text{p}^\text{th}\) only \(23\%\) above \(P_\text{p}^\text{tr}\), even though this laser has end mirrors of very low reflectivities of \(R_1=R_2=4\%\).
Because \(P_\text{p}^\text{th}\) has to be always larger than \(P_\text{p}^\text{tr}\), this situation indicates that the laser threshold cannot be significantly reduced by using coated end mirrors of high reflectivities to reduce the output coupling loss of the laser.
It is possible to minimize the threshold pump power by choosing an optimum fiber length. It is also possible to maximize the output power at a given pumping level by properly choosing the fiber length. This two lengths are different because the former is independent of the input power power while the latter varies with the pump power.
One unique feature of a fiber laser is that the fiber gain medium can be made long to reduce the laser threshold. However, for the same reason as discussed in the rare-earth ion-doped fiber amplifiers tutorial for fiber amplifiers, the effect of increasing the length of the fiber gain medium on the threshold of a fiber laser depends on the nature of the particular fiber gain medium used.
For a four-level fiber laser, the threshold pump power decreases inversely with the length of the rare-earth ion-doped fiber gain medium, assuming that the background attenuation coefficient of the host glass is small.
If the gain medium of a fiber laser functions as a three-level or a quasi-two-level system, however, there is an optimum length for the minimum pump power threshold. Increasing the length of the fiber gain medium beyond the optimum length results in an increase in the threshold.
A rare-earth ion-doped fiber pumped by a properly chosen semiconductor laser is a high-gain device because of the high optical intensity in the fiber waveguide, the long fluorescence lifetime of the rare-earth ions, and the efficient use of the narrow-band pump power matching the absorption band of the rare-earth ions.
This high gain and the fact that a fiber has a large length and a small cross-sectional area make it possible for laser action in a rare-earth ion-doped fiber without a resonant cavity to take place through amplified spontaneous emission (ASE).
The condition for such laser action to occur is when the spontaneous emission originating from one end of the fiber is amplified through the fiber to an intensity that saturates the gain at the other end:
\[\tag{11-111}h\nu\Delta\nu{G}_0=P_\text{sat}\]
where \(h\nu\) is the energy of the spontaneous photon, \(\Delta\nu\) is the bandwidth of the spontaneous emission, \(G_0\) is the integrated unsaturated power gain over the length of the fiber, and \(P_\text{sat}\) is the saturation power as defined in (10-95) [refer to the laser amplifiers tutorial].
Such a device is known as an ASE fiber laser or a mirrorless fiber laser but is often also called a superfluorescent fiber laser. An ASE fiber laser has spatial coherence and , if a single-mode fiber is used, a single transverse mode pattern. However, it does not have a longitudinal mode structure.
Besides, its output has a broad spectrum and very little temporal coherence. It serves as a high-power, broadband light source, which is very useful in many applications where temporal coherence is not needed or is avoided.
Unlike an ordinary resonant laser, an ASE laser has no distinctive threshold. Its gain is never clamped at any particular level.
The next tutorial covers the topic of semiconductors.