An important class of lumped optical amplifiers makes use of rare-earth elements as a gain medium by doping the fiber core during the manufacturing process. Although doped-fiber amplifiers were studied as early as 1964, their use became practical only 25 years later, after their fabrication techniques were perfected. In such amplifiers, properties such as the operating wavelength and the gain bandwidth are determined by the dopants while silica plays the role of a host medium. Among the rare-earth elements, erbium is the most practical element to realize fiber amplifiers operating in the wavelength region near 1.5 μm, and erbium-doped fiber amplifiers (EDFAs) have been studied extensively. Their deployment in WDM systems after 1995 revolutionized fiber-optic communications and led to lightwave systems with capacities exceeding 1 Tb/s.
1. Pumping and Gain Spectrum
The fiber core inside an EDFA contains erbium ions (Er3+), and pumping them at a suitable wavelength provides the optical gain through population inversion. The gain spectrum depends on the pumping scheme as well as on the presence of other dopants, such as germania and alumina, within the fiber core. The amorphous nature of silica broadens the energy levels of Er3+ into bands. Figure (a) below shows a few energy bands of Er3+ in silica glasses.
Many transitions can be used to pump an EDFA. Efficient EDFA pumping is possible using semiconductor lasers operating near 0.98- and 1.48-μm wavelengths. Indeed, the development of such pump lasers was fueled with the advent of EDFAs. It is possible to realize a 30-dB gain with only 15 to 20 mW of absorbed pump power. Efficiencies as high as 11 dB/mW were achieved by 1990 with 0.98-μm pumping. Most EDFAs use 980-nm pump lasers for such lasers are commercially available and can provide more than 100 mW of pump power. Pumping at 1480 nm requires longer fibers and higher powers because it uses the tail of the absorption band shown in figure (b) above.
EDFAs can be designed to operate in such a way that the pump and signal beams propagate in opposite directions, a configuration referred to as backward pumping to distinguish it from the forward-pumping configuration. The performance is nearly the same in the two pumping configurations when the signal power is small enough for the amplifier to remain unsaturated. In the saturation regime, the power-conversion efficiency is generally better in the backward-pumping configuration, mainly because of the important role played by the amplified spontaneous emission (ASE). In the bidirectional-pumping configuration, an EDFA is pumped in both directions simultaneously by using two semiconductor lasers located at the two fiber ends. This configuration requires two pump lasers but has the advantage that the population inversion, and hence the small-signal gain, is relatively uniform along the entire amplifier length.
Figure (b) above shows the gain and absorption spectra of an EDFA whose core was doped with germania. The gain spectrum is quite broad and has a double-peak structure. Its shape is affected considerably by the amorphous nature of silica and by the presence of other dopants within the fiber core such as alumina. The gain spectrum is homogeneously broadened for isolated erbium ions. However, structural disorders lead to inhomogeneous broadening of the gain spectrum, whereas Stark splitting of various energy levels is responsible for homogeneous broadening. Mathematically, the gain should be averaged over the distribution of atomic transition frequencies ω0 resulting in the effective gain of the form is given by
where f(ω0) is the distribution function whose form also depends on the presence of other dopants within the fiber core.
The factor by which a weak input signal is amplified is obtained by integrating over the amplifier length L. If we neglect fiber losses because of small fiber lengths (~10 m) employed to make EDFAs, the amplification factor is given by
Although G(ω) is also often referred to as the gain spectrum, one should not confuse it with g0(z, ω). It can vary from amplifier to amplifier because it also depends on the amplifier length. In practice, both the bandwidth and the flatness of G(ω) are important for WDM systems. This issue is discussed later in this section.
2. Two-Level Model
The gain of an EDFA depends on a large number of device parameters such as erbium-ion concentration, amplifier length, core radius, and pump power. A three-level rate-equation model used commonly for lasers can be adapted for EDFAs. It is sometimes necessary to add a fourth level to include the excited-state absorption. In general, the resulting equations must be solved numerically.
Considerable insight can be gained by using a simple two-level model that is valid when ASE and excited-state absorption are negligible. The model assumes that the top level of the three-level system remains nearly empty because of a rapid transfer of the pumped population to the excited state. It is, however, important to take into account the different emission and absorption cross sections for the pump and signal fields. The population densities of the two states, N1 and N2, satisfy the following two rate equations:
where σja and σje are the absorption and emission cross sections at the frequency ωj with j = p, s. Further, Tl is the spontaneous lifetime of the excited state (about 10 ms for EDFAs). The quantities φp and φs represent the photon flux for the pump and signal waves, defined such that φj = Pj/(ajhνj), where Pj is the optical power, σj is the transition cross section at the frequency νj, and aj is the cross-sectional area of the fiber mode for j = p, s.
The pump and signal powers vary along the amplifier length because of absorption, stimulated emission, and spontaneous emission. If the contribution of spontaneous emission is neglected, Ps and Pp satisfy the simple equations
where α and α' take into account fiber losses at the signal and pump wavelengths, respectively. These losses can be neglected for typical amplifier lengths of 10-20 m but they must be included in the case of distributed amplification. The confinement factors Γs and Γp account for the fact that the doped region within the core provides the gain for the entire fiber mode. The parameter s = ± 1 depending on the direction of pump propagation; s = -1 in the case of a backward-propagating pump.
These four equations can be solved analytically, in spite of their complexity, after some justifiable approximations. For lumped amplifiers, the fiber length is short enough that both α and α' can be set to zero. Noting that N1 + N2 = Nt where Nt is the total ion density, only one equation need be solved. Noting again that the absorption and stimulated-emission terms in the field and population equations are related, the steady-state, obtained by setting the time derivative to zero, can be written as
where ad = Γsas = Γpap is the cross-sectional area of the doped portion of the fiber core. Substituting this solution into the last two equations and integrating them over the fiber length, the powers Ps and Pp at the fiber output can be obtained in an analytical form. This model has been extended to include the ASE propagation in both the forward and backward directions.
Although the preceding approach is essential when the total signal power inside the amplifier is large enough to cause gain saturation, a much simpler treatment is applicable in the so-called small-signal regime in which the EDFA remains saturated. In that case, the φs term can be neglected, and the gain coefficient, g(z) = (σesN2 - σasN1), does not depend on the signal power Ps. The total amplifier gain G for an EDFA of length L is then given by
The following figure shows the small-signal gain at 1.55 μm as a function of the pump power and the amplifier length by using typical parameter values.
For a given amplifier length L, the amplifier gain initially increases exponentially with the pump power, but the increase becomes much smaller when the pump power exceeds a certain value [corresponding to the "knee" in figure (a)]. For a given pump power, the amplifier gain becomes maximum at an optimum value of L and drops sharply when L exceeds this optimum value. The reason is that the latter portion of the amplifier remains unpumped and absorbs the amplified signal. Since the optimum value of L depends on the pump power Pp, it is necessary to choose both L and Pp appropriately. Figure (b) above shows that a 35-dB gain can be realized at a pump power of 5 mW for L = 30 m and 1.48-μm pumping. It is possible to design amplifiers such that high gain is obtained for amplifier lengths as short as a few meters. The qualitative features shown in the figure above are observed in all EDFAs; the agreement between theory and experiment is generally quite good.
The foregoing analysis assumes that both pump and signal waves are in the form of CW beams. In practice, EDFAs are pumped by using CW semiconductor lasers, but the signal is in the form of a pulse train (containing a random sequence of 1 and 0 bits), and the duration of individual pulses is inversely related to the bit rate. The question is whether all pulses experience the same gain or not. It turns out that the gain remains constant with time in an EDFA for even microsecond long pulses. The reason is related to a relatively large value of the fluorescence time associated with the excited erbium ions (Tl ~ 10 ms). When the time scale of signal-power variations is much shorter than Tl , erbium ions are unable to follow such fast variations. As single-pulse energies are typically much below the saturation energy (~10 μJ), EDFAs respond to the average power. As a result, gain saturation is governed by the average signal power, and amplifier gain does not vary from pulse to pulse for a WDM signal. This is an extremely useful feature of EDFAs.
In some applications such as packet-switched networks, signal power may vary on a time scale comparable to Tl. Amplifier gain in that case is likely to become time dependent, an undesirable feature from the standpoint of system performance. A gain-control mechanism that keeps the amplifier gain pinned at a constant value consists of making the EDFA oscillate at a controlled wavelength outside the range of interest (typically below 1.5 μm). Since the gain remains clamped at the threshold value for a laser, the signal is amplified by the same factor despite variations in the signal power. In one implementation of this scheme, an EDFA was forced to oscillate at 1.48 μm by fabricating two fiber Bragg gratings acting as high-reflectivity mirrors at the two ends of the amplifier.
3. Amplifier Noise
Amplifier noise is the ultimate limiting factor for system applications. All amplifiers degrade the signal-to-noise ratio (SNR) of the amplified signal because of spontaneous emission that adds noise to the signal during its amplification. Because of this amplified spontaneous emission (ASE), the SNR is degraded, and the extent of degradation is quantified through a parameter Fn, called the amplifier noise figure. In analogy with the electronic amplifiers, it is defined as
where SNR refers to the electrical power generated when the optical signal is converted into an electric current. In general, Fn depends on several detector parameters that govern thermal noise associated with the detector. A simple expression for Fn can be obtained by considering an ideal detector whose performance is limited by shot noise only.
Consider an amplifier with the gain G such that the output and input powers are related by Pout = GPin. The SNR of the input signal is given by
where ⟨I⟩ = RdPin is the average photocurrent, Rd = q/hν is the responsivity of an ideal photodetector with unit quantum efficiency, and
for shot noise by setting the dark current Id = 0. Here Δf is the detector bandwidth. To evaluate the SNR of the amplified signal, we should add the contribution of ASE to the receiver noise.
The spectral density of ASE is nearly constant (white noise) and can be written as
SASE(ν) = nsphν0(G-1)
where ν0 is the carrier frequency of the signal being amplified. The parameter nsp is called the spontaneous emission factor (or the population-inversion factor) and is given by
nsp = σeN2/(σeN2 - σaN1)
where N1 and N2 are the atomic populations for the ground and excited states, respectively. The effect of spontaneous emission is to add fluctuations to the amplified signal; these are converted to current fluctuations during the photodetection process.
It turns out that the dominant contribution to the receiver noise comes from the beating of spontaneous emission with the signal. The spontaneous emitted radiation mixes with the amplified signal and produces the current
at the photodetector of responsivity R. Noting that Ein and Esp oscillate at different frequencies with a random phase difference, it is easy to see that the beating of spontaneous emission with the signal will produce a noise current
where θ is a rapidly varying random phase. Averaging over the phase, the variance of the photocurrent can be written as
where cos2θ was replaced by its average value 1/2. The SNR of the amplified signal is thus given by
The amplifier noise figure is obtained by substituting (SNR)in and (SNR)out expressions into Fn definition and is given by
where the last approximation is valid for G ≫ 1. This equation shows that the SNR of the amplified signal is degraded by 3 dB even for an ideal amplifier for which nsp = 1. For most practical amplifiers, Fn exceeds 3 dB and can be as large as 6-8 dB.
The preceding analysis assumed that nsp was constant along the amplifier length. In the case of an EDFA, both N1 and N2 vary with z. The spontaneous-emission factor can still be calculated for an EDFA by using the two-level model discussed earlier, but the noise figure depends both on the amplifier length and the pump power Pp, just as the amplifier gain does. Figure (a) below shows the variation of Fn with the amplifier length for several values of Pp/Ppsat when a 1.53-μm signal is amplified with an input power of 1 mW. The amplifier gain under the same conditions is also shown in figure (b). The results show that a noise figure close to 3 dB can be obtained for a high-gain amplifier.
The experimental results confirm that Fn close to 3 dB is possible in EDFAs. A noise figure of 3.2 dB was measured in a 30-m-long EDFA pumped at 0.98 μm with 11 mW of power. A similar value was found for another EDFA pumped with only 5.8 mW of pump power at 0.98 μm. In general, it is difficult to achieve high gain, low noise, and high pumping efficiency simultaneously. The main limitation is imposed by the ASE traveling backward toward the pump and depleting the pump power. Incorporation of an internal isolator alleviates this problem to a large extent. In one implementation, 51-dB gain was realized with a 3.1-dB noise figure at a pump power of only 48 mW.
The measured values of Fn are generally larger for EDFAs pumped at 1.48 μm. A noise figure of 4.1 dB was obtained for a 60-m-long EDFA when pumped at 1.48 μm with 24 mW of pump power. The reason for a larger noise figure for 1.48-μm pumped EDFAs can be understood from figure below, which shows that the pump level and the excited level lie within the same band for 1.48-μm pumping. It is difficult to achieve complete population inversion (N1 ≈ 0) under such conditions. It is nonetheless possible to realize Fn < 3.5 dB for pumping wavelengths near 1.46 μm.
Relatively low noise levels of EDFAs make them an ideal choice for WDM lightwave systems. In spite of low noise, the performance of long-haul fiber-optic communication systems employing EDFAs is often limited by the amplifier noise. The noise problem is particularly severe when the system operates in the anomalous-dispersion region of the fiber because a nonlinear phenomenon known as the modulation instability enhances the amplifier noise and degrades the signal spectrum. Amplifier noise also introduces timing jitter. These issues are discussed later in this tutorial.
4. Multichannel Amplification
The bandwidth of EDFAs is large enough that they have proven to be the optical amplifier of choice for WDM applications. The gain provided by them is nearly polarization insensitive. Moreover, the interchannel crosstalk does not occur in EDFAs because of a relatively large value of Tl (about 10 ms) compared with typical bit durations (0.1 ns at a bit rate of 10 Gb/s) in lightwave systems. The sluggish response of EDFAs ensure that their gain cannot be modulated at frequencies much larger than 10 kHz.
A second source of interchannel crosstalk is cross-gain saturation occurring because the gain of a specific channel is saturated not only by its own power (self-saturation) but also by the power of neighboring channels. This mechanism of crosstalk is common to all optical amplifiers including EDFAs. It can be avoided by operating the amplifier in the unsaturated regime. Experimental results support this conclusion. In a 1989 experiment, negligible power penalty was observed when an EDFA was used to amplify two channels operating at 2 Gb/s and separated by 2 nm as long as the channel powers were low enough to avoid the gain saturation.
The main practical limitation of an EDFA stems from the spectral non-uniformity of the amplifier gain. Even though the gain spectrum of an EDFA is relatively broad, the gain is far from uniform (or flat) over a wide wavelength range. As a result, different channels of a WDM signal are amplified by different amounts. This problem becomes quite severe in long-haul systems employing a cascaded chain of EDFAs. The reason is that small variations in the amplifier gain for individual channels grow exponentially over a chain of in-line amplifiers. Even a 0.2-dB gain difference grows to 20 dB over a chain of 100 in-line amplifiers, making channel powers vary by up to a factor of 100, an unacceptable variation range in practice. To amplify all channels by nearly the same amount, the double-peak nature of the EDFA gain spectrum forces one to pack all channels near one of the gain peaks, thereby reducing the usable gain bandwidth considerably.
The entire bandwidth of 40 nm or so can be used if the gain spectrum is flattened by introducing wavelength-selective losses. The basic idea behind gain flattening is quite simple. If an optical filter whose transmission losses mimic the gain profile (high in the high-gain region and low in the low-gain region) is inserted after the doped fiber, the output power will become nearly constant for all channels. Although fabrication of such a filter is not simple, several gain-flattening techniques have been developed. Among others, thin-film interference filters, Mach-Zehnder filters, acousto-optic filters, and long-period fiber gratings have been used for flattening the gain profile and equalizing channel gains.
The gain-flattening techniques can be divided into active and passive categories. The filter-based methods are passive in the sense that channel gains cannot be adjusted in a dynamic fashion. The location of the optical filter itself requires some thought because of high losses associated with it. Placing it before the amplifier increases the noise, while placing it after the amplifier reduces the output power. Often a two-stage configuration shown in the figure below is used. The second stage acts as a power amplifier but the noise figure is mostly determined by the first stage whose noise is relatively low because of its low gain. A combination of several long-period fiber gratings acting as the optical filter in the middle of two stages resulted by 1997 in an EDFA whose gain was flat to within 1 dB over the 40-nm bandwidth in the wavelength range of 1530-1570 nm.
Ideally, an optical amplifier should provide the same gain for all channels under all possible operating conditions. This is not the case in general. For instance, if the number of channels being transmitted changes, the gain of each channel will change since it depends on the total signal power (because of gain saturation). An active control of channel gains is thus desirable for WDM applications. Many techniques have been developed for this purpose. The most commonly used technique stabilizes the gain dynamically by incorporating within the amplifier a laser that operates outside the used bandwidth. Such devices are called gain-clamped EDFAs (as their gain is clamped by a built-in laser) and have been studied extensively.
WDM lightwave systems capable of transmitting more than 80 channels appeared by 1998. Such systems use the C and L bands simultaneously and need uniform amplifier gain over a bandwidth exceeding 60 nm. Moreover, the use of the L band requires optical amplifiers capable of providing gain in the wavelength range 1570-1610 nm. It turns out the EDFAs can provide gain over this wavelength range, with a suitable design. An L-band EDFA requires long fiber lengths (> 100 m) to keep the inversion level relatively low. The figure below shows an L-band amplifier with a two-stage design.
The first stage is pumped at 980 nm and acts as a traditional EDFA (fiber length 20-30 m) capable of providing gain in the range 1530-1570nm. In contrast, the second stage has 200-m-long doped fiber and is pumped bidirectionally using 1480-nm lasers. An optical isolator between the two stages passes the ASE from the first stage to the second stage (necessary for pumping the second stage) but blocks the backward-propagating ASE from entering the first stage. Such cascaded, two-stage amplifiers can provide flat gain over a wide bandwidth wile maintaining a relatively low noise level. As early as 1996, flat gain to within 0.5 dB was realized over the wavelength range of 1544-1561 nm. The second EDFA was codoped with ytterbium and phosphorus and was optimized such that it acted as a power amplifier. Since then, EDFAs providing flat gain over the entire C and L bands have been made. Raman amplification can also be used for the L band. Combining Raman amplification with one or two EDFAs, uniform gain can be realized over a 75-nm bandwidth covering the C and L bands.
A parallel configuration has also been developed for EDFAs capable of amplifying over the C and L bands simultaneously. In this approach, the incoming WDM signal is split into two branches, which amplify the C-band and L-band signals separately using an optimized EDFA in each branch. The two-arm design has produced a relatively uniform gain of 24 dB over a bandwidth as large as 80 nm when pumped with 980-nm semiconductor lasers while maintaining a noise figure of about 6 dB. The two-arm or two-stage amplifiers are complex devices and contain multiple components, such as optical filters and isolators, within them for optimizing the amplifier performance. An alternative approach to broadband EDFAs uses a fluoride fiber in place of silica fiber as the host medium in which erbium ions are doped. Gain flatness over a 76-nm bandwidth has been realized by doping a tellurite fiber with erbium ions. Although such EDFAs are simpler in design compared with multistage amplifiers, they suffer from the splicing difficulties because of the use of non-silica glasses.
High-capacity lightwave systems as well as coarse WDM systems (channel spacing > 5 nm) are likely to make sue of the short-wavelength region - the so-called S band, extending from 1470 to 1520 nm - in addition to the C and L bands. Thulium-doped fiber amplifiers were initially developed for this purpose, and they are capable of providing flat gain in the wavelength range 1480-1510 nm when pumped using 1420-nm and 1560-nm semiconductor lasers. More recently, attention has focused on EDFAs that can provide gain simultaneously in all three bands with a suitable design. Semiconductor optical amplifiers are also being considered for coarse WDM systems. However, fiber-based Raman amplifiers are also likely to be deployed because they can provide amplifications in any spectral region with a suitable choice of pump lasers; we turn to them in the next tutorial.