## Abstract

This chapter describes the determination of amplifying parameters in rare–earth–doped optical fiber laser amplifiers. In the context of this review, the system will be analyzed under both continuous–wave (CW) and pulse conditions. A comprehensive analysis has been implemented using the set of coupled propagation rate equations based on the atomic energy structure of dopant as well as the absorption and emission cross sections.

### Keywords

- Gain
- Saturation
- ASE parasitic noises
- Broadening
- Amplification relation
- Filling (overlapping) factor

## 1. Introduction

Recently, intense activities have been devoted to characterize rare–earth–doped double–clad fiber laser amplifiers. Owing to their compactness, superb beam quality, great thermal control, and high efficiency, they demonstrate to be important light sources in medicine, modern telecommunication [1], and industries [2, 3]. Progress has been made in developing a fiber–based source in which the mean power scales up to several kilowatts.

The purpose of the fiber amplifier is to intensify a less powerful optical beam, which propagates either inside another fiber or in free space. In this way, the amplifier is “seeded” with a low–power laser beam. As it is known from the main principles of laser physics, the laser oscillator is a device that contains an optical amplifier and puts inside a resonator to provide positive feedback.

The fiber amplifiers are fundamentally divided into core–pumped and cladding–pumped [4] according to the environment of the pump propagation. In the former classification, the amplifying light propagates in the same volume where laser gain media is located, while in the latter, the pump is coupled and propagates outside the core of the doped fiber. The same classification is valid for fiber laser oscillators. Cladding–pumped amplification technique is a hot topic in laser science and technology for high–power regimes, which is the focus of this chapter.

After the invention of the dual–clad fibers, the output powers of the doped fiber lasers have been lifted by several orders of magnitude, and immense activities have been devoted to relevant topics. These systems demonstrate several inherent features including [5] non–uniformly distributed population inversion due to end pumping as well as very high gain even in single–pass amplification by a large ratio of gain length to cross–sectional area.

In addition, both neodymium (Nd) and ytterbium (Yb) are suitable doping elements for high–power regime. Despite the fact that Nd has a relatively low laser threshold, Yb–doped double–clad fiber amplifiers present the prospects for a number of interesting applications due to the broad gain bandwidth with excellent beam quality, renowned as potential sources accessible to high–pump absorption, leading to ultimate efficiency [5, 6]. Furthermore, the benefits from efficient performance, small quantum defects, and superb spectroscopic characteristics as well as free competition processes are competitive with other rare–earth dopants. Particularly, the capability of generating high powers makes them very attractive wherein the output power up to 10/50 kW at CW single–mode/multi–mode schemes has been recently reported.

Besides, a fiber amplifier is characterized by a couple of significant coefficients, i.e., the small signal gain and the saturation quantity. In this chapter, a theoretical consideration of the CW and pulsed fiber arrays is reviewed, followed by the derived main formulas that are important for practical design. In Section 2, a brief description has been devoted to various types of laser amplifiers, operating based on the stimulated emission and optical nonlinearities. An extensive analysis is given in Section 3, regarding the set of coupled propagation rate equations accompanied with the effect of amplified spontaneous emission (ASE) parasitic noises for various pumping modes based on the atomic energy structure of dopants. The serious spectral line broadening mechanisms of gain media is described in Section 4. Eventually, the main issue of relevant amplification relation has been investigated in order to find the corresponding gain saturation parameters.

## 2. Various types of fiber laser amplifiers

Different kinds of laser amplifiers operate based on the stimulated emission amplification principle, or established upon optical nonlinearities, e.g., Raman or Brillouin amplifiers as well as optical parametric amplifiers (OPAs).

Raman–based fiber laser amplifier is one approach that has attracted several groups [7–13]. Here, the governed amplification process is the stimulated Raman scattering (SRS) in the fiber which causes an energy transfer from the pump to the signal. In fact, the vibrational spectrum of core material defines the Raman shift. If the wavelength of laser beam signal is known, then the optimum wavelength for pump signal can be calculated.

Commonly, Raman fiber lasers and amplifiers are not considered as ways for generating high–power narrow–linewidth lasers. For high–efficient operations, typically hundreds of meters of fiber are necessary to provide enough gain.

On the contrary, the optical gain is obtained by stimulated Brillouin scattering (SBS) in the Brillouin fiber amplifiers. It is also pumped optically, and a part of the pump power is transmitted to the signal through SBS. Physically, each pump photon creates a signal photon, and the remaining energy is used to excite an acoustic phonon [14]. Classically, the pump beam gets scattered from an acoustic wave moving through the medium at the speed of sound.

Moreover, the fiber–based OPA is a well–known technique offering a wide gain bandwidth using only a few hundred meters of fiber [15]. This phenomenon is strictly operated based on four–wave mixing (FWM) and phase–matching nonlinearities [16] in which a couple of interacting photons over two wavelength bands, also called pump and idler, usually travel collinearly through a nonlinear optical crystal, creating signal and idler output photons. During this phenomenon, pump light beam becomes weaker and amplifies the idler wave.

For birefringent nonlinear crystals, the collinear incident beams may non–collinearly scatter outside the crystal. Non–collinear OPAs were developed to have additional degree of freedom for central wavelength selection, allowing constant gain up to second order in wavelength.

## 3. Theory

### 3.1. Absorption and emission cross sections

All rare–earth–doped fiber amplifiers intensify the seed signal light through stimulated emission. The local rate equations describe the dynamics of the emission and absorption processes of doping ions within its host material by making use of its atomic energy structure as well as spectroscopic properties [17]. The relation between the emission and absorption cross sections for two energy–level gain media is determined using McCumber theory, which satisfies the following formula [18]:

where

where

Here, * n*is the refractive index of the gain media, and

*denotes the speed of light in vacuum. As an example, Figure 1 demonstrates a model for the Yb*C

^{3+}ions energy–level structure, consisting of two manifolds: a ground state

^{2}F

_{7/2}(with four Stark levels labeled

^{2}F

_{5/2}(with three Stark levels labeled

^{−1}above the ground level. This is the reason why there is no excited state absorption (ESA) at either the pump or laser wavelengths [6]. This large energy gap also precludes the concentration quenching and the nonradiative decay via multiphonon emission from

^{2}F

_{5/2}even in a host of high phonon energy such as silica. The energy band diagram for the lasing material is a quasi–three–level system [20].

The first transition between two Yb manifolds is the absorption and emission of the pump. Afterward, the spontaneous decay from the ^{2}F_{5/2} manifold, as well as the absorption, and stimulated emission of the signal simultaneously appear.

Absorption and stimulated emission cross sections of Yb–doped silica glass optical fibers are shown in the spectroscopic diagram in Figure 2. The absorption or fluorescence peak at 975 nm (A) represents the zero–line transition between the lowest energy levels of the ground state (^{3+} laser systems working around 1 μm.

In addition, the emission spectrum peak (D) corresponds to the energy transfer from level ^{3+} ions enables the easy configuration of the pump wavelength. Depending on the requirement of the laser system, the laser signal wavelength can be configured in the range from 970 nm to 1200 nm due to the wide emission spectrum of Yb.

### 3.2. Rate equations

Amplifier characteristics, comprising the operating wavelength and the gain bandwidth, are determined by the dopants rather than by the host medium. However, because of the tight confinement of light provided by guided modes, fiber amplifiers can provide high optical gains at moderate pump intensity levels over relatively large spectral bandwidths, making them suitable for many telecommunications and signal–processing applications [22].

Currently, the rate equations are known as the most powerful tools to foresee the laser amplifier features associated with the non–uniform pumping along the laser length [23–31]. Here, the rate equations for the quasi–three–energy–level structures are typically introduced with the definition of the relevant parameters. Using the fourth–order Runge–Kutta method, the effect of ASE parasitic noises and gain can be considered to assess the efficiency. The performance of amplifiers also depends on the rates of radiative and nonradiative decays caused by several mechanisms related to lattice vibrations, ion–ion interactions, and cooperative up–conversion to higher levels [32].

In the case of fiber amplifier as a single–pass array without reflectors, the backward signal intensity,

where the superscripts “±” describe the traveling directions of the light waves along the fiber propagation axis (

Furthermore,

In order to solve these equations, the cooperative up–conversion coefficient (

where

On the contrary, the local intensity per unit wavelength,

In low–power communication fiber amplifiers, the core and cladding structures are centrosymmetric having circular cross sections. In order to remove this power limitation, an intelligent solution was proposed in 1988 by Snitzer and coworkers, namely double–clad fibers. At this model, the fiber core is off–centered or the inner cladding form is triangular, rectangular, hexagonal, elliptical, D–shaped, and so on, where the rays scatter in many accessible directions [25, 38, 39]. Hereupon, the pump light helically propagates through the clad and interacts with the active core. The pump overlapping factor (

where

For single–clad fibers, the dimensionless coefficient

Here,

where

where * E*and

*depict the electric and magnet fields of the propagation mode, respectively, and the versor of the*H

*propagation direction is indicated by*z

where

On the one hand, the normalized frequency is defined as * NA*represents the corresponding numerical aperture. For single–mode fibers,

On the other hand, along with the signal, the spontaneous emission is also intensified. In fact, the parasitic ASE noises cause the degradation of the signal–to–noise ratio (SNR) [47–49]. It is due to the lack of reflectors such as fiber Bragg gratins (FBGs) or dichroic mirrors in the amplifier stage, while the selective wavelength oscillates in the oscillator. The evolution formula for the ASE intensities propagating in a given direction is written according to Eq. (7). The total ASE noise at point * z*along the fiber is the consequence of the intensity of ASE from the previous sections of the fiber accompanying the local intensity

The positive scattering parameter ^{4}/km, and

In addition,

It is worth noting that the boundary conditions for the ASE channels are given by

Furthermore, Rayleigh backscattering (RBS) is another important issue that affects the performance of the signal source, unless the optical isolators are well employed [48, 49, 53]. When ASE and RBS are intense enough, these may restrict the amplifier gain leading to a drop in the efficiency in many applications. In the case of strongly pumped condition, RBS at the ASE wavelengths can be ignored mainly due to the reduction of ASE to get significantly weaker than the signal [54]. Here, we assume that

Thus, the rate equations at signal wavelength can be simplified as follows:

However, in the case of CW fiber amplifiers, disregarding ESA and cooperative up–conversion, the set of first–order coupled nonlinear time–independent steady–state (* ∂*/

∂t

It is worth mentioning that there are no significant nonlinear optical effects such as SBS and SRS as well as thermal damages up to 50 W–CW single–mode pump powers [5]. In addition, the dipole–dipole interaction, clustering, and quenching are serious only in highly doping elements.

### 3.3. Gain saturation

Tremendous efforts were made to develop the fiber amplifiers during the recent decade. Indeed, the amplifier is characterized by very important and unique gain and saturation parameters [55–60] that are often used to analyze and compare different gain mediums in certain applications.

The advantages of operating in the gain saturation regime are mainly summarized as follows [4]:

Small fluctuations in the input signal do not reflect the same extent in the output amplified signal.

The fiber amplifier, which has multiple spectrally close input signals with varied intensity, may work as a gain equalizer because smaller input signal powers have higher gain (through less saturation), and higher input powers have lower gain (due to a higher degree of saturation).

A saturated optical amplifier demonstrates a high–energy extraction efficiency; therefore, the overall efficiency of the system is high.

According to theoretical analysis, high–input optical fluence does not increase indefinitely in an amplifier but rather saturates, leading to amplifier gain reduction. Obviously, even from the general consideration of energy conservation, the gain of the amplifier has to saturate, because one cannot extract more power from the amplifier than it was excited (pumped) with.

The emission and absorption cross sections are used to introduce the gain coefficient. According to Eq. (28), the gain for signal radiation after simplification can be defined as follows:

In a fiber amplifier, it depends on the distance * z*from its input end. By substituting Eq. (26) into Eq. (30) and ignoring ASE noises, the gain is rearranged in the following formalism:

Due to high signal power, which saturates the gain, the first term of the dominator is small enough with respect to others. Assuming

where the signal saturation power is introduced as follows:

It decreases the small–signal gain by a factor of one–half equivalent to a 3 dB reduction in gain [49]. It is worth mentioning that not only

From another point of view, the gain coefficient in the steady–state condition can be explained by

where

Let us assume a doped dual–clad fiber amplifier with length

where

Here,

Hence

In the general case of the varied inversion along the gain fiber, the absolute gain factor should be integrated along the entire fiber length [4]:

Accordingly, the gain and saturation properties are strongly dependent on the pump power, dopant concentration, fiber length, as well as the pumping modes (co–/counter–propagation and bidirectional pumping) mainly due to the dominant effect of the filling factors [26, 61, 62].

In return, if optical pulses are amplified, the gain coefficient associates with the elapsed time * t*due to

where

## 4. The spectral line broadening

The nature of spectral line broadening plays a very important role in the performance of the fiber laser amplifiers. The stimulated emission cross section is conventionally rewritten by

where

The spectral line broadening at room temperature (300 K) smooths the overall line shape, which becomes resolved only at low temperatures [4]. With a temperature decrease, Stark level structure becomes more and more evident and determines the line shape characteristic profile.

In the case of rare–earth–doped fiber oscillator and amplifier, different types of broadenings exist. These include strong homogeneous as well as the inhomogeneous broadenings [56, 63]. The designation of whether the transition line shape is homogeneously or inhomogeneously broadened is based on whether the lines are from the same type of centers or from different sorts.

Usually, a homogeneous broadening is explained by the random perturbation of a similar type of optical centers, while that from different kinds is responsible for inhomogeneously broadened lines [4]. Such kinds of interaction result in the shortening of the excited state lifetime of the optical center.

Therefore, the homogeneous broadening is inherent to each atom in a medium as depicted in Figure 3(a). It can be spontaneous broadening, Stark broadening, collision broadening, thermal and dipolar, and so on. The spontaneous broadening is a primary broadening yielded from the lifetime of a pumping level and the frequency uncertainty. The Stark broadening is due to the degeneration of energy level caused by external electric field, making the lifetime very short and the spectral width very large. Moreover, the collision broadening is dominant for gas lasers as excimer lasers operate with high gas pressure discharge.

Conversely, the inhomogeneous broadening of the optical center’s line shape during a transition between energy levels originates from a local site–to–site variation in the optical center’s surrounding field in the lattice environment. The strength and symmetry of the field around the rare–earth ion determine the spectral properties of the optical transitions, as well as the transition strength. According to Figure 3(b), in the case of inhomogeneous broadening, the overall shape of the spectral line is a superposition of all individual, homogeneously broadened lines corresponding to different types of the optical centers. Hence, this is a phenomenon that various frequencies are generated by a variety of influences, and the spectrum is broadened.

The inhomogeneous broadening can be a strain caused by lattice defect or inhomogeneity of magnetic or electric field (crystalline field) for solid–state lasers and Doppler broadening for gas lasers.

In a crystal of solid–state laser, the crystalline field is not uniform since the crystal is imperfect. As a result, a large inhomogeneous broadening is yielded. In a glass doped with rare–earth ions, the inhomogeneous width is relatively largely broadened compared with the crystalline solid–state laser. The amorphous nature of the glass, random distribution of dopants, and host material imperfection lead to site–to–site variations of the local electric field, producing a degree of inhomogeneous broadening in the system which results in spectral hole burning (SHB).

Indeed, the optical transition line shape in the case of homogeneous broadening is described by a Lorentzian function given by

where

Subsequently, the line shape of the inhomogeneously broadened optical transition corresponds to the Gaussian line profile, which can be expressed as the following function of frequency:

where

## 5. Amplification relations

The state–of–the–art fiber amplifiers are significantly related to the determination of amplifying parameters (gain and saturation) which are not explicitly introduced into the rate equations; however, these can be obtained by using a suitable amplification relation. It is applicable for the homogeneous line broadening in Yb:silica fiber amplifiers [63] as well as the Er–doped silica fiber lasers with inhomogeneous line broadening.

In general, the laser amplification consists of a couple of distinct transient (short pulse) and steady–state (long pulse) regimes. The gain and saturation values can be estimated by numerically solving the amplification relations performing the best fitting based on least square method (LSM).

### 5.1. Pulsed regime

In the transient form of the amplification relation, the input pulse is considerably shorter than the fluorescence lifetime of the medium, such as Q–switched or mode–locked pulses with inhomogeneous gain media. The propagation rate equations are the best procedure to simulate the amplified signal energy released from the stimulated emission. This process depends on the energy stored in the upper laser level prior to the synchronization situation of the pumped atoms and incident seed signal. Furthermore, the time–dependent photon transport equation can be utilized to evaluate how an inverted population affects the distribution of a pulse traversing through the amplifier. Hopf [65] and Frontz–Nodvik [66] solved the nonlinear equations for various types of input pulse shapes to obtain the inverted electron population and the photon flux, while the input signal to the amplifier was taken as a square pulse of duration

where

For the pulses whose shapes are sufficiently near the eigen–function for the propagation in the amplifying medium, the change in the average temporal width of the pulse can be reasonably expected to be negligible. In such cases, the approximation

where

Here,

### 5.2. CW regime

For the seed signal, which is long compared to the fluorescent time including the free running oscillator or the CW seed laser [5, 67, 68] under homogeneous line broadening, a Hargrove [69] steady–state gain characterizes the amplification mechanism as follows [5, 6]:

Here,

where

In addition, in the limiting cases

These relations indicate that for low–input powers, the overall gain (

A straightforward integration shows that the CW gain of the amplifier is considered by

The origin of gain saturation lies in the power dependence of the gain coefficient. Since

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