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This is because absorption by the large population of atoms in the lower energy level is more prevalent than stimulated emission by the smaller population of atoms in the upper level.. A

CHAPTER

13 LASER AMPLIFIERS

13.1 THE LASER AMPLIFIER

A Amplifier Gain

B Amplifier Phase Shift

13.2 AMPLIFIER POWER SOURCE

A Rate Equations

B Four- and Three-Level Pumping Schemes

C Examples of Laser Amplifiers

13.3 AMPLIFIER NONLINEARITY AND GAIN SATURATION

A Gain Coefficient

B Gain

*C Gain of Inhomogeneously Broadened Amplifiers

*13.4 AMPLIFIER NOISE

emission of radiation (laser) They received the Nobel Prize in 1964

460

Bahaa E A Saleh, Malvin Carl Teich

Copyright © 1991 John Wiley & Sons, Inc

ISBNs: 0-471-83965-5 (Hardback); 0-471-2-1374-8 (Electronic)

A coherent optical amplifier is a device that increases the amplitude of an optical field while maintaining its phase If the optical field at the input to such an amplifier is monochromatic, the output will also be monochromatic, with the same frequency The output amplitude is increased relative to the input while the phase is unchanged or shifted by a fixed amount In contrast, an amplifier that increases the intensity of an optical wave without preserving the phase is called an incoherent optical amplifier This chapter is concerned with coherent optical amplifiers Such amplifiers are important for various applications; examples include the amplification of weak optical pulses such as those that have traveled through a long length of optical fiber, and the production of highly intense optical pulses such as those required for laser-fusion applications Furthermore, it is important to understand the principles underlying the operation of optical amplifiers as a prelude to the discussion of optical oscillators (lasers) in Chap 14

The underlying principle for achieving the coherent amplification of light is light amplification by the stimulated emission of radiation, known by its acronym as the LASER process Stimulated emission (see Sec 12.2) allows a photon in a given mode

to induce an atom in an upper energy level to undergo a transition to a lower energy level and, in the process, to emit a clone photon into the same mode as the initial photon (viz., a photon with the same frequency, direction, and polarization) These two photons, in turn, can serve to stimulate the emission of two additional photons, and so

on, while preserving these properties The result is coherent light amplification Because stimulated emission occurs when the photon energy is nearly equal to the atomic-transition energy difference, the process is restricted to a band of frequencies determined by the atomic linewidth

Laser amplification differs in a number of respects from electronic amplification Electronic amplifiers rely on devices in which small changes in an injected electric current or applied voltage result in large changes in the rate of flow of charge carriers, such as electrons and holes in a semiconductor field-effect transistor (FET) or bipolar junction transistor Tuned electronic amplifiers make use of resonant circuits (e.g., a capacitor and an inductor) or resonators (metal cavities) to limit the amplifier’s gain to the band of frequencies of interest In contrast, atomic, molecular, and solid-state laser amplifiers rely on their energy-level differences to provide the primary frequency selection These act as natural resonators that select the amplifier’s bandwidth and frequencies of operation Optical cavities (resonant circuits) are often used to provide auxiliary frequency tuning

Light transmitted through matter in thermal equilibrium is attenuated rather than amplified This is because absorption by the large population of atoms in the lower energy level is more prevalent than stimulated emission by the smaller population of atoms in the upper level An essential ingredient for achieving laser amplification is the presence of a greater number of atoms in the upper energy level than in the lower level, which is clearly a nonequilibrium situation Achieving such a population inver- sion requires a source of power to excite (pump) the atoms into the higher energy level,

as illustrated in Fig 13.0-l Although the presentation throughout this chapter is couched in terms of “atoms” and “atomic levels,” these appelations are to be more broadly understood as “active medium” and “laser energy levels,” respectively

461

Atoms

Laser amplifier

Figure 13.0-l The laser amplifier An external power source (called a pump) excites the active medium (represented by a collection of atoms), producing a population inversion Photons interact with the atoms; when stimulated emission is more prevalent than absorption, the medium acts as a coherent amplifier

The properties of an ideal (optical or electronic) coherent amplifier are displayed schematically in Fig 13.0-2(a) It is a linear system that increases the amplitude of the input signal by a fixed factor, called the amplifier gain A sinusoidal input leads to a sinusoidal output at the same frequency, but with larger amplitude The gain of the ideal amplifier is constant for all frequencies within the amplifier spectral bandwidth The amplifier may impart to the input signal a phase shift that varies linearly with frequency, corresponding to a time delay of the output with respect to the input (see Appendix B)

Real coherent amplifiers deliver a gain and phase shift that are frequency depen- dent, typically in the manner illustrated in Fig 13.0-2(b) The gain and phase shift constitute the amplifier’s transfer function For a sufficiently high input amplitude, furthermore, real amplifiers may exhibit saturation, a form of nonlinear behavior in which the output amplitude fails to increase in proportion to the input amplitude Saturation introduces harmonic components into the output, provided that the ampli-

output amplitude

THE LASER AMPLIFIER 463

fier bandwidth is sufficiently broad to pass them Real amplifiers also introduce noise,

so that a randomly fluctuating component is always present at the output, regardless of the input

An amplifier may therefore be characterized by the following features:

13.1 THE LASER AMPLIFIER

A monochromatic optical plane wave traveling in the z direction with frequency v, electric field Re{E(z)exp(j2rvt)}, intensity 1(z) = IE(z)12/217, and photon-flux den- sity 4(z) = I(z)/hv (photons per second per unit area) will interact with an atomic medium, provided that the atoms of the medium have two relevant energy levels whose energy difference nearly matches the photon energy hv The numbers of atoms per unit volume in the lower and upper energy levels are N, and N,, respectively The wave is amplified with a gain coefficient y(z) (per unit length) and undergoes a phase shift q(z) (per unit length) We proceed to determine expressions for y(v) and q(v) Positive y(v) corresponds to amplification; negative y(v), to attenuation

A Amplifier Gain

Three forms of photon-atom interaction are possible (see Sec 12.2) If the atom is in the lower energy level, the photon may be absorbed, whereas if it is in the upper energy level, a clone photon may be emitted by the process of stimulated emission These two processes lead to attenuation and amplification, respectively The third form of interaction, spontaneous emission, in which an atom in the upper energy level emits a photon independently of the presence of other photons, is responsible for amplifier noise as discussed in Sec 13.4

The probability density (s- ‘) that an unexcited atom absorbs a single photon is, according to (12.2-15) and (12.2-ll),

w;: = &(v>, (13.1-1) where a(v) = (h2/&t,,)g(v) is the transition cross section at the frequency v, g(v) is the normalized lineshape function, t,, is the spontaneous lifetime, and h is the wavelength of light in the medium The probability density for stimulated emission is also given by (13.1-1)

The average density of absorbed photons (number of photons per unit time per unit volume) is N,lVi Similarly, the average density of clone photons generated as a result

Since the incident photons travel in the z direction, the stimulated-emission photons will also travel in this direction, as illustrated in Fig 13.1-l An external pump providing a population inversion (N > 0) will then cause the photon-flux density 4(z)

to increase with z Because emitted photons stimulate further emissions, the growth at any position z is proportional to the population at that position; 4(z) will thus increase exponentially

To demonstrate this process explicitly, consider an incremental cylinder of length dz and unit area as shown in Fig 13.1-1 If 4(z) and 4(z) + d4(z) are the photon-flux densities entering and exiting the cylinder, respectively, then d+(z) must be the photon-flux density emitted from within the cylinder This incremental number of photons per unit area per unit time C@(Z) is simply the number of photons gained per unit time per unit volume, NlVi, multiplied by the thickness of the cylinder dz, i.e.,

The coefficient y(v) represents the net gain in the photon-flux density per unit length

of the medium The solution of (13.1-3) is the exponentially increasing function

THE LASER AMPLIFIER 465 Since the optical intensity 1(z) = hv+(z), (13.1-5) can also be written in terms of I as

Thus y(v) also represents the gain in the intensity per unit length of the medium The amplifier gain coefficient y(v) is seen to be proportional to the population difference A/ = N, - N, Although N was considered to be positive in the example provided above, the derivation is valid whatever the sign of N In the absence of a population inversion, N is negative (N2 < N,) and so is the gain coefficient The medium will then attenuate (rather than amplify) light traveling in the z direction, in accordance with the exponentially decreasing function 4(z) = 4(O) exp[ -a(v)z], where the attenuation coefficient (Y(Y) = -y(v) = -No(v) A medium in thermal equilib- rium therefore cannot provide laser amplification

For an interaction region of total length d (see Fig 13.1-l), the overall gain of the laser amplifier G(v) is defined as the ratio of the photon-flux density at the output to the photon-flux density at the input, G(v) = 4(d)/+(O), so that

(13.1-7)

Amplifier Gain

The dependence of the gain coefficient y(v) on the frequency of the incident light v is contained in its proportionality to the lineshape function g(v), as given in (13.1-4) The latter is a function of width Av centered about the atomic resonance frequency

V a = (E2 - E,)/h, where E, and E, are the atomic energies The laser amplifier is therefore a resonant device, with a resonance frequency and bandwidth determined by the lineshape function of the atomic transition This is because stimulated emission and absorption are governed by the atomic transition The linewidth Av is measured either

in units of frequency (Hz) or in units of wavelength (nm) These linewidths are related

by Ah = ]A(c,/v)I = +(c,/v2) Av = (At/c,) Av Thus a linewidth Av = 1012 Hz at

Figure 13.1-2 Gain coefficient y(v) of a Lorentzian-lineshape laser amplifier

EXERCISE 13.1-I

Attenuation and Gain in a Ruby Laser Amplifier

(a) Consider a ruby crystal with two energy levels separated by an energy difference corresponding to a free-space wavelength A, = 694.3 nm, with a Lorentzian lineshape

of width Av = 60 GHz The spontaneous lifetime is t,, = 3 ms and the refractive index

of ruby is II = 1.76 If N, + N, = A!, = lo** cmm3, determine the population differ- ence N = N2 - A/, and the attenuation coefficient at the line center LY(VJ under conditions of thermal equilibrium (so that the Boltzmann distribution is obeyed) at

T = 300 K

(b)

cc>

What value should the population difference

Ykl) = 0.5 cm-’ at the central frequency?

N assume to achieve a gain coefficient How long should the crystal be to provide an overall gain of 4 at the central frequency when y(vO) = 0.5 cm- ‘?

B Amplifier Phase Shift

Because the gain of the resonant medium is frequency dependent, the medium is dispersive (see Sec 5.5) and a frequency-dependent phase shift must be associated with its gain The phase shift imparted by the laser amplifier can be determined by considering the interaction of light with matter in terms of the electric field rather than the photon-flux density or the intensity

We proceed with an alternative approach, in which the mathematical properties of a causal system are used to determine the phase shift For homogeneously broadened media, the phase-shift coefficient cp(v) (phase shift per unit length of the amplifier medium) is related to the gain coefficient y(v) by the Kramers-Kronig (Hilbert transform) relations (see Sec B.l of Appendix B and Sec 5.5), so that knowledge of y(v) at all frequencies uniquely determines q(v)

The optical intensity and field are related by I(z) = lE(~)]~/271 Since I(z) = I(0) exp[y(v)z] in accordance with (13.1-6), the optical field obeys the relation

E(Z) = E(O) exp[+y(+] exp[ -idv>zl~ (13.1-10)

THE LASER AMPLIFIER 467 where <p(v) is the phase-shift coefficient The field evaluated at z + At is therefore

E( z + AZ) = E(z) exp[ &(v) Az] ew[ -k(v) Az]

=E(z)[~ + $y(v)Az -&(v>Az], (13.1-11) where we have made use of a Taylor-series approximation for the exponential func- tions The incremental change in the electric field A E(z) = E(z + AZ) - E(z) there- fore satisfies the equation

A simple example is provided by the Lorentzian atomic lineshape function with narrow width Au +C vo, for which the gain coefficient y(v) is given by (13.1-9) The corresponding phase shift coefficient cp(v) is provided in (B.l-13) of Appendix B,

The Lorentzian gain and phase-shift coefficients are plotted in Fig 13.1-3 as functions

of frequency At resonance, the gain coefficient is maximum and the phase-shift

(al

(b’ LvL

Figure 13.1-3 (a) Gain coefficient y(v) and (b) phase-shift coefficient (P(Y) for a laser amplifier with a Lorentzian lineshape function

coefficient is zero The phase-shift coefficient is negative

nance and positive for frequencies above resonance

for frequencies below reso-

13.2 AMPLIFIER POWER SOURCE

Laser amplifiers, like other amplifiers, require an external source of power to provide the energy to be added to the input signal The pump supplies this power through mechanisms that excite the electrons in the atoms, causing them to move from lower to higher atomic energy levels To achieve amplification, the pump must provide a population inversion on the transition of interest (A/ = A/, - N, > 0) The mechanics

of pumping often involves the use of ancillary energy levels other than those directly involved in the amplification process, however The pumping of atoms from level 1 into level 2 might be most readily achieved, for example, by pumping them from level 1 into level 3 and then by relying on the natural processes of decay from level 3 to popu- late level 2

The pumping may be achieved optically (e.g., with a flashlamp or laser), electrically (e.g., through a gas discharge, an electron or ion beam, or by means of injected electron and holes as in semiconductor laser amplifiers), chemically (e.g., through a flame), or even by means of a nuclear explosion to achieve x-ray laser action For continuous-wave (CW) operation, the rates of excitation and decay of all of the different energy levels participating in the process must be balanced to maintain a steady-state inverted population for the l-2 transition The equations that describe the rates of change of the population densities N, and N, as a result of pumping, radiative, and nonradiative transitions are called the rate equations They are not unlike the equations presented

in Sec 12.3, but selective external pumping is now permitted so that thermal equilib- rium conditions no longer prevail

A Rate Equations

Consider the schematic energy-level diagram of Fig 13.2-1 We focus on levels 1 and 2, which have overall lifetimes r1 and 72, respectively, permitting transitions to lower levels The lifetime of level 2 has two contributions -one associated with decay from 2

to 1 (car), and the other (~~a) associated with decay from 2 to all other lower levels When several modes of decay are possible, the overall transition rate is a sum of the component transition rates Since the rates are inversely proportional to the decay times, the reciprocals of the decay times must be added,

Multiple modes of decay therefore shorten the overall lifetime (i e., they render the decay more rapid) Aside from the radiative spontaneous emission component (of time

-l-

- T&l + 7;; (13.2-1)

Figure 13.2-1 Energy levels 1 and 2 and their decay times

AMPLIFIER POWER SOURCE 469

Figure 13.2-2 Energy levels 1 and 2, together with surrounding higher and lower energy levels

constant ts,) in 721, a nonradiative contribution T,, may also be present (arising, for example, from a collision of the atom with the wall of the container thereby resulting in

a depopulation), so that

If a system like that illustrated in Fig 13.2-1 is allowed to reach steady state, the population densities A/, and A/, will vanish by virtue of all the electrons ultimately decaying to lower energy levels

Steady-state populations of levels 1 and 2 can be maintained, however, if energy levels above level 2 are continuously excited and leak downward into level 2, as shown

in the more realistic energy level diagram of Fig 13.2-2 Pumping can bring atoms from levels other than 1 and 2 out of level 1 and into level 2, at rates R, and R, (per unit volume per second), respectively, as shown in simplified form in Fig 13.2-3 Conse- quently, levels 1 and 2 can achieve nonzero steady-state populations

We now proceed to write the rate equations for this system both in the absence and

in the presence of amplifier radiation (which is the radiation resonant with the 2-l transition)

Rate Equations in the Absence of Amplifier Radiation

The rates of increase of the population densities of levels 2 and 1 arising from pumping and decay are

n Long 72 (but t,,, which contributes to 72 through 7-21, must be sufficiently short

so as to make the radiative transition rate large, as will be seen subsequently)

n Short 71 if R, < (72/~2, JR,

The physical reasons underlying these conditions make good sense The upper level should be pumped strongly and decay slowly so that it retains its population The lower level should depump strongly so that it quickly disposes of its population Ideally, it is desirable to have 721 = t,, -=K T~() so that 72 = fsp, and T, K t,, Under these condi- tions we obtain a simplified result:

a transition probability W Assume that TV = t,, and 71 s t,, so that in steady state

4 - 0 and No = R2tsp If N, is the total population of levels 0, 1, and 2, show that R, = (N, - 2N,W, so that th e population difference is N,, = N,t,,W/(l + 2t,,W)

Rate Equations in the Presence of Amplifier Radiation

The presence of radiation near the resonance frequency v0 enables transitions between levels 1 and 2 to take place by the processes of stimulated emission and absorption as well These are characterized by the probability density Wi = c#w(v), as provided in (13.1-l) and illustrated in Fig 13.2-4 The rate equations (13.2-2) and (13.2-3) must

AMPLIFIER POWER SOURCE 471

Figure 13.2-4 The population densities N, and A!, (cm -3-s-1) of atoms in energy levels 1 and

2 are determined by three processes: decay (at the rates l/~~ and l/~~, respectively, which includes the effects of spontaneous emission), pumping (at the rates -RI and R,, respectively), and absorption and stimulated emission (at the rate wl:)

then be extended to include this source of population loss and gain in each of the levels:

Under steady-state conditions (dN,/dt = dN,/dt = 0), (13.2-5) and (13.2-6) are readily solved for N, and N,, and for the population difference N = N, - N, The result is

Steady-State Population Difference (in Presence

of Amplifier Radiation)

Saturation Time bb”,s:zi

where N, is the steady-state population difference in the absence of amplifier radia- tion, given by (13.2-4) The characteristic time TV is always positive since 72 I 721

In the absence of amplifier radiation, Wi = 0 so that (13.2-7) provides N = No, as expected Because T, is positive, the steady-state population difference in the presence

of radiation always has a smaller absolute value than in the absence of radiation, i.e.,

1 NI I I N,I If the radiation is sufficiently weak so that T,Wi c 1 (the small-signal approximation), we may take N = No As the radiation becomes stronger, Wi increases and N approaches zero regardless of the initial sign of N,, as shown in Fig 13.2-5 This arises because stimulated emission and absorption dominate the interaction when Wi is very large and they have equal probability densities It is apparent that even very strong radiation cannot convert a negative population difference into a positive population difference, nor vice versa The quantity TV plays the role of a saturation time constant,

as is evident from Fig 13.2-S

0

Figure 13.2-5 Depletion of the steady-state population difference N = N, - N, as the rate of absorption and stimulated emission Wi increases When Wi = l/~~, N is reduced by a factor of 2 from its value when Wi = 0

EXERCISE 13.2-2

Saturation Time Constant Show that if t,, +z T,, (the nonradiative part of the lifetime

721 of the 2-1 transition), -=c 720, and z+ TV, then TV = t,,

We now proceed to examine specific (four- and three-level) schemes that are used in practice to achieve a population inversion The object of these arrangements is to make use of an excitation process to increase the number of atoms in level 2 while decreasing the number in level 1

B Four- and Three-Level Pumping Schemes

Four-Level Pumping Schemes

In this arrangement, shown in Fig 13.2-6, level 1 lies above the ground state (which is designated as the lowest energy level 0) In thermal equilibrium, level 1 will be virtually unpopulated, provided that E, x=- k,T, which is of course highly desirable Pumping is accomplished by making use of the energy level (or collection of energy levels) lying

AMPLIFIER POWER SOURCE 473 above level 2 and designated level 3 The 3-2 transition has a short lifetime (decay occurs rapidly) so that there is little accumulation in level 3 For reasons that are made clear in Problem 13.2-1, level 2 is pumped through level 3 rather than directly Level 2

is long-lived, so that it accumulates population, whereas level 1 is short-lived so that it sustains little accumulation All told, four energy levels are involved in the process but the optical interaction of interest is restricted to only two of them (levels 1 and 2)

An external source of energy (e.g., photons with frequency E,/h) pumps atoms from level 0 to level 3 at a rate R If the decay from level 3 to 2 is sufficiently rapid, it may be considered to be instantaneous, in which case pumping to level 3 is equivalent to pumping level 2 at the rate R, = R In this configuration, atoms are neither pumped into nor out of level 1, so that R, = 0 The situation is then the same as that shown in Fig 13.2-4 Thus the expressions in (13.2-7) and (13.2-8) apply In the absence of amplifier radiation <Wj = C$ = 0), the steady-state population difference is given by (13.2-4) with R, = 0, i.e.,

In most four-level systems, the nonradiative decay component in the transition between

2 and 1 is typically negligible (tsp -=c T,,) and 720 Z= t,, B 71 (see Exercise 13.2-2), so that

N= ts,N,w

Finally, the population difference can be written in the generic form of (13.2-7),

rather than by (13.2-10) and (13.2-11) Under conditions of weak pumping (W -=SC l/t,,),

No = t,,N,W is proportional to W (the pumping transition probability density), and 7s = b,, giving rise to the results obtained previously However, as the pumping increases, N, saturates and 7S decreases

Three-Level Pumping Schemes

A three-level pumping arrangement, in contrast, makes use of the ground state (El = 0) as th e 1 ower laser level 1, as shown in Fig 13.2-7 Again, an auxiliary third level (designated 3) is involved The 3-2 decay is rapid so that there is no buildup of population in level 3 The 3-l decay is slow (i.e., ~32 -=x 731) so that the pumping ends

up populating the upper laser level Level 2 is long-lived so that it accumulates population Atoms are pumped from level 1 to level 3 (e.g., by absorbing radiation at the frequency E,/h) at a rate R; their fast (nonradiative) decay to level 2 provides the pumping rate R, = R

It is not difficult to see that under rapid 3-2 decay, the three-level system displayed

in Fig 13.2-7 is a special case of the system shown in Fig 13.2-4 (provided that R is independent of N) with the parameters

R, = R, = R, r1 = 00, 72 = 721

To avoid algebraic problems in connection with the value 71 = a, rather than substitut- ing these special values into (13.2-7) and (13.2-8), we return to the original rate equations (13.2-5) and (13.2-6) In the steady state, both (13.2-5) and (13.2-6) result in

AMPLIFIER POWER SOURCE 475 the same equation,

0 = R - N, - N*M/,: + N,Wi (13.2-18)

It is not possible to determine both N, and N, from a single equation relating them However, knowledge of the total atomic density N, in the system (in levels 1, 2, and 3) provides an auxiliary condition that does permit N, and N, to be determined Since

~32 is very short, level 3 retains a negligible steady-state population; all of the atoms that are raised to it immediately decay to level 2 Thus

Note that rs = t,, for four-level pumping schemes [see (13.2-ll)]

It is of interest to compare these equations with the analogous results (13.2-10) and (13.2-11) for a four-level pumping scheme Attaining a population inversion (N > 0 and therefore No > 0) in the three-level system requires a pumping rate R > N,/2t,, Thus, just to make the population density N, equal to N, (i.e., No = 0) requires a substantial pump power density, given by E,N,/2 t,, The large population in the ground state (which is the lowest laser level) provides an inherent obstacle to achieving

a population inversion in a three-level system that is avoided in four-level systems (in which level 1 is normally empty)

The dependence of the pumping rate R on the population difference N can be included in the analysis of the three-level system by writing R = (N, - N,)W, N, = 0, and N, = i(N, - N), from which R = i( N, - N)W Substituting in the principal equation N = (2Rt,, - NJ/(1 + 2t,W,), and reorganizing terms, we again obtain

N= NO

1 + ?,Wi ’ but now with

N

0

= N,kp - 1)

and

Thus, as in the four-level scheme,

pumping transition probability W

(13.2-25)

A/, and TV are in general nonlinear functions of the

EXERCISE 13.2-3

Pumping Powers in Three- and Four-Level Systems

(a) Determine the pumping transition probability W required to achieve a zero population difference in a three- and a four-level laser amplifier

(b) If the pumping transition probability W = 2/t,, in the three-level system and = 1/2t,,

in the four-level system, show that N, = N,/3 Compare the pumping powers required

to achieve this population difference

Examples of Pumping Methods

As indicated earlier, pumping may be achieved by many methods, including the use of electrical, optical, and chemical means A number of common methods of electrical and optical pumping are illustrated schematically in Fig 13.2-8

It is important to note that R, and R, represent the numbers of atoms/cm3-s that are pumped successfully, The pumping process is generally quite inefficient In optical pumping, for example, many of the photons supplied by the pump fail to raise the atoms to the upper laser level and are therefore wasted

C Examples of Laser Amplifiers

Laser amplification can take place in a great variety of materials The energy-level diagrams for several atoms, molecules, and solids that exhibit laser action were shown

in Sec 12.1A Practical laser systems usually involve many interacting energy levels that influence N, and A/,, the populations of the transition of interest, as illustrated in Fig 13.2-2 Nevertheless, the essential principles of laser amplifier operation may be understood by classifying lasers as either three- or four-level systems

This is illustrated by three solid-state laser amplifiers which are discussed in turn below: the three-level ruby laser amplifier, the four-level neodymium-doped yttrium-aluminum garnet laser amplifier, and the three-level erbium-doped silica fiber laser amplifier Although most laser amplifiers and oscillators operate on the basis of a four-level pumping scheme, two notable exceptions are ruby and Er3+-doped silica fiber Laser amplification can also be achieved with gas lasers and liquid lasers, as indicated briefly near the end of this section All of the laser amplifiers discussed here also operate as laser oscillators (see Sec 14.2E)