Fundamentals of Photonics 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)
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 C. Examples of Laser Amplifiers
13.3
AMPLIFIER NONLINEARITY A. Gain Coefficient B. Gain *C. Gain of Inhomogeneously
*13.4
AMPLIFIER
Townes, emission
Basov, and Prokhorov developed of radiation (laser). They received
Schemes
AND GAIN SATURATION
Broadened
Amplifiers
NOISE
the principle of light amplification the Nobel Prize in 1964.
by the
stimulated
460
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 inversion 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
462
LASER
AMPLIFIERS
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 dependent, 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 Ideal amplifier
output
G,yhaI
T/
w /’
Input
-,
amplitude
(al
output ampy
Azhgi; RFal amplifier
5-
output
++ 4/
#- YJ
u
Input amplitude
161
Figure 13.0-2 (a) An ideal amplifier is linear. It increases the amplitude of signals (whose frequencies lie within its bandwidth) by a constant gain factor, possibly introducing a linear phase shift. (b) A real amplifier typically has a gain and phase shift that are functions of frequency, as shown. For large inputs the output signal saturates; the amplifier exhibits nonlinearity.
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: . Gain . Bandwidth n Phase shift n Power source n Nonlinearity and gain saturation . Noise We proceed to discuss these characteristics in turn. In Sec. 13.1 the theory of laser amplification is developed, leading to expressions for the amplifier gain, spectral bandwidth, and phase shift. The mechanisms by which an amplifier power source can achieve a population inversion are examined in Sec. 13.2. Sections 13.3 and 13.4 are devoted to gain saturation and noise in the amplification process, respectively. This chapter relies on material presented in Chap. 12, especially in Sec. 12.2.
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 density 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
464
LASER
AMPLIFIERS Amplifier
0
z
z+dz
d
Figure 13.1-1 The photon-flux density 4 (photons/cm2-s) containing excited atoms grows to C$ + ~C#Jafter length dz.
entering
an incremental
cylinder
of stimulated emission is N214$ The net number of photons gained per second per unit volume is therefore A/kVi, where A/ = N, - A/, is the population density difference. For convenience, N is simply referred to as the population difference. If N is positive, a population inversion exists, in which case the medium can act as an amplifier and the photon-flux density can increase. If it is negative, the medium acts as an attenuator and the photon-flux density decreases. If N = 0, the medium is transparent. 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., d4 = NWi dz. With the help of (13.1-l), (13.1-2) can be written
in the form of a differential
d4(4 - dz = Y(+P(Z)~
(13.1-2)
equation, (13.1-3)
where
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 4(z)
= 4(O) exP[YWzl*
(13.1-5)
THE LASER
Since the optical intensity 1(z) = hv+(z),
465
AMPLIFIER
(13.1-5) can also be written
I( z>= WQedrWz3.
in terms of I as (13.1-6)
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 equilibrium 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 A, = 0.6 pm corresponds to Ah = 1.2 nm. For example, if the lineshape function is Lorentzian, (12.2-27) provides Av/2~ g(v)
The gain coefficient
=
(v - vo)2 + (Av/~)~
is then also Lorentzian
Y(V) = Y(Vo)(v
(13.1-8)
’
with the same width, i.e.,
wm2 - vo)2 + ( Av/~)~
as illustrated in Fig. 13.1-2, where y(vO) = N(h2/4r2t,, the central frequency vo.
’
(13.1-9)
Av) is the gain coefficient
at
466
LASER
AMPLIFIERS
Figure 13.1-2
EXERCISE Attenuation
Gain coefficient y(v) of a Lorentzian-lineshape
laser amplifier.
13.1-I 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. What value should the population difference N assume to achieve a gain coefficient (b) = 0.5 cm-’ at the central frequency? Ykl) cc>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
where Az],
Az] (13.1-11)
where we have made use of a Taylor-series approximation for the exponential functions. The incremental change in the electric field A E(z) = E(z + AZ) - E(z) therefore satisfies the equation
A%4 - AZ = E(z)[ $Y(v) - b(v)].
(13.1-12)
This incremental amplifier may be regarded as a linear system whose input and output are E(z) and A E(z)/A z, respectively, and whose transfer function is
WV) = +I+> -.i4+).
(13.1-13)
Because this incremental amplifier represents a physical system, it must be causal. But the real and imaginary parts of the transfer function of a linear causal system are related by the Hilbert transform (see Appendix B). It follows that -cp(v> is the Hilbert transform of & [see (5.5ll)] so that the amplifier phase shift function is determined by its gain coefficient. 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.
468
LASER
AMPLIFIERS
coefficient is zero. The phase-shift coefficient is negative for frequencies nance and positive for frequencies above resonance.
13.2
AMPLIFIER
POWER
below reso-
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 populate 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 equilibrium 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, 72
-l-
-
T&l
+
7;;.
(13.2-1)
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
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. Consequently, 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 dN2 -
dt
=R,-
N,
(13.2-2)
72
(13.2-3)
Under steady-state conditions (dN,/dt = dN,/dt = 0), (13.2-2) and (13.2-3) can be solved for N, and N,, and the population difference N = N, - N, can be found. The
Figure 13.2-3 Energy levels 1 and 2 and their decay times. By means of pumping, the population density of level 2 is increased at the rate R, while that of level 1 is decreased at the rate R,.
470
LASER
AMPLIFIERS
result is
of Amplifier
Radiation)
where the symbol N, represents the steady-state population difference N in the absence of amplifier radiation. A large gain coefficient clearly requires a large population difference, i.e., a large positive value of N,. Equation (13.2-4) shows that this may be achieved by: n n
n
Large R, and R,. which contributes to 72 through 7-21,must be sufficiently short Long 72 (but t,,, so as to make the radiative transition rate large, as will be seen subsequently). 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 conditions we obtain a simplified result: ( 13.2-4a)
In the absence of depumping (R, = O), or when R, -=x (tJ7,)R2, simplifies to
No =
EXERCISE
R,t,,.
this result further
(13.2-4b)
13.2- 1
Optical Pumping. Assume that R, = 0 and that R, is realized by exciting atoms from the ground state E = 0 to level 2 using photons of frequency E,/h absorbed with 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, = + 2t,,W). (N, - 2N,W, so that th e population difference is N,, = N,t,,W/(l
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 levels: dN2
-
dt
= R, - N,
dN1 =
-
dt
loss and gain in each of the
- N2Wi + N,Wi
(13.2-5)
72
-RI
- 5 71
+ 3
+ N2Wi - N,Wi.
(13.2-6)
721
The population density of level 2 is decreased by stimulated emission from level 2 to level 1 and increased by absorption from level 1 to level 2. The spontaneous emission contribution is contained in 721. 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
N=
No 1 + TsWi
(13.2-7) 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 radiation, 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., 1NI I IN,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.
472
LASER
AMPLIFIERS
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. 721 of the 2-1 transition),
Show that if t,, +z T,, (the nonradiative
-=c 720, and z+ TV, then
TV
part of the lifetime
= 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
3 I Rapid decay
Short-lived
level
Short-lived
level
1 T32
I ’
Figure 13.2-6
Ground
state
Energy levels and decay rates for a four-level system.
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 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 = N,kp 0
1+
- 1) t,,W
’
(13.2-24)
476
LASER
AMPLIFIERS
and
(13.2-25)
Thus, as in the four-level scheme, A/, and TVare in general nonlinear pumping transition probability W.
EXERCISE Pumping
functions of the
13.2-3
Powers
in Three- and Four-Level
(a) Determine the difference in a (b) If the pumping in the four-level to achieve this
Systems
pumping transition probability W required to achieve a zero population three- and a four-level laser amplifier. transition probability W = 2/t,, in the three-level system and = 1/2t,, system, show that N, = N,/3. Compare the pumping powers required 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).
AMPLIFIER
ld’(
Lens
Nds+:YAG
POWER
477
SOURCE
LaG
rod
diode
Er3+
:stlica
fiber
Figure 13.2-8 Examples of electrical and optical pumping. (a) Direct current (dc) is often used to pump gas lasers. The current may be passed either along the laser axis, to give a longitudinal discharge, or transverse to it. The latter configuration is often used for high-pressure pulsed lasers, such as the transversely excited atmospheric (TEA) CO, laser. (b) Radio-frequency (RF) discharge currents are also used for pumping gas lasers. (c) Flashlamps are effective for optically pumping ruby and rare-earth solid-state lasers. (d) A semiconductor injection laser diode (or array of laser diodes) can be used to optically pump Nd3+:YAG or Er3+:silica fiber lasers.
Ruby Ruby (Cr 3+-A1,03) . is sapphire (Al,O,), in which chromium ions (Cr3’) replace a small percentage of the aluminum ions (see Sec. 12.1A). As with most materials, laser action can take place on a variety of transitions. The energy levels pertinent to the well-known red ruby laser transition are shown in Fig. 13.2-9 (these are labeled in group-theory notation). Ruby is the first material in which laser action was observed. In
ev
Ruby 4
Pump I
1
I
694.3-nm
i
laser
I
1
i
Figure 13.2-9 Energy levels pertinent to the 694.3-nm red ruby laser transition. interacting levels are indicated in circles.
The three
478
LASER Ruby
AMPLIFIERS Flashlamp
Capacitor
I
Elliptical mirror
POL-NP
SUPPlY
W
(a)
Figure 13.2-l 0 The ruby laser amplifier. (a) Geometry used in the first laser oscillator built by Maiman in 1960 (see Chap. 14). (b) Cross section of a high-efficiency geometry using a rod-shaped flashlamp and a reflecting elliptical cylinder.
essence, ruby is a three-level system in which level 1 is the ground state, level 2 consists of a pair of closely spaced discrete levels (the lower of which corresponds to the red laser transition at A, = 694.3 nm), and level 3 comprises two bands of energies centered at about 550 nm (green) and 400 nm (violet). These absorption bands are responsible for the pink color of the material. The material may be optically pumped from level 1 to level 3 by surrounding the ruby rod by a flashlamp
or enclosing
it with
a rod-shaped
flashlamp
within
a reflecting
cylinder of elliptical cross section, as shown in Fig. 13.2-10. The flashlamp emits broad-spectrum radiation, some fraction of which is absorbed and results in the excitation of the Cr3 + ions to level 3. The broad nature of level 3 is useful in maximizing the percentage of pump light absorbed. Excited Cr3+ ions rapidly decay from level 3 to level 2 (r3* is of the order of picoseconds), whereas the spontaneous lifetime for the 2-l transition is relatively long (tSp = 3 ms), in agreement with the scheme shown in Fig. 13.2-7. Nonradiative decay is negligible (r2r = t,,). The transition has a homogeneously broadened linewidth Au = 60 GHz, arising principally from elastic collisions with lattice phonons. Commercially available ruby laser amplifiers use rods that are typically 5 to 20 cm in length. They can deliver a small-signal gain of about 20 in the pulsed mode. The properties of a typical ruby laser oscillator are provided in Table 14.2-1. Nd 3 >YAG and Nd 3 ?Glass A useful near-infrared four-level laser amplifier makes use of neodymium in the form of impurity ions in a crystal of yttrium-aluminum garnet (Nd,Y,-,Al,O,,, usually written as Nd3+*YAG). . The crystal is pale purple in color. The energy levels pertinent to the A, = 1.064~pm transition are shown in Fig. 13.2-11; spectroscopic notation is used. Level 1 has an energy = 0.2 eV above the ground state. This energy is substantially larger than k,T = 0.026 eV at room temperature, so that the thermal population of the lower laser level is negligible. Level 3 is a collection of four = 30-nm-wide absorption bands centered at about 810, 750, 585, and 525 nm. The 2-l transition is homogeneously broadened (as a result of collisions with lattice phonons), with a room-temperature linewidth Au = 120 GHz. The excited ions rapidly decay from level 3 to level 2 (732 = 100 ns), the spontaneous lifetime t,, is 1.2 ms, and r1 is short (= 30 ns), in agreement with the four-level scheme shown in Fig. 13.2-6. The gain is substantially greater than that of ruby by virtue of it being a four-level system. Nd3 +:YAG can also be optically pumped directly to the upper laser level; an efficient laser system making use of this scheme has recently been developed in which the pump is a semiconductor injection laser [see Fig. 13.2-8(d)].
AMPLIFIER
POWER
SOURCE
479
Nds+:YAG
03
-2
EA
T32
I
\
0’2 II I
1.064-pi
Pump
4F3,2
1
laser
I @cL4rll,2
00
41 912
J
0
13.2-11 Energy levels pertinent to the 1.064~pm Nd3+:YAG laser transition. The energy levels for Nd 3f:glass are similar but the absorption bands are broader.
Figure
The neodymium in glass laser amplifier (Nd3+:glass) has characteristics that are quite similar to those of Nd 3+*YAG, . with the notable exception that it is inhomogeneously broadened; this is a result of the amorphous nature of glass, which presents a different environment at each ionic location. Nd3+:glass therefore has a far larger room-temperature linewidth, Av = 3000 GHz, which turns out to be desirable for mode-locked pulsed lasers (see Chap. 14). Nd3+:glass amplifiers can be made in very large sizes and have been used extensively in laser fusion experiments (particularly in the lo-beam NOVA laser system at the Lawrence Livermore National Laboratory in California, which is capable of delivering 10’ J in a 1-ns pulse and in the GEKKO system at Osaka University in Japan). The characteristics of typical Nd3+:YAG and Nd3+:glass laser oscillators are provided in Table 14.2-1. Er 3 +Silica . Fiber Rare-earth-doped silica fibers can serve as useful laser amplifying media while offering the advantages of single-mode guided-wave optics (see Chaps. 7 and 8). In particular they offer polarization-independent gain and low insertion loss. The core of the silica fiber may be doped with any of a number of rare-earth ions (e.g., Nd, Er, Yb, Pr, Sm). Pumping is achieved by transmitting laser light (e.g., light from a semiconductor injection laser, dye laser, color-center laser, Ti3+:Al,03 laser, or At-+ ion laser) through the fiber [see Fig. 13.2-8(d)]. Fiber laser amplifiers can be made to operate over a broad range of wavelengths (e.g., 1.3 pm, 1.55 pm, 2 to 3 pm). Er3+:silica fibers, in particular, have a broad laser transition (Au = 4000 GHz) near h = 1.55 ,um, which coincides with the wavelength of maximum transmission for silica fibers (see Fig. 8.3-2). Because of their high gain, erbium-doped silica fibers offer substantial promise for use as optical amplifiers and repeaters in fiber-optic communication systems. In one configuration, an 807-nm semiconductor laser pump is used to drive a l-m-long SiO,:GeO, fiber (typical fiber lengths lie in the range between 0.5 and 10 m) doped with = 500 parts per million (ppm) erbium. This wavelength, as well as 980 nm, are convenient because of the presence of strong pumping bands in Er3+. However, pumping at 807 nm gives rise to undesirable excited-state absorption. The laser transition can instead be directly pumped at 1.48 pm by light from an InGaAsP semiconductor laser in which case excited-state absorption does not occur. Efficient
480
LASER
AMPLIFIERS
TABLE 13.2-l
Characteristics
Laser Medium He-Ne Ruby Nd3 +:YAG Nd3+:glass Er 3 +:siIica fiber Rhodamine-6G dye Ti3+:A1203 co2
Arf ‘H and I indicate respectively.
of a Number
of Important
Transition Wavelength A, (pm)
Transition Cross Section
0.6328 0.6943 1.064 1.06 1.55 0.56-0.64 0.66-1.18 10.6 0.515
1 x lo-‘3 2 x 10-u’ 4 x lo-l9
line
broadening
co (cm2)
“6 ; ;;:‘2: 2 x 10-16 ; ; :;I:; 3 x lo-t2 dominated
Laser Transitions
Spontaneous Lifetime
Transition Linewidth” AV
t SP 0.7 ps 3.0 ms 1.2 ms 0.3 ms 10.0 ms 3.3 ns 3.2 ps 2.9 s 10.0 ns
by homogeneous
1.5 GHz 60 GHz 120 GHz 3 THz 4 THz 5 THz 100 THz 60 MHz 3.5 GHz
Refractive Index n
1 1.76 1.82 1.5 1.46 1.33 1.76 z 1 z 1
I
z
H H I WI WI H I I
and inhomogeneous
mechanisms,
light amplification is possible because of the frequency shift that exists between the fluorescence and absorption bands of this transition. Currently, gains = 30 dB are available by launching = 5 mW of pump power (from a diode laser pump operated at either 980 nm or 1.48 pm) into a roughly 50-m length of fiber containing = 300 ppm Er,O,. Optical bandwidths = 30 nm can be obtained, although larger bandwidths are possible with reduced gain. The Er3+:silica fiber system behaves as a three-level laser at T = 300 K and as a four-level laser when cooled to T = 77 K. The broadening is a mixture of homogeneous (phonon mediated) and inhomogeneous (arising from local field variations in the glass). Other Laser Amplifieers The transition cross section, spontaneous lifetime, transition linewidths, and refractive indices of several important laser transitions are provided in Table 13.2-1. The free-space wavelength A, shown in the table represents the most commonly used transition in each laser medium. The He-Ne gas laser system, for example, is most often used on its red-orange line at 0.633 pm, but it is also extensively used at 0.543, 1.15, and 3.39 pm (it also has laser transitions at hundreds of other wavelengths). CO, is a commonly used laser amplifying medium in the middle-infrared region of the spectrum. The values reported in the table are typical for low-pressure operation (the atomic linewidth in a gas depends on its pressure because of the role of collision broadening, which is a homogeneous broadening mechanism). The tunable rhodamine-6G dye laser, which is usually pumped by an Ar+ laser, provides gain over a continuous band of wavelengths stretching from 560 to 640 nm. Other dyes cover different wavelength regions. Dye laser amplifiers enjoy broad application and are effective for the amplification of femtosecond optical pulses. The Ti3+:Al,03 laser enjoys even broader tunability than the rhodamine-6G dye laser and at the same time is far easier to operate. Free-electron laser systems are also often used for amplification. The semiconductor laser amplifier is discussed in Chap. 16.
13.3 A.
AMPLIFIER
NONLINEARITY
AND GAIN
SATURATION
Gain Coefficient
It has been established that the gain coefficient y(v) of a laser medium depends on the population difference A/ [see (13.1-4)]; that N depends on the transition rate wl: [see (13.2-7)]; and that PVi, in turn, depends on the radiation photon-flux density 4 [see
AMPLIFIER NONLINEARITY AND GAIN SATURATION
481
(13.1-l)]. It follows that the gain coefficient of a laser medium is dependent on the photon-flux density that is to be amplified. This is the origin of gain saturation and laser amplifier nonlinearity, as we now show. Substituting (13.1-l) into (13.2-7) provides (13.3-1)
where
Density
This represents the dependence of the population difference N on the photon-flux density 4. Now, substituting (13.3-1) into the expression for the gain coefficient (13.1-4) leads directly to the saturated gain coefficient for homogeneously broadened media:
Y(V) =
Ycl(4 1 + 4/4,(v) ’
(13.3-3) Saturated Gain Coefficient
where
y&> = &+>
A2 = ~oj-&v). SP
1
(13.3-4) Small-Signal Gain Coefficient
The gain coefficient is a decreasing function of the photon-flux density 4, as illustrated in Fig. 13.3-l. The quantity 4,(v) = l/ 7s(T( v ) re p resents the photon-flux density at which the gain coefficient decreases to half its maximum value; it is therefore called the saturation photon-flux density. When 7s = t,, the interpretation of 4,(v) is straightforward: Roughly one photon can be emitted during each spontaneous emission time into each transition cross-sectional area [a(v)~#~,(v)t,, = 11. YW Y&4 -I
13.3-1 Dependence of the normalized saturated gain coefficient ~(v)/Y&v) on the normalized photon-flux density +/4&v). When C$ equals its saturation value 4,(v), the gain coefficient is reduced by a factor of 2.
Figure
482
LASER AMPLIFIERS
EXERCISE
13.3- 1
Saturation Photon-Flux Density for Ruby. Determine the saturation photon-flux density, and the corresponding saturation intensity, for the A, = 694.3-nm ruby laser transition at v = vo. Use the parameters provided in Table 13.2-1 on page 480. Assume that 7.9= 2tsp, in accordance with (13.2-23)
EXERCISE
13.3-2
Spectral Broadening of a Saturated Amplifier. Consider a homogeneously broadened amplifying medium with a Lorentzian lineshape of width Av [see (13.1-8)]. Show that when the photon-flux density is 4, the amplifier gain coefficient y(v) assumes a Lorentzian lineshape with width:
.,_,,jl
+ &i”‘.
(13.3-5) Linewidth of Saturated Amplifier
I
This demonstrates that gain saturation is accompanied by an increase in bandwidth reduced frequency selectivity), as shown in Fig. 13.3-2.
“0
Figure 13.3-2 Gain coefficient reduction tion when 4 = 2&,(v,).
B.
and bandwidth
(i.e.,
V
increase resulting from satura-
Gain
Having determined the effect of saturation on the gain coefficient (gain per unit length), we embark on determining the behavior of the overall gain for a homogeneously broadened laser amplifier of length d. For simplicity, we suppress the frequency dependencies of y(v) and 4,(v), using the symbols y and 4, instead. If the photon-flux density at position z is 4(z), then in accordance with (13.3-3) the gain coefficient at that position is also a function of z. We know from (13.1-3) that the incremental increase of photon-flux density at the position z is d4 = 74 dz, which
AMPLIFIER
leads to the differential
NONLINEARITY
AND
GAIN
483
SATURATION
equation (13.3-6)
Rewriting
this equation as (l/4
+ l/4,)
d4 = y. dz, and integrating,
we obtain (13.3-7)
The relation between the input photon-flux output 4(d) is therefore [In(Y)
+ Y] = [In(X)
density to the amplifier
+ X] + rod,
4(O) and the
(13.3-8)
where X = 4(O)/+, and Y = 4(d)/4, are the input and output photon-flux densities normalized to the saturation photon-flux density, respectively. The solution for the gain G = +(d)/+(O) = Y/X can be examined in two limiting cases: . If both X and Y are much smaller much smaller than the saturation negligible in comparison with ln( X) mate relation In(Y) = In(X) + rod,
than unity (i.e., the photon-flux densities are photon-flux density), then X and Y are and In(Y), whereupon we obtain the approxifrom which
Y = Xexp(y,d).
(13.3-9)
In this case the relation between Y and X is linear, with a gain G = Y/X = exp(yod). This accords with (13.1-7) which was obtained under the small-signal approximation, valid when the gain coefficient is independent of the photon-flux density, i.e., y = yo. . When X B- 1, we can neglect In(X) in comparison with X, and In(Y) in comparison with Y, whereupon Y = x
+ rod
Under these heavily saturated conditions, the atoms of the medium are “busy” emitting a constant photon-flux density Nod/~,. Incoming input photons therefore simply leak through to the output, augmented by a constant photon-flux density that is independent of the amplifier input. For intermediate values of X and Y, (13.3-8) must be solved numerically. A plot of the solution is shown as the solid curve in Fig. 13.3-3(b). The linear input-output
484
LASER
AMPLIFIERS Amplifier
(al
------- ew(yod) L Y =Xew(yO
cd
1 4
2
Input X = W)/#, (6)
6
01 0.01
0.1
1
10
Input 4(O)/& (cl
Figure 13.3-3 (a) A nonlinear (saturated) amplifier. (b) Relation between the normalized output photon-flux density Y = &Cd)/&, and the normalized input photon-flux density X = &(0)/4,. For X < 1, the gain Y/X = exp(y,d). For X B 1, Y = X + rod. (c) Gain as a function of the input normalized photon-flux density X in an amplifier of length d when y,d
= 2.
relationship obtained for X s 1, and the saturated relationship for X z+ 1, are evident as limiting cases of the numerical solution. The gain G = Y/X is plotted in Fig. 13.3-3(c). It achieves its maximum value exp(yad) for small values of the input photon-flux density (X =z l), and decreases toward unity as X + 00. Saturable Absorbers If the gain coefficient y. is negative, i.e., if the population is normal rather than inverted (A/, < 0), the medium provides attenuation rather than amplification. The attenuation coefficient a(v) = -y(v) also suffers from saturation, in accordance with the relation a(v) = ao(v)/[l + +/c$,(v)]. Th is indicates that there is less absorption for large values of the photon-flux density. A material exhibiting this property is called a saturable absorber. The relation between the output and input photon-flux densities, 4(d) and 4(O), for an absorber of length d is governed by (13.3-8) with negative yo. The overall transmittance of the absorber Y/X = c$(d)/4(0) is presented as a function of X = ~$(0)/4, in the solid curve of Fig. 13.3-4. The transmittance increases as 4(O) increases, ultimately reaching a limiting value of unity. This effect occurs because the population difference N --) 0, so that there is no net absorption.
AMPLIFIER
NONLINEARITY
AND GAIN
SATURATION
485
1
10
0.8
Saturable
absorber output photons
0.1
InwtX=W)h, Figure 13.3-4 The transmittance of a saturable absorber Y/X = 4(d)/+(O) versus the normalized input photon-flux density X = &(0)/4,, for yod = - 2. The transmittance increases with increasing input photon-flux density.
*C.
Gain of Inhomogeneously
Broadened
Amplifiers
An inhomogeneously broadened medium comprises a collection of atoms with different properties. As discussed in Sec. 12.2D, the subset of atoms labeled p has a homogeneously broadened lineshape function 9,&v). The overall inhomogeneous average lineshape function of the medium is described by g(v) = (g@(v)), where ( * ) represents an average with respect to p. Because the small-signal gain coefficient yO(v) is proportional to g(v), as provided in (13.3-4), different subsets j3 of atoms have different gain coefficients yap(v). The average small-signal gain coefficient is therefore
A2 j+,(v) = Nog.,,,iW
(13.3-11)
SP
Obtaining the saturated gain coefficient is more subtle, however, because the saturation photon-flux density 4$v), being inversely proportional to g(v) as provided in (13.3-2) is itself dependent on the subset of atoms p. An average gain coefficient may be defined by using (13.3-3) and (13.3-2),
where Yop( 4
ydv)
= 1 + 4/&(v)
&3(v) = b 1 + +a2ga(v)
’
(13.3-13)
with b = N,(A2/8r t,) and a 2 = (A2/87rX~Jtsp). Evaluating the average of (13.3-13) requires care because the average of a ratio is not equal to the ratio of the averages.
486
LASER
AMPLIFIERS
Doppler-Broadened Medium Although all of the atoms in a Doppler-broadened medium share a g(v) of identical shape, the center frequency of the subset /3 is shifted by an amount vP proportional to the velocity vP of the subset. If g(v) is Lorentzian with width Au, (13.1-g) provides g(v) = (Av/~T)/[(v - vJ2 + (Av/2j2] and gp(v) = g(v - v&. Substituting gp(v) into (13.3-13) provides b(Av/2r) = (v _ vs - vo)2 + (Av,/2j2
Q(V)
,,=A.(1
’
+ &)“2
(13.3-14)
(13.3-15)
and
2 =-
,^,
(13.3-16)
f%vo). SP
Equation (13.3-H) was obtained for the homogeneously broadened saturated amplifier considered in Exercise 13.3-2 [see (13.3-511. It is evident that the subset of atoms with velocity vP has a saturated gain coefficient 7/p(v) with a Lorentzian shape of width Au, that increases as the photon-flux density becomes larger. The average of y&v) specified in (13.3-12) is obtained by recalling that the shifts up follow a zero-mean Gaussian probability density function p(vP) = (27x7$ 1’2 exp (i.e., the Doppler broadening is much wider than Av$, we may regard the broad function p(v,) as constant and remove it from the integral when evaluating $~a). Setting v = v. and up = 0 in the exponential provides ‘y(vo)
=
bP (0)
=
(1 + 2~#~2~,‘7rAv)~‘~
where the average small-signal
gain coefficient ‘y. = No-
,R
70
[1+ 4/4s(vo)l1’2’
(13.3-18)
To is
(27TcT;y2.
(13.3-19)
SP
Equation (13.3-18) provides an expression for the average saturated gain coefficient of a Doppler broadened medium at the central frequency vo, as a function of the photon-flux density C#Jat v = vo. The gain coefficient saturates as 4 increases in
AMPLIFIER
Figure 13.3-5 ened media.
Comparison
of gain saturation
NONLINEARITY
AND GAIN
in homogeneously
SATURATION
and inhomogeneously
407
broad-
accordance with a square-root law. The gain coefficient in an inhomogeneously broadened medium therefore saturates more slowly than the gain coefficient in a homogeneously broadened medium [see (13.3-3)], as illustrated in Fig. 13.3-5. Ho/e Burning When a large flux density of monochromatic photons at frequency v1 is applied to an inhomogeneously broadened medium, the gain saturates only for those atoms whose lineshape function overlaps v ,. Other atoms simply do not interact with the photons and remain unsaturated. When the saturated medium is probed by a weak monochromatic light source of varying frequency v, the profile of the gain coefficient therefore exhibits a hole centered around vl, as illustrated in Fig. 13.3-6. This phenomenon is known as hole burning. Since the gain coefficient y&v) of the subset of atoms with velocity vP has a Lorentzian shape with width Av,, given by (13.3-H), it follows that the width of the hole is Av,. As the flux density of saturating photons at v1 increases, both the depth and the width of the hole increase.
Figure 13.3-6 The gain coefficient of an inhomogeneously broadened medium saturated by a large flux density of monochromatic photons at frequency vl.
is locally
488
LASER
AMPLIFIERS
“13.4
AMPLIFIER
NOISE
The resonant medium that provides amplification by the process of stimulated emission also generates spontaneous emission. The light arising from the latter process, which is independent of the input to the amplifier, represents a fundamental source of laser amplifier noise. Whereas the amplified signal has a specific frequency, direction, and polarization, the amplified spontaneous emission CASE) noise is broadband, multidirectional, and unpolarized. As a consequence it is possible to filter out some of this noise by following the amplifier with a narrow bandpass optical filter, a collection aperture, and a polarizer. The probability density (per second) that an atom in the upper laser level spontaneously emits a photon of frequency between v and v + dv is (see Exercise 12.2-1). p,,(v)
dv = ;g(
v) dv.
(13.4-1)
SP
The probability density of spontaneously emitting a photon of any frequency is, of course, Psp = l/t,,. If N, is the atomic density in the upper energy level, the average spontaneously emitted photon density is /&P,,(v). The average spontaneously emitted power per unit volume per unit frequency is therefore hvN,P,,(v). This power density is emitted uniformly in all directions and is equally divided between the two polarizations. If the amplifier output is collected from a solid angle dR (as illustrated in Fig. 13.4-Q and from only one of the polarizations, it contains only a fraction $ dfi/4r of the spontaneously emitted power. Furthermore, if the receiver is sensitive only to photons within a narrow frequency band B centered about the amplified signal frequency v, the number of photons added by spontaneous emission from an incremental volume of unit area and length dz is [sp(v) dz, where !&p(v) = N*~y(v)B~
(13.4-2)
SP
is the noise photon-flux density per unit length. In determining the noise photon-flux density contributed by the amplifier, it is incorrect to simply multiply the photon-flux density per unit length by the length of the amplifier. The spontaneous-emission noise itself is amplified by the medium; noise generated near the input end of the amplifier provides a greater contribution than noise generated near the output end. One way in which spontaneous-emission noise
Input photo1 flux
output photon flux
Figure 13.4-1 Spontaneous emission is a source of amplifier noise. It is broadband, radiated in all directions, and unpolarized. Only light within a narrow optical band, solid angle dR, and a single polarization is collected by the optics at the output of the amplifier.
READING
may be accounted for is to replace the differential photon-flux density (13.1-3) by
-dz =
YbM
Equation (13.4-3) permits the calculation amplified signal and spontaneous-emission
EXERCISE Amplified
LIST
489
equation governing the growth of
(13.4-3)
+ sspw.
of the photon-flux photons.
density arising from the
13.4-7 Spontaneous
Emission
(ASE)
(a) Use (13.4-3) to show that in the absence of any input signal, spontaneous emission produces a photon-flux density at the output of an unsaturated amplifier [Y(Y) = y&)1 of length d given by &Cd) = &,texp[r&)dl - 11,where &, = ~&)/Yo(~). (b) Since both ,$&I) and y&) are proportional to g(v), &, is independent of g(v) so that the frequency dependence of 4(d) is governed by the factor kxp[roWdl - 1). If yO(v) is Lorentzian with width Av, i.e., yO(v) = y&,)(A~/2)~/[(v - vO)2 + (Av/~)~], show that the bandwidth of the factor {exp[yO(v)dl - 11 is smaller than Av, i.e., that the amplification of spontaneous emission is accompanied by spectral narrowing.
In the process of amplification, the photon-number statistics of the incoming light are altered (see Sec. 11.20 A coherent signal presented to the input of the amplifier has a number of photons counted in time T that obeys Poisson statistics, with a variance U: equal to the mean signal photon number fis. The ASE photons, on the other hand, obey Bose-Einstein statistics exhibiting aisE = A,, + fi& and are therefore considerably noisier than Poisson statistics. The photon-number statistics of the light after amplification, comprising both signal and spontaneous-emission contributions, obey a probability law intermediate between the two. If the counting time is short and the emerging light is linearly polarized, these statistics can be well approximated by the Laguerre-polynomial photon-number distribution (see Problem 13.4-2), which has a variance given by @n2
= A, + (iiASE + ii&)
-+ 2n,n,,,.
(13.4-4)
The photon-number fluctuations are seen to contain contributions from the signal alone and from the spontaneous emission alone, as well as added fluctuations from the interference of the two components.
READING
LIST
Books on Laser Theory A. Yariv, Quantum Electronics, Wiley, New York, 3rd ed. 1989. J. T. Verdeyen, Laser Electronics, Prentice-Hall, Englewood Cliffs, NJ, 2nd ed. 1989. 0. Svelto, Principles of Lasers, Plenum Press, New York, 3rd ed. 1989.
490
LASER
AMPLIFIERS
J. Wilson and J. F. B. Hawkes, Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 2nd ed. 1989. P. W. Milonni and J. H. Eberly, Lasers, Wiley, New York, 1988. W. Witteman, The Laser, Springer-Verlag, New York, 1987. K. A. Jones, Introduction to Optical Electronics, Harper & Row, New York, 1987. J. Wilson and J. F. B. Hawkes, Lasers: Principles and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1987. A. E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986. to Laser Physics, Springer-Verlag, Berlin, 2nd ed. 1986. K. Shimoda, Introduction B. B. Laud, Lasers and Nonlinear Optics, Wiley, New York, 1986. A. Yariv, Optical Electronics, Holt, Rinehart and Winston, New York, 3rd ed. 1985. H. Haken, Light: Laser Light Dynamics, vol. 2, North-Holland, Amsterdam, 1985. H. Haken, Laser Theory, Springer-Verlag, Berlin, 1984. R. Loudon, The Quantum Theory of Light, Oxford University Press, New York, 2nd ed. 1983. B. E. A. Saleh, Photoelectron Statistics, Springer-Verlag, New York, 1978. D. C. O’Shea, W. R. Callen, and W. T. Rhodes, Introduction to Lasers and Their Applications, Addison-Wesley, Reading, MA, 1977. M. Sargent III, M. 0. Scully, and W. E. Lamb, Jr., Laser Physics, Addison-Wesley, Reading, MA, 1974. F. T. Arecchi and E. 0. Schulz-Dubois, eds., Laser Handbook, vol. 1, North-Holland/Elsevier, Amsterdam/New York, 1972. A. E. Siegman, An Introduction to Lasers and Masers, McGraw-Hill, New York, 1971. B. A. Lengyel, Lasers, Wiley, New York, 2nd ed. 1971. A. Maitland and M. H. Dunn, Laser Physics, North-Holland, Amsterdam, 1969. W. S. C. Chang, Principles of Quantum Electronics, Addison-Wesley, Reading, MA, 1969. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics, Wiley, New York, 1969. D. Ross, Lasers, Light Amplifiers, and Oscillators, Academic Press, New York, 1969. E. L. Steele, Optical Lasers in Electronics, Wiley, New York, 1968. A. K. Levine, ed., Lasers, ~01s. 1-4, Marcel Dekker, New York, 1966-1976. G. Birnbaum, Optical Masers, Academic Press, New York, 1964. G. Troup, Masers and Lasers, Methuen, London, 2nd ed. 1963. Articles R. Baker, Optical Amplification, Physics World, vol. 3, no. 3, pp. 41-44, 1990. D. O’Shea and D. C. Peckham, Lasers: Selected Reprints, American Association of Physics Teachers, Stony Brook, NY, 1982. M. J. Mumma, D. Buhl, G. Chin, D. Deming, F. Espenak, and T. Kostiuk, Discovery of Natural Gain Amplification in the 10 pm CO, Laser Bands on Mars: A Natural Laser, Science, vol. 212, pp. 45-49, 1981. F. S. Barnes, ed., Laser Theory, IEEE Press Reprint Series, IEEE Press, New York, 1972. A. L. Schawlow, ed., Lasers and Light-Readings from Scientific American, W. H. Freeman, San Francisco, 1969. Journal of Physics, vol. 36, pp. J. H. Shirley, Dynamics of a Simple Maser Model, American 949-963, 1968. J. Weber, ed., Lasers: Selected Reprints with Editorial Comment, Gordon and Breach, New York, 1967. C. Cohen-Tannoudji and A. Kastler, Optical Pumping, in Progress in Optics, vol. 5, E. Wolf, ed., North-Holland, Amsterdam, 1966. W. E. Lamb, Jr., Theory of an Optical Maser, Physical Review, vol. 134, pp. A1429-A1450, 1964. A. Yariv and J. P. Gordon, The Laser, Proceedings of the IEEE, vol. 51, pp. 4-29, 1963. T. H. Maiman, Stimulated Optical Radiation in Ruby, Nature, vol. 187, pp. 493-494, 1960. A. L. Schawlow and C. H. Townes, Infrared and Optical Masers, Physical Review, vol. 112, pp. 1940-1949, 1958.
PROBLEMS
491
Historical J. Hecht, ed., Laser Pioneer Interviews, High Tech Publications, Torrance, CA, 1985. A. Kastler, Birth of the Maser and Laser, Nature, vol. 316, pp. 307-309, 1985. M. Bertolotti, Masers and Lasers: An Historical Approach, Adam Hilger, Bristol, England, 1983. C. H. Townes, Science, Technology, and Invention: Their Progress and Interactions, Proceedings vol. 80, pp. 7679-7683, 1983. of the National Academy of Sciences (USA), D. C. O’Shea and D. C. Peckham, Resource Letter L-l: Lasers, American Journal of Physics, vol. 49, pp. 915-925, 1981. C. H. Townes, The Laser’s Roots: Townes Recalls the Early Days, Laser Focus Magazine, vol. 14, no. 8, pp. 52-58, 1978. A. L. Schawlow, Masers and Lasers, IEEE Transactions on Electron Devices, vol. ED-23, pp. 773-779, 1976. B. Kursunoglu A. L. Schawlow, From Maser to Laser, in Impact of Basic Research on Technology, and A. Perlmutter, eds., Plenum Press, New York, 1973. W. E. Lamb, Jr., Physical Concepts in the Development of the Maser and Laser, in Impact of Basic Research on Technology, B. Kursunoglu and A. Perlmutter, eds., Plenum Press, New York, 1973. A. Kastler, Optical Methods for Studying Hertzian Resonances, in Nobel Lectures in Physics, 1963-1970, Elsevier, Amsterdam, 1972. C. H. Townes, Production of Coherent Radiation by Atoms and Molecules, in Nobel Lectures in Physics, 1963-1970, Elsevier, Amsterdam, 1972. N. G. Basov, Semiconductor Lasers, in Nobel Lectures in Physics, 1963-1970, Elsevier, Amsterdam, 1972. A. M. Prokhorov, Quantum Electronics, in Nobel Lectures in Physics, 1963-1970, Elsevier, Amsterdam, 1972. C. H. Townes, Quantum Electronics and Surprise in the Development of Technology, Science, vol. 159, pp. 699-703, 1968. B. A. Lengyel, Evolution of Masers and Lasers, American Journal of Physics, vol. 34, pp. 903-913, 1966. R. H. Dicke, Molecular Amplification and Generation Systems and Methods, U.S. Patent 2,851,652, Sept. 9, 1958. J. P. Gordon, H. J. Zeiger, and C. H. Townes, The Maser-New Type of Microwave Amplifier, Frequency Standard, and Spectrometer, Physical Review, vol. 99, pp. 1264-1274, 1955. N. G. Basov and A. M. Prokhorov, Possible Methods of Obtaining Active Molecules for a Molecular Oscillator, Soviet Physics-JETP, vol. 1, pp. 184-185, 1955 [Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki (USSR), vol. 28, pp. 249-250, 19551. V. A. Fabrikant, The Emission Mechanism of Gas Discharges, Trudi Ksyesoyuznogo Efektrotekhnicheskogo Instituta (Reports of the All-Union Electrotechnical Institute, Moscow), vol. 41, Elektronnie i Ionnie Pribori (Electron and Ion Devices), pp. 236-296, 1940.
13.1-1
Amplifier Gain and Rod Length. A commercially available ruby laser amplifier using a 15cm-long rod has a small-signal gain of 12. What is the small-signal gain of a 20-cm-long rod? Neglect gain saturation effects.
13.1-2
Laser Amplifier Gain and Population Difference. A H-cm-long rod of Nd3+:glass used as a laser amplifier has a total small-signal gain of 10 at A, = 1.06 pm. Use the data in Table 13.2-1 on page 480 to determine the population difference N required to achieve this gain (Nd3+ ions per cm3).
13.1-3
Amplification of a Broadband exhibits a Lorentzian lineshape
Signal. The transition between two energy levels of central frequency vc = 5 X lOI with a linewidth
492
LASER AMPLIFIERS
Av = 1012 Hz. The population is inverted so that the maximum gain coefficient y(vO) = 0.1 cm-‘. The medium has an additional loss coefficient cu, = 0.05 cm-‘, which is independent of v. Approximately how much loss or gain is encountered by a light wave in 1 cm if it has a uniform power spectra1 density centered about v0 with a bandwidth 2Av? Pumping System. Write the rate equations for a two-level system, showing that a steady-state population inversion cannot be achieved by using direct optical pumping between levels 1 and 2.
13.2-1
The Two-Level
13.2-2
Two Laser Lines. Consider an atomic system with four levels: 0 (ground state), 1,
2, and 3. Two pumps are applied: between the ground state and level 3 at a rate R,, and between ground state and level 2 at a rate R,. Population inversion can occur between levels 3 and 1 and/or between levels 2 and 1 (as in a four-level laser). Assuming that decay from level 3 to 2 is not possible and that decay from levels 3 and 2 to the ground state are negligible, write the rate equations for levels 1, 2, and 3 in terms of the lifetimes TV, car, and r2,. Determine the steady state populations N,, N,, and N, and examine the possibility of simultaneous population inversions between 3 and 1, and between 2 and 1. Show that the presence of radiation at the 2-l transition reduces the population difference for the 3-l transition. 13.3-1 Significance of the Saturation Photon-Flux Density. In the general two-level atomic system of Fig. 13.2-3, 72 represents the lifetime of level 2 in the absence of stimulated emission. In the presence of stimulated emission, the rate of decay from level 2 increases and the effective lifetime decreases. Find the photon-flux density 4 at which the lifetime decreases to half its value. How is that flux density related to the saturation photon-flux density 4,? 13.3-2 Saturation Optical Intensity. Determine the saturation photon-flux density 4s(v0) and the corresponding saturation optical intensity Is(vJ, for the homogeneously broadened ruby and Nd 3+:YAG laser transitions provided in Table 13.2-1. 13.3-3 Growth of the Photon-Flux Density in a Saturated Amplifier. The growth of the photon-flux density 4(z) in a laser amplifier is described by (13.3-7). Use a computer to plot +(z)/4, versus yOz for +(0)/4, = 0.05. Identify the onset of saturation in this amplifier. 13.3-4 Resonant Absorption of a Medium in Thermal Equilibrium. A unity refractive index medium of volume 1 cm3 contains N, = 1O23atoms in thermal equilibrium. The ground state is energy level 1; level 2 has energy 2.48 eV above the ground state (A, = 0.5 km). The transition between these two levels is characterized by a spontaneous lifetime t,, = 1 ms, and a Lorentzian lineshape of width .Av = 1 GHz. Consider two temperatures, T, and T2, such that k,T, = 0.026 eV and k,T, = 0.26 eV. (a) Determine the populations N, and N2. (b) Determine the number of photons emitted spontaneously every second. (c) Determine the attenuation coefficient of this medium at h, = 0.5 km assuming that the incident photon flux is small. (d) Sketch the dependence of the attenuation coefficient on frequency, indicating on the sketch the important parameters. (e) Find the value of photon-flux density at which the attenuation coefficient decreases by a factor of 2 (i.e., the saturation photon-flux density). (f) Sketch th e d ependence of the transmitted photon-flux density 4(d) on the incident photon-flux density 4(O) for v = v. and v = v. + Au when +(0)/4, K 1.
PROBLEMS
493
13.3-5 Gain in a Saturated Amplifying Medium. Consider a homogeneously broadened laser amplifying medium of length d = 10 cm and saturation photon flux density 4, = 4 x 1018 photons/cm2-s. It is known that a photon-flux density at the input c)(O) = 4 x 1o15 photons/cm2-s produces a photon-flux density at the output 4(d) = 4 x 1016 photons/cm2-s. (a) Determine the small-signal gain of the system G,. (b) Determine the small-signal gain coefficient yo. (c) What is the photon-flux density at which the gain coefficient decreases by a factor of 5? (d) Determine the gain coefficient when the input photon-flux density is 4(O) = 4 x 1019 photons/cm2-s. Under these conditions, is the gain of the system greater than, less than, or the same as the small-signal gain determined in part (a)? *13.4-1
An unsaturated laser amplifier of length d and gain coefficient ye(v) amplifies an input signal 4,(O) of frequency v and introduces amplified spontaneous emission (ASE) at a rate tsp (per unit length). The amplified signal photon-flux density is t),(d) and the ASE at the output is on the amplifier gain 4 ASE. Sketch the dependence of the ratio 4S(d)/+ASE coefficient-length product yo(v)d. Ratio of Signal Power to ASE Power.
* 13.4-2 Photon-Number Distribution for Amplified Coherent Light. A linearly polarized superposition of interfering thermal and coherent light serves as a suitable model for the light emerging from a laser amplifier. This superposition is known to have random energy fluctuations %V that obey the noncentral-chi-square probability distribution = -2
p(w)
exp( - z)Io[
2(>,j”2],
%SE
provided that the measurement time is sufficiently short.+ Here I, denotes the modified Bessel function, %jsn is the mean energy of the ASE, and ws is the (constant) energy of the amplified coherent signal. (a) Calculate the mean and variance of V. (b) Use (11.2-26) and (11.2-27) to determine the photon-number mean A and variance ai, confirming the validity of (13.4-4). (c) Use (11.2-25) to show that the photon-number distribution is given by
p(n) = (l
n'SE + 'AS,)
"+,
eq(
-
l +'iAsE)Ln(
-
:LE),
where L, represents the Laguerre polynomial
and A, and AAsn are the mean signal and amplified-spontaneous-emission photon numbers, respectively. (d) Use a computer to plot p(n) for A,/A = 0, 0.5, 0.8, and 1, when ii = 5, demonstrating that it reduces to the Bose-Einstein distribution for As/A = 0 and to the Poisson distribution for ?i,/n = 1. ‘See, for example,
B. E. A. Saleh,
Photoelectron
Statistics,
Springer-Verlag,
New York,
1978.
