Lecture 8: Laser amplifiers

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Optical transitions
Optical absorption and amplification
Population inversion
Coherent optical amplifiers: gain,
nonlinearity, noise
References: This lecture follows the materials from Photonic Devices, Jia-Ming Liu,
Chapter 10. Also from Fundamentals of Photonics, 2nd ed., Saleh & Teich,
Chapter 14.
1
Intro


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
The word laser is an acronym for light amplification by stimulated
emission of radiation.
However, the term laser generally refers to a laser oscillator, which
generates laser light without an input light wave.
A device that amplifies a laser beam by stimulated emission is called a
laser amplifier.
Laser light is generally highly collimated with a very small divergence
and highly coherent in time and space. It also has a relatively narrow
spectral linewidth and a high intensity in comparison with light generated
from ordinary sources (e.g. light-emitting diodes)
Due to the process of stimulated emission, an optical wave amplified by a
laser amplifier preserves most of the characteristics including the
frequency spectrum, the coherence, the polarization, the divergence and
the direction of propagation of the input wave.
Here, we discuss the characteristics of laser amplifiers. We will discuss
laser oscillators in Lecture 9.
2
Optical transitions
3
Optical transitions



Optical absorption and emission occur through the
interaction of optical radiation with electrons in a material
system that defines the energy levels of the electrons.
Depending on the properties of a given material, electrons
that interact with optical radiation can be either those
bound to individual atoms or those residing in the energyband structures of a material such as a semiconductor.
The absorption or emission of a photon by an electron is
associated with a resonant transition of the electron
between a lower energy level |1> of energy E1 and an
upper energy level |2> of energy E2.
4
Photon-matter interaction processes

There are three fundamental processes electrons make transitions
between two energy levels upon a photon of energy.
E = h12 = E2 – E1
• A two-level system is a model system that only contains
two energy levels with which the photon interacts.
5
Three basic photon-matter interaction processes

Absorption – when the quantum energy h equals the energy
difference between the two energy levels (a resonant
condition); the atom gains a quantum of energy

Stimulated emission – the emission of a photon is triggered
by the arrival of another, resonant photon

Spontaneous emission – when an atom emits a photon,
losing a quantum of energy in the process

Einstein in 1917 first pointed out that stimulated emission is
essential in the overall balance between emission and
absorption, about reaching thermal equilibrium for a system
of atoms. (Einstein Relations)
Stimulated emission was demonstrated in 1953 in the
microwave frequency by Basov, Prokhorov and Townes
(Nobel 1964)

6
Three basic photon-matter interaction processes




A photon emitted by stimulated emission has the same
frequency, phase, polarization and propagation direction as
the optical radiation that induces the process.
Spontaneously emitted photons are random in phase and
polarization and are emitted in all directions, though their
frequencies are still dictated by the separation between the
two energy levels, subject to a degree of uncertainty
determined by the linewidth of the transition.
Therefore, stimulated emission results in the amplification
of an optical signal, whereas spontaneous emission adds
noise to an optical signal.
Absorption leads to the attenuation of an optical signal.
7
Spontaneous emission

An electron spontaneously falls from a higher energy level E2
to a lower one E1, the emitted photon has frequency
 = (E2 – E1) / h


|2>
|1>
This photon is emitted in a random direction with arbitrary
polarization.
The probability of such a spontaneous jump is given
quantitatively by the Einstein coefficient for spontaneous
emission (known as the “Einstein A coefficient”) defined as
A21 = “probability” per second of a spontaneous jump from
level |2> to level |1>.
8
8
Probability per second of a spontaneous emission

For example, if there are N2 population per unit volume in
level |2> then N2A21 per second make jumps to level |1>.

The total rate at which jumps are made between the two
levels is
dN2/dt = -N2A21
A negative sign because the population of level 2 is decreasing

Generally an electron can make jumps to more than one lower
level, unless it is in the first (lowest) excited level.
9
9
Natural lifetime

The population of level |2> falls exponentially with time as
electrons leave by spontaneous emission.
N2 = N20 exp(-A21t)

The time in which the population falls to 1/e of its initial
value is called the natural lifetime of level |2>,
2 = 1/A21

The magnitude of this lifetime is determined by the actual
probabilities of jumps from level |2> by spontaneous
emission.
10
10
Spectral lineshape
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
The spectral characteristic of a resonant transition is therefore
never infinitely sharp.
Any allowed resonant transition between two energy levels
has a finite relaxation time constant because at least the
upper level has a finite lifetime due to spontaneous emission.
From Quantum Mechanics, the finite spectral width of a
resonant transition is dictated by the uncertainty principle of
quantum mechanics.
Intuitively, any response that has a finite relaxation time in
the time domain must have a finite spectral width in the
frequency domain. (recall the impulse response discussed in
Lecture 2)
We will see that the rate of the induced transitions between
two energy levels in a given system is directly proportional to
the spontaneous emission rate from the upper to the lower
level.
11
Spectral lineshape


For each particular resonant transition between two energy
^
levels, there is a characteristic lineshape function g()
of
finite linewidth  that characterizes the optical processes
associated with the transition.
The lineshape function is generally normalized as


0
0
 ĝ( )d   ĝ( )d  1
ĝ( )  2 ĝ( )

Area = 1


12
12
Homogeneous broadening
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If all of the atoms in a material that participate in a resonant
interaction associated with the energy levels |1> and |2> are
indistinguishable, their responses to an electromagnetic field
are characterized by the same resonance frequency 21 and
the same relaxation constant 21.
In such a homogeneous system, the physical mechanisms that
contribute to the linewidth of the transition affect all atoms
equally.
Spectral broadening due to such mechanisms is called
homogeneous broadening.
Previously (in Lect. 2), we discussed that such
homogeneously broadened systems can be described as the
damped response characterized by a single resonance
frequency and a single relaxation constant.
13
Homogeneous broadening



In the interaction of a material with an optical field, the
absorption and emission of optical energy are characterized
by the imaginary part ” of the susceptibility of the material.
Therefore, the spectral characteristics of optical absorption
and emission due to a resonant transition in a homogeneously
broadened medium are described by the Lorentzian lineshape
function of ”(). (recall from Lect 2)
Using the normalization condition, we find that the resonant
transition between |1> and |2> has the following normalized
Lorentzian lineshape function:
 h
ĝ( ) 
2 [(   21 )2  ( h / 2)2 ]
where h is the FWHM of the lineshape
14
Inhomogeneous broadening
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However, in many practical situations, the simple picture that
gives rise to Lorentzian lineshape is not adequate.
For example, because of the Doppler effect, gas atoms with
different velocities have different effective resonance
frequencies even if they are otherwise identical.
In solids the slightly different environments in which the
resonant atoms find themselves, such as random dislocations,
impurities and strain fields, also give rise to different
effective resonance frequencies for differently located but
otherwise identical atoms.
Thus, in many cases the actual emission line must be thought
of as a superposition of a large number of Lorentzian lines,
each with homogeneous width k and each with a distinct
center frequency k.
Ref: Optical resonance and two-level atoms, L. Allen and J. H. Eberly, pp. 7-10
15
Spectral lineshape
Spread in
frequencies
Gaussian lineshape




The origin of inhomogeneous broadening. The individual Lorentzian
emission lines associated with different atomic dipoles are oscillating at
multiple distinct frequencies.
If a dielectric material is made up of those atoms with such individual
lines, its emission line will be the sum of the curves. When the individual
lines are densely spaced over a frequency range large compared with their
own individual widths, the total lineshape is termed inhomogeneously
broadened.
Ref: Optical resonance and two-level atoms, L. Allen and J. H. Eberly, pp. 7-10
16
Transition rates
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The transition rate of a resonant optical process measures the
probability per unit time for the process to occur.
The transition rate of an induced process is a function of the
spectral distribution of the optical radiation and the spectral
characteristics of the resonant transition.
The spectral distribution of an optical field is characterized by
its spectral energy density u() – the energy density of the
optical radiation per unit frequency interval at the optical
frequency .
The total energy density of the radiation u 

 u( )d
0


The spectral intensity distribution I() = (c/n) u(), n is the
refractive index of the medium
The total intensity I 

 I( )d
0
17
Spectral energy density


The energy density of a radiation field u() (joules per unit
volume per unit frequency interval) can be simply related to
the intensity of a plane electromagnetic wave.
If the intensity of the wave is I() (watts per unit area per
frequency interval)
u() c = I()
where c is the velocity of light in free space (in the medium
of refractive index n, u() c/n = I())
V
Length c in a second
A
18
18
Transition rates

For the upward transition from |1> to |2> associated with
absorption in the frequency range between  and +d is
W12 ( )d  B12 u( )ĝ( )d

For the downward transition from |2> to |1> associated with
stimulated emission
W21 ( )d  B21u( )ĝ( )d

(s-1)
(s-1)
The spontaneous emission rate is independent of the energy
density of the radiation and is solely determined by the
transition lineshape function
Wsp ( )d  A21ĝ( )d
(s-1)
The A and B constants are the Einstein A and B coefficients.
19
Radiative processes connecting two energy levels in
thermal equilibrium
Population N2
Spontaneous emission
Population N1

E2
Stimulated
emission
absorption
h
E1
Einstein (1917) demonstrated that the rates of the three
transition processes of absorption (B12), stimulated emission
(B21) and spontaneous emission (A21) are related.
20
20
Transition rates

The total induced transition rates
W12 
W21 


W
12
( )d  B12  u( )ĝ( )d
0
0


W
21
( )d  B21  u( )ĝ( )d
0

0
The total spontaneous emission rate is
Wsp 

W
sp
( )d  A21
N2
0
u()
|2>
W12 = B12u() W21 = B21u() Wsp = A21
|1>
N1
21
Transition rates
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

The induced and the spontaneous transition rates for a given
system are directly proportional to one another.
The relationship can be obtained by considering the
interaction of blackbody radiation with an ensemble of
identical atomic systems in thermal equilibrium.
The spectral energy density of the blackbody radiation at a
temperature T (known as thermal radiation or blackbody
radiation) is given by Planck’s formula:
1
8 n 3h 3
u( ) 
c3
e h /kBT 1
where kB is the Boltzmann constant, kBT is the thermal
energy (kBT = 26 meV @ T = 300 K)
22
Blackbody radiation
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
A system under thermal equilibrium produces a radiation
energy density u() (J Hz-1m-3) which is identical to
blackbody radiation.
A blackbody absorbs 100% all the radiation falling on it,
irrespective of the radiation frequency.
If the inside of this body is in thermal equilibrium it must
radiate as much energy as it absorbs and the emission from
the body is therefore characteristic of the equilibrium
temperature T inside the body
=> this type of radiation is often called “thermal” radiation or
blackbody radiation
Thermal radiation
23
23
Planck’s law of blackbody radiation

Planck showed that the radiation energy density for a
blackbody radiating within a frequency range  to +d is given
by
u = (8nh3/c3) [exp(h/kBT) – 1]-1
= (8n2/c3) h [exp(h/kBT) – 1]-1
Photon energy
Photon density of states in a medium
Of refractive index n
(number of photon modes per
volume per frequency interval)
Photon probability of occupancy
(average number of photons in each mode according to Bose‐Einstein
distribution)
24
24
Planck’s law of blackbody radiation
Radiation energy density u() (JHz‐1m‐3)
180
160
140
1500 K
120
100
80
60
1000 K
40
20
0
0
5E+13
1E+14
1.5E+14
2E+14
2.5E+14
3E+14
3.5E+14
Frequency  (Hz)
25
25
Transition rates

If N2 and N1 are the population densities per unit volume of
the atoms in levels |2> and |1>, the number of atoms per unit
volume making the downward transition per unit time
accompanied by the emission of radiation in a frequency
range from  to +d
N 2 (W21 ( )  Wsp ( ))d

The number of atoms per unit volume making the upward
transition per unit time
N1W12 ( )d
26
Transition rates

In thermal equilibrium, both the blackbody radiation spectral
density and the atomic population density in each energy
level should reach a steady state
N 2 (W21 ( )  Wsp ( ))  N1W12 ( )

This is the principle of detailed balance in thermal
equilibrium. The steady-state population distribution in
thermal equilibrium:
W12 ( )
B12 u( )
N2


N1 W21 ( )  Wsp ( ) B21u( )  A21
27
Transition rates

In thermal equilibrium at temperature T, the population ratio
of the atoms in the upper and the lower levels follows the
Boltzmann distribution.
N 2 g2
 exp(h / kBT )
N1 g1
where g2 and g1 are the degeneracy factors* of these energy
levels, and the energy density
A21 / B21
u( ) 
 g1B12 

 exp(h / kBT ) 1
 g2 B21 
*In an atomic or molecular system, a given energy level
usually consists of a number of degenerate quantummechanical states, which have the same energy.
28
Boltzmann distribution
Energy E
E2
N2 << N1 in thermal equilibrium
 exp (-h/kBT)
1 eV
[For T = 300 K, kBT = 26 meV]
E1
N2
(=1)
N1
(=5 x 1016)
Population N
N2 << N1 in thermal equilibrium
29
Transition rates
Identify u() with Planck’s formula:
A21 8 n 3h 3

B21
c3
g1B12  g2 B21
The spontaneous radiative lifetime of the atoms in the level |2>
associated with the radiative spontaneous transition from |2> to
|1> is
1
1
 sp 

Wsp A21
30
Transition rates

Therefore, the spectral dependence of the spontaneous
emission rate
Wsp ( ) 

1
 sp
ĝ( )
The transition rates of both of the induced processes of
absorption and stimulated emission are directly proportional
to the spontaneous emission rate.
c3
c2
W21 ( ) 
u( )ĝ( ) 
I( )ĝ( )
3
3
2
3
8 n h  sp
8 n h  sp
g2
W12 ( )  W21 ( )
g1
31
Transition cross section

We often express the transition probability of an atom in its
interaction with optical radiation at a frequency  in terms of the
transition cross section, () [m2, cm2].
I( )
W21 ( ) 
 21 ( )
h
I( )
 12 ( )
W12 ( ) 
h
hphoton-flux
density)
The emission cross section
c2
 e ( )   21 ( ) 
ĝ( )
2 2
8 n   sp
The absorption cross section
g2
g2
 a ( )   12 ( )   21 ( )   e ( )
g1
g1
32
Transition cross section

For the ideal Lorentzian lineshape in a homogeneously
broadened medium
 h
ĝ( ) 
2 [(   21 )2  ( h / 2)2 ]

The peak value of the lineshape occurs at the center of the
spectrum and is a function of linewidth h only.
2
gˆ ( 21 ) 
 h
Thus, the peak value of the emission cross section at the
center wavelength  of the spectrum

e 
2
4 2 n 2  h sp
33
Characteristics of some laser materials
Gain medium
Wavelength
(m)
System
Peak cross
section e
(m2)
Spontaneous
linewidth
(gain
bandwidth)

sp
2
HeNe
0.6328
I, 4
3.0x10-17
1.5 GHz
300 ns
30 ns
Ruby (Cr3+:Al2O3)
0.6943
H, 3
1.25-2.5 x
10-24
330 GHz
3 ms
3 ms
Nd:YAG
1.064
H, 4
2-10 x 10-23
150 GHz
515 s
240 s
Nd:glass
1.054
I, 4
4.0 x 10-24
6 THz
330 s
330 s
Er:fiber
1.53
H/I, 3
6.0 x 10-25
5 THz
10 ms
10 ms
Ti:sapphire
0.66-1.1
H,Q2
3.4x10-23
100 THz
3.9 s
3.2 s
Semiconductor
0.37-1.65
H/I, Q2
1-5 x 10-20
10-20 THz
~1 ns
~1 ns
34
H: homogeneously broadened; I: inhomogeneously broadened
34
Optical absorption and
amplification
35
Optical absorption and amplification

For a monochromatic optical field at frequency  and intensity
I() = I(’-)
W21 = (I/h) e() and W12 = (I/h) a()

The net power (time-averaged) that is transferred from the optical
field to the material is the difference between that absorbed by the
material and that emitted due to stimulated emission:
Wp = hW12N1 – hW21N2
= [N1a() – N2e()]I


Wp > 0 => net power absorption from the optical field
Wp < 0 => net power flows from the medium to the optical field
36
Optical absorption and amplification

absorption coefficient [m-1, cm-1]
() = N1a() – N2e() = (N1 – (g1/g2)N2) a()

gain coefficient [m-1, cm-1]
() = N2e() – N1a() = (N2 – (g2/g1)N1) e()
() > 0 and () < 0 if N1 > (g1/g2) N2
() > 0 and () < 0 if N2 > (g2/g1) N1


A material absorbs optical energy in its normal state of thermal
equilibrium when the lower energy level is more populated than the upper
energy level.
A material must be in a nonequilibrium state of population inversion with
the upper energy level more populated than the lower energy level in
order to provide a net optical gain to the optical field.
37
Optical absorption and amplification



For simplicity, in some later discussion we can assume the
degeneracy of levels 1 and 2 are equal, i.e. g1 = g2
absorption coefficient [m-1, cm-1]
() = N1a() – N2e() = (N1 – N2) ()
gain coefficient [m-1, cm-1]
() = N2e() – N1a() = (N2 – N1) ()
() > 0 and () < 0 if N1 > N2
() > 0 and () < 0 if N2 > N1
And e() = a() = ()
38
Resonant optical susceptibility

For resonant interaction of an isotropic medium with a
monochromatic plane optical field at a frequency  = 2,
we have
E(t)  Eeit  E * ei t
Pr es (t)   0 (  res ( )Eeit   *res ( )E * eit )

where Pres is the polarization contributed by the resonant
transitions and res is the resonant susceptibility.
The time-averaged power density absorbed by the medium is
P
Wp  E 
t
 2 0  "res ( ) | E | 
2
t

nc
 "res ( ) I
39
Resonant optical susceptibility

Relate the time-averaged power density absorbed by the
medium to the population relation
Wp 


 "res ( ) I  [ N 1 a ( )  N 2 e ( )] I
nc
The imaginary part of the susceptibility contributed by the
resonant transitions between energy levels |1> and |2> is
nc
 "res ( )  [N1 a ( )  N 2 e ( )]




The real part ’res() can then be found through the KramersKronig relations (recall from Lect. 2)
Recall from Lect. 2 that a medium has an optical loss if ” >
0, and it has an optical gain if ” < 0.
It is also clear that there is a net power loss from the optical
field to the medium if ”res > 0, but there is a net power gain
for the optical field if ”res < 0.
40
Resonant optical susceptibility

The medium has an absorption coefficient given by
 ( ) 

nc
 "res ( )
in the case of normal population distribution when ”res>0,
whereas it has a gain coefficient given by
 ( )  



nc
 "res ( )
In the case of population inversion when ”res<0
Note that the material susceptibility characterizes the response of a material
to the excitation of an electromagnetic field. Therefore, the resonant
susceptibility res accounts for only the contributions from the induced
processes of absorption and stimulated emission, but not that from the
process of spontaneous emission.
41
Resonant optical susceptibility

When the phase information of the optical wave is of no
interest, we can find the evolution of the intensity of the
optical wave as it propagates through the medium.
dI/dz = -I
(-ve sign represents attenuation)
in the case of optical attenuation when ”res > 0, and
dI/dz = I
in the case of optical amplification when ”res < 0
42
Population inversion
43
Population inversion and optical gain





Population inversion is the basic condition for the presence of
an optical gain.
In the normal state of any system in thermal equilibrium, a
low-energy state is always more populated than a high-energy
state – no population inversion
Population inversion in a system can only be accomplished
through a process called pumping – actively exciting the
atoms in a low-energy state to a high-energy state.
Population inversion is a nonequilibrium state that cannot be
sustained without active pumping. To maintain a constant
optical gain we need continuous pumping to keep the
population inversion at a constant level.
Many different pumping techniques depending on the gain
media: optical excitation, current injection, electric
discharge, chemical reaction, and excitation with ion beams
44
Population inversion

A nonequilibrium distribution showing population inversion
Energy E
E2
E1
N1
N2
Population N
45
Population inversion and optical gain




The use of a particular pumping technique depends on the
properties of the gain medium being pumped.
The lasers and optical amplifiers are often made of either
dielectric solid-state media doped with active ions, such as
Nd:YAG and Er:glass fiber, or direct-gap semiconductors,
such as GaAs and InP.
For dielectric media, the most commonly used pumping
technique is optical pumping either with incoherent light
sources, such as flashlamps and light-emitting diodes, or
with coherent light sources from other lasers.
Semiconductor gain media can also be optically pumped,
but they are usually pumped with electric current
injection.
46
Rate equations



The net rate of change of population density in a given energy
level is described by a rate equation.
Here we only write the rate equations for the upper laser level
|2> and the lower laser level |1>.
In the presence of a monochromatic, coherent optical wave of
intensity I at a frequency ,
dN2/dt = R2 – N2/2 – (I/h) (N2e – N1a)
dN1/dt = R1 – N1/1 + N2/21 + (I/h) (N2e – N1a)
where R2 and R1 are the total rates of pumping into energy
levels |2> and |1>, and 2 and 1 are the fluorescence lifetimes
(total lifetimes) of levels |2> and |1>. The rate of population
decay, including radiative and nonradiative spontaneous
relaxation from |2> to |1> is 1/21.
47
Rate equations

Because it is possible for the population in level |2> to relax
to other energy levels, the total population decay rate of level
|2> is 1/2  1/21.
2  21  sp



In an optical gain medium, level |2> is known as the upper
laser level and level |1> is known as the lower laser level.
The fluorescence lifetime 2 of the upper laser level is an
important parameter that determines the effectiveness of a
gain medium.
In general, the upper laser level has to be a metastable state
with a relatively large 2 for a gain medium to be useful.
48
Population inversion

Population inversion in a medium is generally defined as
N2 > (g2/g1) N1


(N2 > N1 for g1 = g2)
However, this condition does not guarantee an optical gain at a particular
optical frequency  when the population in each level, |1> or |2>, is
distributed unevenly among its sublevels.
A better condition for population inversion to guarantee an optical gain at
a given frequency 
N2e() – N1a() > 0

The pumping requirement for the condition to be satisfied depends on the
properties of a medium. For atomic and molecular media, there are three
different basic systems. Each has a different pumping requirement to
reach effective population inversion for an optical gain. The pumping
requirement can be found by solving the coupled rate equations.
49
Two-level systems
|2>
hp
h
pump
|1>



When the only energy levels involved in the pumping and the
relaxation processes are the upper and the lower laser levels
|2> and |1>, the system can be considered as a two-level
system. (i.e. p = )
Level |1> is the ground state with 1 = ∞, and level |2>
relaxes only to level |1> so that 21 = 2.
The total population density is Nt = N1 + N2.
50
Two-level systems


No matter how a true two-level system is pumped, it is not
possible to achieve population inversion for an optical gain
in the steady state.
The optical pump for a two-level system has to be in
resonance with the transition between the two levels –
inducing both downward and upward transitions.
|2>
hp
hp
pump
|1>

While a pumping mechanism excites atoms from the lower
energy level to the upper energy level, the same pump also
stimulates atoms in the upper energy level to relax to the
lower energy level.
51
Two-level systems

While a pumping mechanism excites atoms from the lower
laser level to the upper laser level, the same pump also
stimulates atoms in the upper laser level to relax to the lower
laser level.
R2 = -R1 = W12pN1 - W21pN2,
where W12p and W21p are the pumping rates from 1 to 2 and
from 2 to 1.


Under these conditions, dN2/dt and dN1/dt are equivalent to
each other (N1 + N2 = Nt = constant).
The upward (W12p) and downward (W21p) pumping rates are
not independent of each other but are directly proportional to
each other because both are associated with the interaction of
the same pump source with a given set of energy levels.
52
Two-level systems


Take the upward pumping rate W12p = Wp and the downward
pumping rate to be W21P= p Wp, where p is a constant that depends
on the detailed characteristics of the two-level atomic system and
the pump source.
In the steady state when dN2/dt = dN1/dt = 0,
N2e – N1a = [Wp2(e-pa)-a]Nt [1+(1+p)Wp2 + (I2/h)(e+a)]-1

For optical pumping
p = ep/ap = e(p)/a(p),
where ap and ep are the absorption and emission cross sections at
the pump wavelength.
53
Two-level systems

In a true two-level system, the energy levels |2> and |1> can
each be degenerate with degeneracies g2 and g1, but the
population densities in both levels are evenly distributed
among the respective degenerate states.
In this situation, p = ep/ap = g1/g2 = e/a

N2e – N1a = -aNt [1+(e+a)(I/hWp/a)2]-1 < 0

No matter how a true two-level system is pumped, it is clearly
not possible to attain population inversion for an optical gain
in the steady state.

54
Two-level systems

Intuitively, the pump for a two-level system has to be in
resonance with the transition between the two levels, thus
inducing downward transitions and upward transitions.

In the steady state, the two-level system would reach thermal
equilibrium with the pump at a finite temperature T, resulting
in a Boltzmann population distribution
N2/N1 = (g2/g1) exp(-h/kBT) without population inversion.
55
Quasi-two-level systems
|2>
hp
h
pump
|1>



However, many laser gain media including laser dyes,
semiconductor gain media, and some solid-state gain media,
are often pumped as a quasi-two-level system.
An energy level is split into a band of closely spaced, but not
exactly degenerate, sublevels with its population density
unevenly distributed among these sublevels.
A system is a quasi-two-level system if either or both of the
two levels involved are split in such a manner.
56
Quasi-two-level systems



By pumping such a quasi-two-level system properly, it is
possible to reach the needed population inversion in the steady
state for an optical gain at a particular laser frequency .
Now the ratio p = ep/ap at the pump frequency p can be
made different from the ratio e/a at the laser frequency  due
to the uneven population distribution among the sublevels
within an energy level.
The pumping requirements for a steady-state optical gain from
a quasi-two-level system (see p.53)
p = ep/ap < e/a;

Wp > (1/2) a/(e – pa)
Because the absorption spectrum is generally shifted to the
short-wavelength side of the emission spectrum, these
conditions can be satisfied by pumping sufficiently strongly at a
higher transition energy than the photon energy corresponding 57
to the peak of the emission spectrum.
Three-level systems
|3>
hp
pump
Nonradiative
relaxation
|2>
h
|1>


Population inversion in steady state is possible for a threelevel system.
The lower laser level |1> is the ground state (or is very close
to the ground state, within an energy separation of << kBT
from the ground state, s.t. it is normally populated). The
atoms are pumped to an energy level |3> above the upper
laser level |2>.
58
Population inversion in three-level systems

Over a period the population in the metastable state N2
increases above those in the ground state N1.
=>The population inversion is obtained between levels |2>
and |1>.

Drawback: the three-level system generally requires very
high pump powers because the terminal state of the
stimulated transition is the ground state. More than half the
ground state atoms must be pumped into the metastable state
to attain population inversion.
59
Three-level systems

An effective three-level system satisfies the following
conditions:
 Population relaxation from level |3> to level |2> is very
fast and efficient, ideally 2 >> 32 ≈ 3
s. t. the atoms excited by the pump quickly end up in level
|2>
 Level |3> lies sufficiently high above level |2> with E32
>> kBT s. t. the population in level |2> cannot be
thermally excited back to level |3>
 The lower laser level |1> is the ground state, or its
population relaxes very slowly if it is not the ground state.

Under these conditions
R2 ≈ WpN1, R1 ≈ -WpN1, and N1+N2 ≈ Nt
1 ≈ ∞ and 21 ≈ 2
60
Three-level systems


The parameter WP is the effective pumping rate for exciting an atom in the
ground state to eventually reach the upper laser level. It is proportional to
the power of the pump.
In the steady state with a constant pump, Wp is a constant and dN2/dt =
dN1/dt = 0
N2e – N1a = (Wp2e-a)Nt [1+Wp2 + (I2/h)(e+a)]-1

The pumping condition for a constant optical gain under steady-state
population inversion
Wp > a/2e

This condition sets the minimum pumping requirement for effective
population inversion to reach an optical gain in a three-level system.
Note that almost all of the population initially resides in the lower laser
level |1>. To attain population inversion, the pump has to be strong
enough to depopulate sufficient population density from level |1>, while
the system has to be able to keep it in level |2>. In the case when a = e
(i.e. g1 = g2), no population inversion occurs before at least one-half of
the total population is transferred from level |1> to level |2>.

61
Erbium-doped silica fibers

Er3+:silica fiber amplifier is a three-level system.
3
32
pump
2
1.55 m
1
• Pumping at 980 nm using InGaAs laser diodes; a mixture of
homogeneous/inhomogeneous broadening;  ~ 5.3 THz
• The laser transition can also be directly pumped at 1.48 m by
light from InGaAsP laser diodes – like a quasi-two-level scheme
62
62
Four-level systems
Nonradiative
relaxation
hp
|3>
|2>
h
pump
|1>
Nonradiative
relaxation


|0>
A four-level system is more efficient than a three-level
system.
The lower laser level |1> lies sufficiently high above the
ground level |0>, with E10 >> kBT. Thus, in thermal
equilibrium, the population in |1> is negligibly small
compared with that in |0>. Pumping takes place from level
|0> to level |3>.
63
Four-level systems



Levels |3> and |2> need to satisfy the same conditions as in a three-level
system.
The population in level |1> relaxes very quickly back to the ground level,
ideally 1 ≈ 10 << 2, s. t. level |1> remains relatively unpopulated in
comparison with level |2> when the system is pumped.
Under these conditions,
N1 ≈ 0; R2 ≈ Wp(Nt – N2)

where the effective pumping rate Wp is proportional to the pump power.
In the steady state when Wp is held constant, by taking dN2/dt = 0,
(ignoring dN1/dt because N1 ≈ 0)
N2e – N1a ≈ N2e = (Wp2e)Nt [1 + Wp2 + (I2/h)e]-1
=> No minimum pumping requirement for an ideal four-level system because
level |1> is initially empty. A practical four-level system is much more
efficient than a three-level system.
64
Neodymium-doped glass

Nd3+:glass amplifier is a four-level system.
3
32
2
1.053 m
pump
0
1
1
65
65
Neodymium-doped glass

Level 1 is 0.24 eV above the ground state. This is substantially
larger than the thermal energy 0.026 eV at room temperature, so
that the thermal population of the lower laser is negligible.

Level 3 is a collection of four absorption bands, centered at 805,
745, 585, and 520 nm.

The excited ions decay rapidly from level 3 to level 2 and then
remain in level 2 for a substantial time sp = 330 s. 1 is very
short (~ 300 ps)

The 2→1 transition is inhomogeneously broadened because of the
amorphous nature of the glass, which presents a different
environment at each ionic location. This material therefore has a
large spontaneous linewidth (gain bandwidth)  ≈ 6 THz
66
66
Coherent optical amplifiers
Gain, nonlinearity, noise
Coherent optical amplifiers





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 remains unchanged or is shifted by a fixed amount.
In contrast, an incoherent optical amplifier increases the
intensity of an optical wave without preserving its phase.
Coherent optical amplifiers are important, for example, in the
amplification of weak optical pulses that have traveled
through a long length of optical fiber, and as a basis to
understanding laser oscillators.
68
Coherent light amplification



As seen earlier, stimulated emission allows a photon in a
given mode to induce an atom whose electron is 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. A clone photon has the same
frequency, direction and polarization as the initial photon.
These two photons in turn 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 only when the photon energy is
nearly equal to the transition energy difference, the process is
restricted to a band of frequencies determined by the
transition linewidth.
69
Laser amplification vs. electronic amplifiers



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 (electrons and
holes in a semiconductor field-effect transistor). Tuned
electronic amplifiers make use of resonant circuits (e.g. a
capacitor and an inductor) to limit the gain of the amplifier to
the band of frequencies of interest.
In contrast, atomic, molecular, and solid-state laser amplifiers
rely on differences in their allowed energy levels to provide
the principal frequency selection. These entities act as
natural resonators that select the frequency of operation and
bandwidth of the device.
70
Population inversion




Light transmitted through matter in thermal equilibrium is
attenuated.
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 attaining laser amplification is the
presence of a greater number of atoms in the upper energy
level than in the lower level. This is a nonequilibrium
situation.
Attaining such a population inversion requires a source of
power to excite (pump) the atoms to the higher energy level.
71
Ideal coherent amplifiers



An ideal coherent amplifier is a linear system that increases
the amplitude of the input signal by a fixed factor, 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
at the output with respect to the input).
gain
phase


Output amp.

Input amplitude
72
Real coherent amplifiers





Real coherent amplifiers deliver a gain and phase shift that are frequency
dependent. The gain and phase shift determine the amplifier’s transfer
function.
For a sufficiently large input amplitude, real amplifiers generally exhibit
saturation, a form of nonlinear behavior in which the output amplitude
does not increase in proportion to the input amplitude.
Saturation introduces harmonic components into the output, provided that
the amplifier bandwidth is sufficiently broad to pass them.
Real amplifiers also introduce noise, s.t. a random fluctuating component
is present at the output, regardless of the input.
An amplifier may therefore be characterized by the following features:

Gain

Bandwidth

Phase shift

Power source

Nonlinearity and gain saturation

Noise
73
Theory of laser amplification

A monochromatic optical plane wave traveling in the z
direction with frequency , electric field
E ( z )  Re[ E ( z ) exp(i 2t )]

Intensity

Photon-flux density (photons per second per unit area)
I ( z ) | E ( z ) |2
 ( z )  I ( z ) / h


Consider the atomic medium (gain or active medium) with
two relevant energy levels whose energy difference nearly
matches the photon energy h.
The numbers of atoms per unit volume in the lower and upper
energy levels are denoted N1 and N2. (assume g1 = g2)
74
Gain and bandwidth


The wave is amplified with a gain coefficient () (per unit
length) and undergoes a phase shift () (per unit length).
() > 0 corresponds to amplification, () < 0 corresponds
to attenuation.
Recall that the probability density (s-1) that an unexcited
atom absorbs a single photon is
Wi   ( )
where the transition cross section at the frequency 
 ( ) 

c2
8n   sp
2
2
gˆ ( )
Here we assume the probability density for stimulated
emission is the same as that for absorption. (a() = e() =
())
75
Gain coefficient




The average density of absorbed photons (number of photons
per unit time per unit volume) is N1Wi.
Similarly, the average density of clone photons generated as a
result of stimulated emission is N2Wi.
The net number of photons gained per second per unit
volume is therefore NWi, where N = N2 – N1 is the
population density difference.
N is referred to as the population difference.
 If N > 0, a population inversion exists, in which case the
medium acts as an amplifier and the photon-flux density
can increase.
 If N < 0, the medium acts as an attenuator and the photonflux density decreases.
 If N = 0, the medium is transparent.
76
Gain coefficient



As the incident photons travel in the z direction, the
stimulated-emission photons also travel in this
direction.
An external pump providing a population inversion
N > 0 then causes the photon-flux density (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. (z) thus increases
exponentially.
77
Gain coefficient

Consider the incremental number of photons per unit area per
unit time, d(z), is the number of photons gained per unit
time per unit volume, NWi, multiplied by the thickness dz
d  NWi dz

In the form of a differential equation
d
  ( ) ( z )
dz
where the gain coefficient
 ( )  N ( )  N
c2
8n   sp
2
2
gˆ ( )
78
Gain coefficient


The coefficient () represents the net gain in the photon-flux
density per unit length of the medium.
The photon-flux density therefore is given as
 ( z )   (0) exp[ ( ) z ]

The optical intensity I(z) = h(z)
I ( z )  I (0) exp[ ( ) z ]

Thus, () also represents the gain in the intensity per unit
length of the medium.
79
Gain

For an interaction region of total length d, the overall gain of
the laser amplifier G() is defined as the ratio of the photonflux density at the output to the photon-flux density at the
input,
G ( )   (d ) /  (0)


G ( )  exp[ ( )d ]
Note that in the absence of a population inversion, N is
negative (N2 < N1) and so is the gain coefficient. The
medium will then attenuate light traveling in the z direction.
A medium in thermal equilibrium cannot provide laser
amplification.
80
Gain bandwidth





The dependence of the gain coefficient () on the frequency
of the incident light  is contained in its proportionality to the
lineshape function g().
The latter is a function of width  centered about the atomic
resonance frequency 0 = (E2 – E1)/h.
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  in frequency (Hz) and  in wavelength
(nm) are related by
 = |(c/)| = (c/2) = (2/c)
81
Gain bandwidth

If the lineshape function is Lorentzian, the gain coefficient is
then also Lorentzian with the same width
( / 2) 2
 ( )   ( 0 )
(  0 ) 2  ( / 2) 2
where the peak gain coefficient at the central frequency 0
 ( 0 )  N
c2
4 2 n 2 2 sp 
82
Phase‐shift coefficient


The laser amplification process also introduces a phase shift.
When the lineshape is Lorentzian with linewidth 
g() = (/2) / [( – 0)2 +(/2)2]

The amplifier phase shift per unit length turns out to be () = [( – 0)/] ()

This phase shift is in addition to that introduced by the medium
hosting the laser atoms. 83
Gain coefficient and phase-shift coefficient for a laser
amplifier with a Lorentzian lineshape function
gain
coefficient
()



Phase-shift
coefficient
()


(Compare these with the ideal coherent amplifier gain and phase responses on p. 72)84
Rate equations
R2
2
1/21 = 1/sp + 1/nr
sp
R1


1
1
nr
1/2 = 1/21 + 1/20
20
1/nr: non-radiative decay rate
Steady-state populations of levels 1 and 2 can be maintained only if
the energy levels above level 2 are continuously excited by
pumping and ultimately populate level 2.
Pumping serves to populate level 2 at rate R2 and depopulate level
1 at rate R1 (per unit volume per second)
=> levels 1 and 2 can attain non-zero steady-state populations.
85
85
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 = R2 – N2/2
dN1/dt = -R1 – N1/1 + N2/21

Steady-state population difference in the absence of amplifier
radiation (dN1/dt = dN2/dt = 0)
N0 = N2 – N1 = R22(1-1/21) + R11

A large gain coefficient requires a large population difference
(0() = N0e())
86
86
Rate equations in the absence of amplifier radiation

To increase population difference N0

Increase pumping and de-pumping rate (R2 and R1)
Long 2, but sp must be sufficiently short so as to make
the radiative transition rate large

Short 1 (if R1 < (2/21)R2)



The physical picture:

the upper level should be pumped strongly and decay
slowly so that it retains its population.

The lower level should be de-pumped strongly so that it
quickly disposes of its population.
Ideally, 21 ≈ sp << 20 so that 2 ≈ sp, and 1 << sp
87
87
Rate equations in the presence of amplifier radiation


The presence of radiation near the resonance frequency 0 enables
transitions between levels 2 and 1 to occur via stimulated emission and
absorption.
These processes are characterized by the probability density
Wi = ()
where  = I/h (assuming g1 = g2 and thus e() = a())
2
R2
Wi-1
1
R1
sp
1
nr
20
88
88
Rate equations in the presence of amplifier radiation
dN2/dt = R2 – N2/2 – N2Wi + N1Wi
dN1/dt = -R1 – N1/1 + N2/21 + N2Wi - N1Wi

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.

Under steady-state conditions (dN1/dt = dN2/dt = 0), the
population difference in the presence of amplifier radiation
(assuming g1 = g2)
N = N2 – N1 = N0/(1 + sWi)

The characteristic time s (saturation time constant) is always
positive (2 ≤ 21) is given by
s = 2 + 1(1 – 2/21)
89
89
Population difference N
Depletion of the steady-state population difference



N0
N0/2
0
0.01
0.1
1
10
sWi
If the radiation is sufficiently weak so that sWi << 1 (the small signal
approximation), we may take N ≈ N0
As the amplifier radiation becomes stronger, Wi increases and ultimately the
population difference N  0 (transparency). This arises because stimulated
emission and absorption dominate the interaction when Wi is very large and
they have equal probability densities. Even very strong radiation cannot
convert a negative population difference into a positive one, nor vice versa.
The quantity s plays the role of a saturation time constant, i.e. when Wi = 1/s,
90
90
N is reduced by a factor of 2 from its value when Wi = 0.
Four-level pumping

Pump R
Laser Wi-1
Rapid
decay

 Rapid
decay

Short-lived |3>
Long-lived |2>

Short-lived |1>
Ground state |0>
Here we assume that the rate of pumping into level 3, and out
of level 0, are the same.
91
91
Four-level pumping




An external source of energy pumps atoms from level 0 to
level 3 at a rate R.
If the decay from level 3 to level 2 is sufficiently fast, it may
be taken to be instantaneous, in which case pumping to level
3 is equivalent to pumping level 2 at the rate R2 = R.
However, in this case, atoms are neither pumped into nor out
of level 1, s.t. R1 = 0.
Thus, in the absence of amplifier radiation (Wi =  = 0), the
steady-state population difference is given by (see p.86)
 1 
N 0  R 2 1 
  21 
92
Four-level pumping

In most four-level systems, the nonradiative decay component
in the 2 to 1 transition is negligible (sp << nr) and 20 >> sp
>> 1, s.t.
N 0  R sp
 s   sp
And therefore

R sp
N
1  spWi
We have assumed that the pumping rate R is independent of
the population difference N = N2 – N1.
93
93
Four-level pumping

This is not always the case because the population densities
of the ground state and level 3, Ng and N3, are related to N1
and N2 by
N g  N1  N 2  N 3  N a

where the total atomic density in the system, Na, is a constant.
If the pumping involves a transition between the ground state
and level 3 with transition probability W, then
R  (N g  N 3 )W

If levels 1 and 3 are short-lived, then N1 ≈ N3 ≈ 0, Ng + N2 ≈
Na s.t. Ng ≈ Na - N2 ≈ Na - N
94
94
Four-level pumping

Under these conditions, the pumping rate can be
approximated as
R  ( N a  N )W


which reveals that the pumping rate is a linearly decreasing
function of the population difference N and is thus not
independent of it.
This arises because the population inversion established
between levels 2 and 1 reduces the number of atoms available
to be pumped.
We obtain
 sp N aW
N
1  spW   spWi
95
95
Four-level pumping

The population difference can be written in the general form
N0
N
1   sWi
 sp N aW
N0 
1  spW
 sp
s 
1  spW
96
96
Three-level pumping

Pump R
Laser
Wi-1
Rapid
decay
Short-lived |3>
Long-lived |2>

Ground state |1>

Here we assume that the rate of pumping into level 3
is the same as the rate of pumping out of level 1.
97
97
Three-level pumping

Under rapid 3 to 2 decay, the three-level system (assumed R
is independent of N)
1  
R1  R2  R

In steady state, both the rate equations provide the same result
0  R

 2   21
N2
 21
 N 2Wi  N1Wi
As 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.
N1  N 2  N a
98
98
Three-level pumping

The population difference N can be cast in the form:
N0
N
1  sWi

Where
N 0  2R 21  N a
 s  2 21

When nonradiative decay from level 2 to level 1 is negligible
(sp << nr), 21 may be replaced by sp
N 0  2 R sp  N a
 s  2 sp
99
99
Three-level pumping



Attaining a population inversion N0 > 0 in the three-level
system requires a pumping rate R > Na/2sp. A substantial
pump power density given by E3Na/2sp.
The large population in the ground state (which is the lowest
laser level) is an inherent obstacle to attaining a population
inversion in a three-level system that is avoided in a fourlevel system (in which level 1 is normally empty as 1 is
short).
The saturation time constant s ≈ sp for the four-level
pumping scheme is half that for the three-level scheme.
100
100
Three-level pumping

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  (N1  N 3 )W

N3 ≈ 0, N1 = (Na-N)/2,
1
R  (N a  N )W
2

Substituting this in the principal equation
2R sp  N a
N
1 2 spWi
101
101
Three-level pumping

We can write the population difference in the usual form
N0
N
1  sWi
N a ( spW 1)
N0 
1  spW
2 sp
s 
1  spW
As in the four-level scheme, N0 and s saturates as the
pumping transition probability W increases.
102
102
Saturated gain in homogeneously broadened media




The gain coefficient () of a laser medium depends on the
population difference N.
N is governed by the pumping level.
N depends on the transition rate Wi.
Wi depends on the photon-flux density .
=> 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.
103
Saturation photon-flux density
Wi   ( )

Substituting

Into steady-state population difference (in the presence of
amplifier radiation)
=>
N0
N
1  sWi
N0
N
1  /  s ( )
1
s
c
  s ( ) 
ĝ( )
2 2
 s ( )
8 n   sp
2
where
104
Gain coefficients

This represents the dependence of the population difference N on
the photon-flux density .

Substituting N into the expression for the gain coefficient,
 ( )  N ( )  N

c2
8n   sp
2
2
gˆ ( )
We obtain the saturated gain coefficient
 0 ( )
 ( ) 
1  / s ( )
Where the small-signal gain coefficient
c2
 0 ( )  N 0 ( )  N 0
ĝ( )
2 2
8 n   sp
105
105
Gain coefficients


The gain coefficient is a decreasing function of the photonflux density .
When  equals its saturation value s() = s, the gain
coefficient is reduced to half its unsaturated value.
)
1
0.5
0
0.01
0.1
1
10 s()
106
106
Amplified spontaneous emission



The resonant medium that provides amplification via
stimulated emission also generates spontaneous emission.
The light arising from the spontaneous emission, 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 noise associated with
amplified spontaneous emission (ASE) is broadband,
multidirectional, and unpolarized.
=> it is possible to filter out some of this noise by following the
amplifier with a narrowband optical filter, a collection
aperture, and a polarizer.
107
Amplified spontaneous emission
Spontaneous
photon flux
Filter and
polarizer
Input
photon flux

Spontaneous emission is a source of amplifier noise. It is
relatively broadband, radiated in all directions, and
unpolarized. Optics can be used at the output of the amplifier
to limit the spontaneous emission noise to a narrow optical
band, solid angle d and a single polarization.
108
Amplifier noise




The ASE of a laser amplifier is directly proportional to the optical
bandwidth of the amplifier.
To increase the signal-to-noise ratio (SNR) at the amplifier output, the
total noise power can be reduced to a minimum by placing at the output
end of an amplifier an optical filter that has a narrow bandwidth matching
the bandwidth of the optical signal.
Because of the spontaneous emission noise, the SNR of an optical signal
always degrades after the optical signal passes through an amplifier.
The degradation of the SNR of the optical signal passing through an
amplifier is measured by the optical noise figure of the amplifier defined
as
SNRin
Fo 
SNRout
where SNRin and SNRout represent the values of the optical SNR at the
input and output ends of the amplifier.
109