11

Lecture 8: Laser amplifiers! 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.

2

Intro! 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

3

Optical transitions

3

4

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 energy-band 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

55

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.

66

! Absorption – when the quantum energy h! equals the energy difference between the two energy levels (a resonantcondition); 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 isessential in the overall balance between emission andabsorption, 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)

Three basic photon-matter interaction processes

7

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

88

Spontaneous emission! An electron spontaneously falls from a higher energy level E2to a lower one E1, the emitted photon has frequency

! = (E2 – E1) / h

! This photon is emitted in a random direction with arbitrarypolarization.

! 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>.

|2>

|1>

99

Probability per second of a spontaneous emission

! For example, if there are N2 population per unit volume inlevel |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

! Generally an electron can make jumps to more than one lower level, unless it is in the first (lowest) excited level.

A!negative!sign!because!the!population!of!level!2!is!decreasing

1010

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.

1111

Spectral lineshape! 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.

12

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

12

!!$

#!Area = 1

g(% )0

&

' d% ( g())0

&

' d) (1

g(% ) ( 2* g())

^

13

Homogeneous broadening! 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 andthe 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

14

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:

14

g(% ) ( #%h

2*[(% -%21)2 . (#%h / 2)2 ]

where #!h (/+012*/is the FWHM of the lineshape3

15

Inhomogeneous broadening! 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

16

Spectral 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 inhomogeneouslybroadened.

Ref: Optical resonance and two-level atoms, L. Allen and J. H. Eberly, pp. 7-10

! !

Spread in frequencies Gaussian lineshape

1717

Transition rates! 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 spectralcharacteristics 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

! The spectral intensity distribution I(!) = (c/n) u(!), n is the refractive index of the medium

! The total intensity

u ( u(% )d%0

&

'

I ( I(% )d%0

&

'

1818

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 perfrequency 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(!))

Length!c!in!a!second

AV

19

Transition rates! For the upward transition from |1> to |2> associated with

absorption in the frequency range between ! and !+d! is

! For the downward transition from |2> to |1> associated with stimulated emission

! The spontaneous emission rate is independent of the energy density of the radiation and is solely determined by the transition lineshape function

The A and B constants are the Einstein A and B coefficients. 19

W12 (% )d% ( B12u(% )g(% )d%

W21(% )d% ( B21u(% )g(% )d%

Wsp(% )d% ( A21g(% )d%

(s-1)

(s-1)

(s-1)

2020

Radiative processes connecting two energy levels in thermal equilibrium

Population!N1

Population!N2

Spontaneous!emission

Stimulatedemission

absorption

E2

E1

h!

! Einstein (1917) demonstrated that the rates of the three transition processes of absorption (B12), stimulated emission(B21) and spontaneous emission (A21) are related.

2121

Transition rates! The total induced transition rates

W12 ( W12 (% )d0

&

' % ( B12 u(% )g(% )d0

&

' %

W21 ( W21(% )d0

&

' % ( B21 u(% )g(% )d0

&

' %

! The total spontaneous emission rate is

Wsp ( Wsp(% )d0

&

' % ( A21

u(!)

N2

N1

|2>

|1>

Wsp = A21W21 = B21u(!)W12 = B12u(!)

2222

Transition rates! 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 blackbodyradiation) is given by Planck’s formula:

where kB is the Boltzmann constant, kBT is the thermal energy (kBT = 26 meV @ T = 300 K)

u(% ) ( 8*n3h% 3

c31

eh% /kBT -1

2323

Blackbody radiation! 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

2424

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

u4!5 = (8*n6h!3/c3) [exp(h!/kBT) – 1]-1

= (8*n6!2/c3) h! [exp(h!/kBT) – 1]-1

Photon!density!of!states in!a!mediumOf!refractive!index!n(number!of!photon!modes!pervolume!per!frequency!interval)

Photon!probability!of!occupancy(average!number!of!photons!in!each!mode!according!to!Bose"Einsteindistribution)

Photon!energy

2525

Planck’s law of blackbody radiation

0

20

40

60

80

100

120

140

160

180

0 5E+13 1E+14 1.5E+14 2E+14 2.5E+14 3E+14 3.5E+14

Radiation!en

ergy!den

sity!!u(!)!(JHz"1m

"3)

Frequency!! (Hz)

1500!K

1000!K

26

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!

! The number of atoms per unit volume making the upwardtransition per unit time

26

N2 (W21(% ).Wsp(% ))d%

N1W12 (% )d%

2727

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

! This is the principle of detailed balance in thermal equilibrium. The steady-state population distribution in thermal equilibrium:

N2 (W21(% ).Wsp(% ))( N1W12 (% )

N2

N1

( W12 (% )W21(% ).Wsp (% )

( B12u(% )B21u(% ). A21

28

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.

where g2 and g1 are the degeneracy factors* of these energy levels, and the energy density

*In an atomic or molecular system, a given energy level usually consists of a number of degenerate quantum-mechanical states, which have the same energy.

28

N2

N1

( g2

g1

exp(-h% / kBT )

u(% ) ( A21 / B21

g1B12

g2B21

7

89

:

;<exp(h% / kBT )-1

29

Boltzmann distribution

E1

E2

Energy E

Population NN2 N1

= exp (-h!/kBT)1 eV

(=1) (=5 x 1016)

[For T = 300 K, kBT = 26 meV]

N2 << N1 in thermal equilibrium

N2 << N1 in thermal equilibrium

30

Transition rates

30

Identify u(!) with Planck’s formula:

A21

B21

( 8*n3h% 3

c3

g1B12 ( g2B21

The spontaneous radiative lifetime of the atoms in the level |2> associated with the radiative spontaneous transition from |2> to|1> is

" sp (1

Wsp

( 1A21

31

Transition rates! Therefore, the spectral dependence of the spontaneous

emission rate

! The transition rates of both of the induced processes of absorption and stimulated emission are directly proportional to the spontaneous emission rate.

31

Wsp (% ) ( 1" sp

g(% )

W21(% ) ( c3

8*n3h% 3" sp

u(% )g(% ) ( c2

8*n2h% 3" sp

I(% )g(% )

W12 (% ) ( g2

g1

W21(% )

3232

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].

The emission cross section

The absorption cross section

W21(% ) ( I(% )h%

> 21(% )

W12 (% ) ( I(% )h%

>12 (% )

> e(% ) (> 21(% ) ( c2

8*n2% 2" sp

g(% )

> a (% ) (>12 (% ) ( g2

g1

> 21(% ) ( g2

g1

> e(% )

4?2h!@//photon-fluxdensity)

33

Transition cross section! For the ideal Lorentzian lineshape in a homogeneously

broadened medium

! The peak value of the lineshape occurs at the center of the spectrum and is a function of linewidth #!h only.

! Thus, the peak value of the emission cross section at the center wavelength A of the spectrum

g(% ) ( #%h

2*[(% -%21)2 . (#%h / 2)2 ]

h

g%*

%#

( 2)(ˆ 21

sphe n "%*

A>#

( 22

2

4

3434

Characteristics of some laser materialsGain medium Wavelength

(Bm)System Peak cross

section >e(m2)

Spontaneouslinewidth(gainbandwidth)#!

"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 Bs 240 Bs

Nd:glass 1.054 I, 4 4.0 x 10-24 6 THz 330 Bs 330 Bs

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 Bs 3.2 Bs

Semiconductor 0.37-1.65 H/I, Q2 1-5 x 10-20 10-20 THz ~1 ns ~1 ns

H: homogeneously broadened; I: inhomogeneously broadened

35

Optical absorption and amplification

35

3636

Optical absorption and amplification! For a monochromatic optical field at frequency ! and intensity

I(!) = IC(!’-!)

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

37

Optical absorption and amplification! absorption coefficient [m-1, cm-1]

D(!) = N1>a(!) – N2>e(!) = (N1 – (g1/g2)N2) >a(!)

! gain coefficient [m-1, cm-1]+(!) = N2>e(!) – N1>a(!) = (N2 – (g2/g1)N1) >e(!)

D(!) > 0 and +(!) < 0 if N1 > (g1/g2) N2

+(!) > 0 and D(!) < 0 if N2 > (g2/g1) N1

E 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.

E A material must be in a nonequilibrium state of population inversion withthe upper energy level more populated than the lower energy level inorder to provide a net optical gain to the optical field. 37

38

! 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]D(!) = N1>a(!) – N2>e(!) = (N1 – N2) >(!)

! gain coefficient [m-1, cm-1]+(!) = N2>e(!) – N1>a(!) = (N2 – N1) >(!)

D(!) > 0 and +(!) < 0 if N1 > N2

+(!) > 0 and D(!) < 0 if N2 > N1

And >e(!) = >a(!) = >(!)

Optical absorption and amplification

39

Resonant optical susceptibility! For resonant interaction of an isotropic medium with a

monochromatic plane optical field at a frequency ) = 2*!,we have

where Pres is the polarization contributed by the resonanttransitions and ,res is the resonant susceptibility.

! The time-averaged power density absorbed by the medium is

39

E(t) ( Ee-i)t .E *ei)t

Pr es (t) (F0 (, res ())Ee-i)t . , *res ())E *ei)t )

Inc

EtPEW resres

tp )("||)("2 2

0 ),)),)F ((GGH(

42

Kramers-Kronig Relations! A dispersive material, i.e. a material whose refractive index is

wavelength dependent, must be absorptive and must exhibit an absorption coefficient that is also wavelength dependent.

! The relation between the absorption coefficient and the refractive index is a result of the Kramers-Kronig relations, which relate the real and imaginary parts of the susceptibility of a medium, !’(") and !”(")

! Given the real or the imaginary component of !(") for all ", Kramers-Kronig relations allow the complementary component to be determined for all ". The Kramers-Kronig relations connecting !”(") and !’(") translate into relations between the absorption coefficient #(") and the refractive index n(").

! The Kramers-Kronig relations are a special Hilbert-transform pair, as can be understood from linear systems theory. They are applicable for all linear, shift-invariant, casual systems with real impulse responsefunctions.

dss

ss$%

&'

022)("2)('

(!

)(! ds

ss

$%

&'

022)('2)("

((!

)(!

40

Resonant optical susceptibility! Relate the time-averaged power density absorbed by the

medium to the population relation

! The imaginary part of the susceptibility contributed by the resonant transitions between energy levels |1> and |2> is

! The real part ,’res()) can then be found through the Kramers-Kronig 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 opticalfield to the medium if ,”res > 0, but there is a net power gain for the optical field if ,”res < 0.

40

, "res ()) ( nc)

[N1> a ())-N2> e())]

INNInc

W earesp )]()([)(" 21 )>)>),) -((

41

Resonant optical susceptibility! The medium has an absorption coefficient given by

in the case of normal population distribution when ,”res>0,whereas it has a gain coefficient given by

! 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 inducedprocesses of absorption and stimulated emission, but not that from the process of spontaneous emission.

41

D()) ( )nc, "res ())

)(")( ),))+ resnc-(

42

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 = -DI

in the case of optical attenuation when ,”res > 0, and

dI/dz = +I

in the case of optical amplification when ,”res < 0

42

(-ve sign represents attenuation)

43

Population inversion

43

4444

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

45

Population inversion! A nonequilibrium distribution showing population inversion

E1

E2

Energy E

Population NN1 N2

46

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

4747

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 spontaneousrelaxation from |2> to |1> is 1/"21.

4848

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 I 1/"21.

"2 J "21 J "sp

! In an optical gain medium, level |2> is known as the upperlaser 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.

49

Population inversion! Population inversion in a medium is generally defined as

N2 > (g2/g1) N1

! However, this condition does not guarantee an optical gain at a particularoptical 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 ata 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 threedifferent 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.

(N2 > N1 for g1 = g2)

5050

Two-level systems

! 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.

|2>

|1>

h!p

pumph!

51

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.

! 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.

|2>

|1>

h!p

pumph!p

5252

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 - W21

pN2,

where W12p and W21

p are the pumping rates from 1 to 2 and from 2 to 1.

E Under these conditions, dN2/dt and dN1/dt are equivalent toeach other (N1 + N2 = Nt = constant).

E The upward (W12p) and downward (W21

p) 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.

5353

Two-level systems! Take the upward pumping rate W12

p = Wp and the downwardpumping rate to be W21

P= 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/>a

p = >e(Ap)/>a(Ap),

where >ap and >e

p are the absorption and emission cross sections at the pump wavelength.

54

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 distributedamong the respective degenerate states.

" In this situation, p = >ep/>a

p = g1/g2 = >e/>a

E N2>e – N1>a = ->aNt [1+(>e+>a)(I/h!/./Wp/>a)"2]-1 < 0

E 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.

5555

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.

5656

Quasi-two-level systems

! 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.

h!p

pumph!

|1>

|2>

5757

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/>a

p at the pump frequency !p can be made different from the ratio >e/>a at the laser frequency ! dueto the uneven population distribution among the sublevels within an energy level.

! The pumping requirements for a steady-state optical gain froma quasi-two-level system (see p.53)

p = >ep/>a

p < >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 to the peak of the emission spectrum.

5858

Three-level systems

! Population inversion in steady state is possible for a three-level system.

! The lower laser level |1> is the ground state (or is very close to the ground state, within an energy separation of << kBTfrom the ground state, s.t. it is normally populated). The atoms are pumped to an energy level |3> above the upper laser level |2>.

h!p

pump h!

Nonradiativerelaxation

|3>

|2>

|1>

59

Population inversion in three-level systems

! Over a period the population in the metastable state N2increases above those in the ground state N1.

=>The population inversion is obtained between levels |2> and |1>.

! Drawback: the three-level system generally requires veryhigh pump powers because the terminal state of the stimulated transition is the ground state. More than half theground state atoms must be pumped into the metastable state to attain population inversion.

6060

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 "3s. 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 conditionsR2 WpN1, R1 -WpN1, and N1+N2 Nt

"1 and "21 "2

6161

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

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>.

6262

Erbium-doped silica fibers! Er3+:silica fiber amplifier is a three-level system.

3

2

1

pump

"32

1.55 Bm

• 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 Bm by light from InGaAsP laser diodes – like a quasi-two-level scheme

6363

Four-level systems

! 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 smallcompared with that in |0>. Pumping takes place from level |0> to level |3>.

h!p h!

Nonradiativerelaxation

Nonradiativerelaxation

pump

|0>

|1>

|2>

|3>

6464

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 becauselevel |1> is initially empty. A practical four-level system is much more efficient than a three-level system.

6565

Neodymium-doped glass! Nd3+:glass amplifier is a four-level system.

0

3

2

1pump

"32

"1

1.053 Bm

6666

! 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 Bs. "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

Neodymium-doped glass

Coherent optical amplifiersGain, nonlinearity, noise

68

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.

69

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 samefrequency, 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 isrestricted to a band of frequencies determined by the transition linewidth.

70

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 tothe 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.

71

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 thepresence of a greater number of atoms in the upper energy level than in the lower level. This is a nonequilibriumsituation.

! Attaining such a population inversion requires a source of power to excite (pump) the atoms to the higher energy level.

72

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

Input amplitude

Out

put a

mp.

73

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

74

Theory of laser amplification! A monochromatic optical plane wave traveling in the z

direction with frequency !, electric field

! Intensity

! Photon-flux density (photons per second per unit area)

! 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)

)]2exp()(Re[)( tizEzE *%-(

2|)(|)( zEzI =

%K hzIz /)()( (

75

Gain and bandwidth! The wave is amplified with a gain coefficient +(!) (per unit

length) and undergoes a phase shift L(!) (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

where the transition cross section at the frequency !

! Here we assume the probability density for stimulated emission is the same as that for absorption. (>a(!) = >e(!) = >(!))

)(%K>(iW

)(ˆ8

)( 22

2

%"%*

%> gn

csp

(

76

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 photon-flux density decreases.

# If N = 0, the medium is transparent.

77

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 K(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. K(z) thus increases exponentially.

78

Gain coefficient! Consider the incremental number of photons per unit area per

unit time, dK(z), is the number of photons gained per unit time per unit volume, NWi, multiplied by the thickness dz

! In the form of a differential equation

where the gain coefficient

dzNWd i(K

)()( zdzd K%+K (

)(ˆ8

)()( 22

2

%"%*

%>%+ gn

cNNsp

((

79

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

! The optical intensity I(z) = h!K(z)

! Thus, +(!) also represents the gain in the intensity per unit length of the medium.

])(exp[)0()( zz %+KK (

])(exp[)0()( zIzI %+(

80

Gain! For an interaction region of total length d, the overall gain of

the laser amplifier G(!) is defined as the ratio of the photon-flux density at the output to the photon-flux density at the input,

! 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.

)0(/)()( KK% dG (

])(exp[)( dG %+% ((M

81

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 #A in wavelength (nm) are related by

#A = |#(c/!)| = (c/!2)#! = (A2/c)#!

82

Gain bandwidth

! If the lineshape function is Lorentzian, the gain coefficient isthen also Lorentzian with the same width

where the peak gain coefficient at the central frequency !0

220

2

0 )2/()()2/()()(%%%

%%+%+#.-

#(

%"%*%+

#(

spncN 222

2

0 4)(

83

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!!

L(!)!=![(! – !0)/#!]!+(!)

! This!phase!shift!is!in!addition!to!that!introduced!by!the!mediumhosting!the!laser!atoms.!!

84

Gain coefficient and phase-shift coefficient for a laser amplifier with a Lorentzian lineshape function

Phase-shiftcoefficientL(!)

gaincoefficient+(!)

!

!!$

#!

!$

(Compare these with the ideal coherent amplifier gain and phase responses on p. 72)

Rate equations

85

! Steady-state populations of levels 1 and 2 can be maintained only ifthe 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 level1 at rate R1 (per unit volume per second)

=> levels 1 and 2 can attain non-zero steady-state populations. 85

2

1"sp "nr

"1 "20

R2

R1

1/"21 = 1/"sp + 1/"nr

1/"2 = 1/"21 + 1/"20

1/"nr: non-radiative decay rate

8686

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(!))

8787

! 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 decayslowly 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

Rate equations in the absence of amplifier radiation

8888

Rate equations in the presence of amplifier radiation

! The presence of radiation near the resonance frequency !0 enablestransitions between levels 2 and 1 to occur via stimulated emission andabsorption.

! These processes are characterized by the probability density

Wi = K>(!)

where K = I/h! (assuming g1 = g2 and thus >e(!) = >a(!))

2

1"sp "nr

"1 "20

R2

R1

Wi-1

8989

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 stimulatedemission from level 2 to level 1 and increased by absorptionfrom 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)

Rate equations in the presence of amplifier radiation

9090

Depletion of the steady-state population difference

! 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 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,N is reduced by a factor of 2 from its value when Wi = 0.

0.01 0.1 1 10

N0/2

N0

0 "sWi

Popu

latio

n di

ffere

nce

N

91

Four-level pumping

! Here we assume that the rate of pumping into level 3, and out of level 0, are the same.

91

Pump R

Ground state |0>

Short-lived |1>

Long-lived |2>

Short-lived |3>"60

Laser Wi-1

"1

Rapiddecay

Rapiddecay

"01 "0$

92

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 = K = 0), the steady-state population difference is given by (see p.86)

N0 ( R" 2 1- "1

" 21

7

89

:

;<

93

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.

And therefore

! We have assumed that the pumping rate R is independent of the population difference N = N2 – N1.

93

N0 O R" sp

" s O " sp

N OR" sp

1." spWi

94

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 N1and N2 by

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

! If levels 1 and 3 are short-lived, then N1 N3 0, Ng + N2Na s.t. Ng Na - N2 Na - N

94

Ng .N1 . N2 .N3 ( Na

R ( (Ng -N3)W

95

Four-level pumping! Under these conditions, the pumping rate can be

approximated as

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

95

WNNR a )( -O

N O" spNaW

1." spW ." spWi

96

Four-level pumping

! The population difference can be written in the general form

96

isWNN".

(1

0

N0 O" spNaW1." spW

" s O" sp

1." spW

97

Three-level pumping

! Here we assume that the rate of pumping into level 3 is the same as the rate of pumping out of level 1.

97

Pump R

Ground state |1>

Long-lived |2>

Short-lived |3>"60

Laser Wi-1

Rapiddecay

"01

98

Three-level pumping! Under rapid 3 to 2 decay, the three-level system (assumed R

is independent of N)

! In steady state, both the rate equations provide the same result

! 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.

98

R1 ( R2 ( R "1 (& " 2 ( " 21

0 ( R- N2

" 21

- N2Wi .N1Wi

N1 .N2 ( Na

99

Three-level pumping! The population difference N can be cast in the form:

! Where

! When nonradiative decay from level 2 to level 1 is negligible ("sp << "nr), "21 may be replaced by "sp

99

N ( N0

1." sWi

N0 ( 2R" 21 -Na

" s ( 2" 21

asp NRN -O "20

sps "" 2O

100

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 four-level system (in which level 1 is normally empty as "1 isshort).

! The saturation time constant "s "sp for the four-level pumping scheme is half that for the three-level scheme.

100

101

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

! N3 0, N1 = (Na-N)/2,

! Substituting this in the principal equation

101

R ( (N1 -N3)W

R O 12

(Na - N )W

N (2R" sp - Na

1. 2" spWi

102

Three-level pumping! We can write the population difference in the usual form

As in the four-level scheme, N0 and "s saturates as the pumping transition probability W increases.

102

N ( N0

1." sWi

N0 (Na (" spW -1)

1." spW

" s (2" sp

1." spW

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 K.

=> the gain coefficient +(!) of a laser medium is dependent on the photon-flux density K that is to be amplified. This is the origin of gain saturation and laser amplifier nonlinearity.

103

Saturation photon-flux density! Substituting

! Into steady-state population difference (in the presence of amplifier radiation)

=>

where104

Wi (K> (% )

N ( N0

1." sWi

N ( N0

1.K /Ks (% )1

Ks (% )( " s> (% ) ( c2

8*n2% 2" s

" sp

g(% )

105105

Gain coefficients! This represents the dependence of the population difference N on

the photon-flux density K.

! Substituting N into the expression for the gain coefficient,

! We obtain the saturated gain coefficient

Where the small-signal gain coefficient

)(ˆ8

)()( 22

2

%"%*

%>%+ gn

cNNsp

((

+ (% ) ( +0 (% )1.K /Ks (% )

+0 (% ) ( N0> (% ) ( N0c2

8*n2% 2" sp

g(% )

106106

! The gain coefficient is a decreasing function of the photon-flux density K.

! When K equals its saturation value Ks(!) = 12"s>4!5, the gain coefficient is reduced to half its unsaturated value.

0.01 0.1 1 10

0.5

1

0K2Ks(!)

+2+$4!)

Gain coefficients

107

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.

108

Amplified spontaneous emission

! 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 dP and a single polarization.

Inputphoton flux

Spontaneousphoton flux

Filter and polarizer

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

where SNRin and SNRout represent the values of the optical SNR at the input and output ends of the amplifier.

109

Fo (SNRin

SNRout