Lecture 5

Optical gain

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+ 1 Ray and wave optics
+ 2 Light beams and beam optics
+ 3 Optical resonators
+ 4 Photon optics
+ 5 Optical gain
6 Laser oscillation
7 Pulsed lasers
8 Electromagnetic optics and crystal optics
9 Second-order nonlinear optical effects
10 Third-order nonlinear optical effects
11 Electro-optics
12 Acousto-optics
13 Ultrafast optics
14 Laser spectroscopy and microscopy
15 Laser fabrication and materials processing
2
Interactions between photons and atoms
EM states (optical Electronic states
modes) E and k (material)
conservation Level 2 (excited state)
rules E2

cavity modes Occupancies of

EM mode and E1
Level 1 (ground state)
electronic
levels

Amplification: increase in the photon number

We need to exploit light-matter interactions in order to obtain
amplification
3
Optical transitions and statistics
Spontaneous emission E E
En
population
E3 inversion

E2
Stimulated absorption
E1
N(E), P(E)
✦N(E) number of occupied levels
✦P(E) probability of level occupation
✦ N2 > N1 is needed in order to obtain
Stimulated emission
amplification. This condition is called
“population inversion”
✦ We need to consider probabilities of the
optical transitions, density of available
optical modes, and level occupancy 4
Electronic level occupancy at equilibrium
熱平衡状態のエネルギー準位の占有

For energy levels in isolated atoms (ensembles of non-interacting systems)

Boltzman distribution 分布

N1 > N2 > N3

Under thermodynamic equilibrium conditions number of occupied higher energy

levels will be always lower than that of the lower energy levels

In interacting systems (e.g., semiconductors), probability of level occupancy is

described by Fermi-Dirac function.

5
Modes of a 3D cavity
Atoms can exchange energy with photons only at the allowed
mode frequencies. We need to estimate number of available
frequencies
Boundary condition 限界条件
integer # of half-wavelengths fit
between the cavity walls, E=0 at the walls

dz
dy
dx
resonance frequencies of the cavity

frequency spacings
6
 Density of EM modes
number of resonator modes [0…k]
part of total
volume
possible
volume of
polarizations
sphere

volume per
k-point

Density of modes M(ν)

7
 Spontaneous emission 自然放出
Emission into any direction (e.g., Probability density 選移率密度[s-1]of
luminescence), if there is a suitable
optical mode spontaneous emission into the given mode

Level 2 (excited state)

EM
mode
V - cavity volume
σ(ν) - transition cross-section [cm2]
Level 1 (ground state)
遷移断面

Oscillator
line shape function
strength
振動子強度

full width at half maximum

FWHM

line shape is often described by a Lorentz function (similar as in the harmonic oscillator model)
8
Spontaneous lifetime 自然遷移寿命
Spontaneous emission probability density

direction averaged value

N - population of
excited level 2 is enhanced by availability of EM modes
<— avg. osc.
strength

Spontaneous emission into any mode

- spontaneous lifetime 寿命

Oscillator strength can be estimated from

the measured N(t) dependence 9
Lifetime broadening 寿命による広がり
Limited lifetime of the excited state decaying oscillating
envelope carrier
t=0 t=t1 N2
E2 time N2(0)

radiative
non-radiative τ-lifetime
E1

Fourier transform of time-dependent Lorentz function

signal

can estimate
lifetime from
spectrum

10
Inhomogeneous broadening
Homogeneous broadening: all atoms in the medium have identical
lineshapes

• Lifetime broadening 2 2 2
• Collisional broadening 1 1 1

Inhomogeneous broadening: different atoms in the medium have

different lineshapes. The total lineshape is the average over all atoms
in the medium

homogeneous inhomogeneous
2
2 2
1 1 1

11
Doppler broadening
Frequency shift due
to atom moving with
velocity v:

homogeneous lineshape velocity distribution

Gaussian inhomogeneous lineshape

Gaussian distribution in
ideal gas: hom << inh

Gaussian inhomogeneous line shape is common also in solids 12

Absorption 吸収
Probability density
per one photon
EM (same as for spontaneous emission)
mode

if there are n photons in the cavity mode

Probability
I ~ nphot
Beer-Lambert law can be obtained from
simple rate equation for number of
photons N in the system

Homogeneous/inhomogeneous absorption line shape is similar to that

of spontaneous emission 13
Stimulated emission 誘導放
Probability density of stimulated emission 
(per photon in a cavity mode),
same as for absorption process
EM EM
mode mode

V - cavity volume
σ(ν) - transition cross-section [cm2]
• Two photons of the same
frequency, phase, polarization, and
propagation direction are obtained
Probability of stimulated emission
from a single photon;
• Probability increases with photon
number (~intensity);
• Possibility to obtain optical
n - number of photons in a cavity mode
amplification (=negative
to which emission occurs
absorption)
14
Optical amplification (gain) 増幅
Stimulated emission results in two photons with identical frequency
and phase. It enables optical gain (negative absorption).

Prerequisites
✦Gain medium
✦Energy supply to the gain medium from external source
✦A method to create population inversion (occupancy of upper
levels higher than that of the ground state)
15
Building an optical amplifier(not laser) 
1. Gain medium is prepared 2. Weak beam (signal) is passed
by excitation (pumping) through the gain medium
N2-1
N2
N2 >N1
N1
N1+1
pumping

input output

16
 Optical transition rates
ind. absorption stim. emission ind. emission
E2, N2
B12 A21 B21
energy/(Hz cm3)
⇢(⌫)

E1, N1
A21- probability of spontaneous emission transition
Einstein
B12- probability of induced absorption transition
coefficients B21- probability of induced emission transition

Emission rate:
; Absorption rate:
[cm-3s-1]
steady-state conditions
Number of emission transitions = Number of absorption transitions
17
 Rate equation
Under steady-state conditions:

we obtain: compare with Planck formula

We can see some similarities between the two expressions.

Assuming Boltzmann distribution of level population
we can express , and the two equations become identical
when Einstein
and
relations

Spontaneous emission rate is always

much faster than induced emission
rate (bad news for amplification) 18
Monochromatic and broadband light*
photon

photon
spectral
energy atomic transition
density
ρ(ν)

For monochromatic light, probability of induced (absorption/emission)

transition is
Wi ⌘ Pab = Pst = (⌫)
- photon flux density [photons/cm2s]
19
Broadband light in a cavity*
Average # of photons in the probability density of transition
frequency range ⌫ . . . ⌫ + d⌫ by single photon

spectral energy
density
Z Z 1
1
⇢(⌫)V h c i ⇢(⌫0 ) ⇢(⌫0 )
Wi = (⌫) d⌫ = c (⌫)d⌫ = cS
0 h⌫ V h⌫0 0 h⌫0

3 3
n avg # of
Wi = ⇢(⌫0 ) = n= ⇢(⌫0 ) photons/mode
8⇡htsp tsp 8⇡h
Spont. Psp = A Einstein coefficients A, B
3
1
Wi = B⇢(⌫0 ) A= B=
Induced tsp 8⇡htsp

This result is similar to that obtained before for narrowband light 20

Absorption and gain
Photon density N:
1. increases due to induced emission
2. decreases due to induced absorption
3. is nearly independent of spontaneous emission
2 1

Photon density is related to energy density

Intensity is also related to the energy density
Intensity of light propagating in absorbing/gain medium can be expressed as

From Beer-Lambert
(absorption) law 21
Continued
Using expression for dN/dt obtained from rate equation (first equation on
the previous page) we obtain

Which relates absorption coefficient (macroscopic parameter) with

microscopic parameters of the two-level system

Because of Boltzmann population distribution

In thermodynamic equilibrium conditions N2 < N1 and α > 0, which means
dominant role of absorption. In order to obtain gain, we must create
conditions where N2 > N1 (population inversion, negative absorption α < 0).
The gain(amplification) coefficient is often denoted by γ = -α

We must violate the Boltzman distribution to enable gain 22

 Obtaining population inversion反転分布
excited state is de-populated —> available for pumping
Fast non-optical relaxation to
E2 level 4 creates large population
2
level 4 is highly
Optical excitation populated
(pumping) to level 2
E4
stimulated emission
(other than levels population
4 and 3 responsible 1 inversion 3 is dominant
—> gain
for gain
E3 level 3 is almost
empty

Fast non-optical relaxation to

E1 4 level 1 clears out level 3
ground state is highly populated —> available for pumping 23
Pumping schemes
Isolated two-level system N2 ≤ N1
Population inversion is impossible.
2 No amplification.
relaxation
pumping rate
time Rate equations
1/Wi τ12
1

Non-isolated two-level system

From now, N denotes population
difference N=N2-N1
decay
p ing st.em
um abs Population
p
difference
g decay
p in
p um

Population difference in the absence of amplifier radiation 24

Four-level pumping
Purpose: to increase the difference N = N2-N1

pump
W

Population difference Population difference in the Characteristic

absence of amplifier radiation relaxation time

N0 and τs saturate as the pumping transition probability W increases 25

 Three-level pumping
Purpose: to increase the difference N = N2-N1

pump
W

ground state <=> lower level for stimulated emission;

naturally more populated than in a 4-level system;
needs harder pumping to create inversion

Population difference Population difference in the Characteristic

absence of amplifier radiation relaxation time
atomic density

N0 and τs saturate as the pumping transition probability W increases.

Three-level pumping scheme is less efficient than four-level scheme 26
Gain and attenuation
external “pump” excitation provides population inversion

coordinate z

amplifier gain N= N2-N1

photon flux variation with coordinate z
coefficient

amplification and attenuation is included, depending on the sign of N

amplifier gain
27
Gain bandwidth and dispersion

For comparison:
absorption coefficient faster φ > 0
and refractive index from
Lorentz oscillator model slower, φ< 0

28
Gain saturation N0 - population
difference without
amplifier radiation
N=N2-N1
N0
Wi = (⌫) N=
1 + ⌧s W i

saturation photon-flux density

saturated gain

29
Practical amplifiers
Electrically-pumped semiconductor amplifier

Optically pumped fiber amplifier

30
Important points

❖ Level occupancy and population inversion

❖ Line broadening mechanisms
❖ Main requirements for achievement of optical gain
❖ Einstein relations and Einstein coefficients
❖ Role of population inversion for optical gain
❖ Three- and Four-level pumping schemes, their main
differences
❖ Gain dispersion and saturation