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Lecture 2.
Basic Lasers Principles
 The
phenomenon of stimulated emission
 Einstein coefficients relationship with the
construction of the particle emitting
 The shape of the emission lines
 Broadening of the emission lines
 Population inversion as a precondition
for amplifying of radiation

In 1915, Neils Bohr proposed a model of the atom that
explained a wide variety of phenomena that were puzzling
scientists in the late 19th century. This simple model became
the basis for the field of quantum mechanics and, although
not fully accurate by today’s understanding, still is useful for
demonstrating laser principles.

Under the right circumstances an electron can go from its
ground state (lowest-energy orbit) to a higher (excited)
state, or it can decay from a higher state to a lower state, but
it cannot remain between these states. The allowed energy
states are called “quantum” states and are referred to by the
principal “quantum numbers” 1, 2, 3, etc. The quantum states
are represented by an energy-level diagram.

For an electron to jump to a higher quantum state, the
atom must receive energy from the outside world.
This can happen through a variety of mechanisms
such as inelastic or semielastic collisions with other
atoms and absorption of energy in the form of
electromagnetic radiation (e.g., light).
 When
an electron drops from a higher state
to a lower state, the atom must give off
energy, either as kinetic activity
(nonradiative transitions) or as
electromagnetic radiation (radiative
transitions).

Each photon has an intrinsic energy determined by
the equation:
E=hc/λ
where λ is the wavelength of the light, c is the speed
of light in a vacuum and h is Planck’s constant


Lets consider two energy
levels of an atom or
molecule and interacting
with the system of such
particles beam of photons
with energies
corresponding to the
difference between these
two levels.
Photons are absorbed and
transfer particles into an
excited state
 Probability
dW of particle transitions
between states 1 and 2 is described by
the coefficients introduced by Einstein in
the following way :
dW12 = B12 ρν dt
where ρν means the density of radiation
in the system

In general, when an electron
is in an excited energy state,
it must eventually decay to a
lower level, giving off a
photon of radiation. This
event is called “spontaneous
emission,” and the photon is
emitted in a random direction
and a random phase. The
average time it takes for the
electron to decay is called the
time constant for spontaneous
emission, and is represented
by τ.

When an electron is in energy
state E2, and its decay path is to
E1, but, before it has a chance to
spontaneously decay, a photon
happens to pass by whose
energy is approximately E2 - E1,
there is a probability that the
passing photon will cause the
electron to decay in such a
manner that a photon is emitted
at exactly the same wavelength,
in exactly the same direction,
and with exactly the same phase
as the passing photon. This
process is called “stimulated
emission.”
 Excited
particle may return to a base level
spontaneously emitting a photon
dWs21= A21 dt

The beam of photons may also interact with
the excited particles and the particle
returns to the ground state by emitting a
photon identical in all respects with that
which forced emission
dW21 = B21 ρν dt

Balance of transitions increasing and decreasing the
cast on level 2 is as follows :
n1 dW12 = n2 dWs21 + n2 dW21
n1 B12 ρν = n2 A21 + n2 B21 ρν

Accordingly to Boltzman distribution the cast per one
level is given by:
n2 = (g1/g2) n1 exp{-(E2-E1) / kT}
where g1 and g2 means the degree of degeneracy of
levels
 Thus:
n1 B12 ρν = (g1/g2) n1 exp{-(E2-E1) / kT} (A21 +
B21 ρν)
 After
transformations we obtain the
expression for the energy density :
A 21
1
ρν 
B21 B12g1 exp E  E /kT
2
1
B21g 2

Comparing to the Planck’s formula:


ρ  8ππ /c hνν 21
2
3
we get the following equations:
B12g1 = B21g2
A21 = (8πν2 / c3) hνB21

Now consider the group of atoms
in exactly the same excited state,
and most are effectively within the
stimulation range of a passing
photon. We also will assume that t
is very long, and that the
probability for stimulated
emission is 100 percent. The
incoming (stimulating) photon
interacts with the first atom,
causing stimulated emission of a
coherent photon; these two
photons then interact with the
next two atoms in line, and the
result is four coherent photons. At
the end of the process, we will
have eleven coherent photons, all
with identical phases and all
traveling in the same direction.


Note that the energy to put these
atoms in excited states is
provided externally by some
energy source which is usually
referred to as the “pump” source.
Of course, in any real population
of atoms, the probability for
stimulated emission is quite small.
Furthermore, not all of the atoms
are usually in an excited state; in
fact, the opposite is true.

Pumping
medium (electrical, optical etc) for population
inversion


Define Ni , the population of level i
Ni(t)= the # of electrons, per unit of volume,
occupying energy level i at time t

Boltzmann’s principle, a
fundamental law of
thermodynamics, states that, when
a collection of atoms is at thermal
equilibrium, the relative
population of any two energy
levels is given by:
where N2 and N1 are the
populations of the upper and lower
energy states, respectively, T is the
equilibrium temperature, and k is
Boltzmann’s constant.


Substituting hν for the E2 – E1 we obtain:
For a normal population of atoms, there
will always be more atoms in the lower
energy levels than in the upper ones.
Since the probability for an individual
atom to absorb a photon is the same as
the probability for an excited atom to
emit a photon via stimulated emission,
the collection of real atoms will be a net
absorber, not a net emitter, and
amplification will not be possible.
Achieving population inversion in a two-level atom is not very
practical. Such a task would require a very strong pumping transition that
would send any decaying atom back into its excited state.
 Consequently, to
make
a laser, we have to
create a “population
inversion.”
 For
photon flux Φ

The electron is pumped (excited)
into an upper level E4 by some
mechanism (for example, a collision
with another atom or absorption of
high-energy radiation). It then
decays to E3, then to E2, and finally to
the ground state E1. Let us assume
that the time it takes to decay from
E4 to E3 is much longer than the time
it takes to decay from E2 to E1. In a
large population of such atoms, at
equilibrium and with a continuous
pumping process, a population
inversion will occur between the E3
and E2 energy states, and a photon
entering the population will be
amplified coherently.

Although with a population
inversion we have the ability
to amplify a signal via
stimulated emission, the
overall single-pass gain is
quite small, and most of the
excited atoms in the
population emit
spontaneously and do not
contribute to the overall
output.
 We
use semi-classic approach:
electromagnetic radiation which a source
of disturbance will be decribed by
classics theory thus the atom with the aid
of quantum mechanics
 Solving Schröding’s equation we can find
expression for Einstein’s coefficients
 Within
the spectral lines of light emitted by
any body the light intensity changes you can
therefore speak of the power spectral
density.
 The primary cause of the spectral lines of
finite thickness is the Heisenberg
uncertainty principle:
E    
 Such
broadening of the line is always
present and is called the natural broadening
 Doppler
Broadening also affects the
width of the spectral line for a group of
light-emitting particles
 Broadening of the spectral lines can also
occur as a result of particle collisions
 Kryterium
pozwalające stwierdzić czy
dana linia emisyjna jest jednorodnie czy
niejednorodnie poszerzona: jeśli ma
miejsce zróżnicowanie warunków emisji
dla różnych cząstek to dają one różne
wkłady do całkowitego poszerzenia linii
widmowej i linia emisyjna jest
niejednorodna, w przeciwnym wypadku
uważamy linię za jednorodną
 Skończona
szerokość linii (nawet tylko
naturalnie poszerzonej) zmniejsza
prawdopodobieństwo emisji
wymuszonej.

To turn this system into a
laser, we need a positive
feedback mechanism that
will cause the majority of
the atoms in the population
to contribute to the
coherent output. This is the
resonator, a system of
mirrors that reflects
undesirable (off-axis)
photons out of the system
and reflects the desirable
(on-axis) photons back into
the excited population
where they can continue to
be amplified.

The lasing medium is
pumped continuously to
create a population
inversion at the lasing
wavelength. As the
excited atoms start to
decay, they emit photons
spontaneously in all
directions. Some of the
photons travel along the
axis of the lasing
medium, but most of the
photons are directed out
the sides.

The lasing medium is
pumped continuously to
create a population
inversion at the lasing
wavelength. As the
excited atoms start to
decay, they emit photons
spontaneously in all
directions. Some of the
photons travel along the
axis of the lasing
medium, but most of the
photons are directed out
the sides.
 In
the simplest case consists of two flat
resonator, fully reflecting mirrors, which
are arranged perfectly parallel to each
other at a distance
L = n λ/2 ,
where (n is an integer, λ is laser light
wavelength), it allows the creation of
standing waves inside resonator.
 Warunek
rezonansu spełniać również mogą
fale rozchodzące się pod określonymi
kątami Θ w stosunku do osi rezonatora: 2L =
n λ cosΘ, Takie zjawisko najłatwiej można
zaobserwować w rezonatorach o wklęsłych
zwierciadłach np. rezonatorze konfokalnym
(współogniskowym).
 It is therefore possible formation of resonant
vibration, which, due to misaligned
propagation direction is called the
transverse or non-axial mods
Laser resonator:
 Provides positive feedback (amplifier
becomes a generator)
 Force oscillations in the resonant
frequencies - standing waves (modes)
are formed
 Interferes with the emission line halfwidth (the better resonator-the narrower
line and the better monochromaticity)
 Interferes with the geometry of the beam
 Resonator’s
quality is defined as the the
ratio of the energy contained in the
resonator Wst to the energy P lost during
one period
 Finesse
determines the quality of the
resonator
 The
smaller the total loss of the resonator,
the higher the finesse and the narrower is
the resonance line
4R
F 
2
1  R 
 The
intensity of transmitted light
 K. Łukaszewski, Wstęp
do elektroniki
kwantowej i optyki nieliniowej, skrypt
WFTIMS, Łódź 1999.
 F. Kaczmarek, Wstęp do fizyki laserów,
PWN, Warszawa 1979