Ambo University, Institute of Technology

Applied Modern Physics

Lecture 05 : Atomic Structure

Miressa M. Electrical and Computer
Engineering Department
Fundamentals of matter

• Atom is a basic unit of matter - “invisible” in Greek, postulated

more than 2000 years ago
• Structure of matter: atoms
• Structure of atoms: electrons, protons, neutrons
• Experimental evidence

E. Rutherford
nucleus = 10 m

1 ̇ ̈ (= 100,00 )

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Atoms

• Early evidence for existence of atoms:

– chemical reactions (Dalton 1800):
C + O ---> CO, 2H + O ---> H O
– Brownian motion (Brown 1827, Einstein 1905,
Perrin 1909)

– kinetic theory of gases: =

• (Avagadro number = 6 10 = 1 mole)

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Atoms

• Atoms form molecules:

– H O, O , C H O , DNA … ,

• Atoms form solids, liquids, gases:

• Atoms form crystals

• …and much, much more

– magnets, superconductors, superfluids, liquid crystals, rubber,

colloids, glasses, conductors, insulators,…

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J. J. Thomson experiment

• discovery of the electron (1897)

• measured e/m ratio:

• = v → = v/
–v = / from balance of electric and magnetic forces (eE = evB)
– (and inadvertently invents the cathode ray (TV) tube)
• Plum pudding atomic model.
• J. J. Thomson suggested that the number of
electrons in an atom was about the same as the
atomic number.
• Thomson proposed that all charges were equally
distributed over the volume of the atom with
radius R.
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The Rutherford atom

• In 1911 Ernest Rutherford performed some experiments shooting

Experimental setup
alpha particles into a gold foil.

• The α particles were emitted by

radon gas in the tube T, and were
collimated by the narrow channel
D. The nearly parallel beam of α
particles then passes through a
thin gold foil F and the scattered
α particles produced faint light
flashes on a phosphorous screen
S, which were observed through a
microscope.

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The Rutherford atom

• Based on these results he could set up a model of the atom in which

the atom consisted of a heavy nucleus with a positive charge

surrounded by negatively charged electrons like a small solar system.
Also this model was in conflict with the laws of classical physics.

• nucleus = 10 m

• Rutherford's alpha scattering experiments were the first experiments

in which individual particles were systematically scattered and
detected. This is now the standard operating procedure of particle
physics.

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The Rutherford atom

• Rutherford’s model suffered from many disadvantages.

– it did not say anything about the location or movement of the
electrons or the nucleus.
– It gave no explanation on how these electrons could remain in
equilibrium about the nucleus.
– If, as in planetary motion, electrons were assumed to be moving in
an orbit, then classical theory would lead to loss of energy through
radiation which would cause the electron to spiral towards the
nucleus and finally fall into the nucleus.
– Another defect of the Rutherford model was that it did not
explain the observed spectroscopic effect.

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Atomic spectra

• Isaac Newton showed that white light from the sun is composed of
a continuum of colors (frequencies). Newton introduced the term
“spectrum" to describe this phenomenon.
• Joseph Fraunhofer made the first observation of the discrete
nature of emission and absorption from atomic systems.

• Gustav Kirchoff and Robert Bunsen studied the origin of the solar
spectral lines. Solar spectral lines were due to absorption of light by
particular atomic species in the solar atmosphere.
• Each atom and molecule has its own characteristic spectrum.

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Absorption and emission spectra

Spectrum from a white

light source

Emission spectrum from

a hot atomic gas vapor

Absorption spectrum observed

when white light is passed
through a cold atomic gas
To understand the emission spectral lines, atomic model was required!

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Atomic spectra

• Atoms only absorb or emit light at certain discrete wavelengths λi.

These specific wavelengths that are characteristic of each chemical
element, are called the absorption or emission spectra of the
atom.
• These spectra are like a fingerprint of the atom, since every atomic
species can be unambiguously recognized by its spectrum.
• Experiments from emission and absorption spectra brought
about the following results:
(1) Each wavelength observed in an absorption spectrum also
appears in the emission spectrum of the same kind of atoms if the
atoms have been excited into the emitting state by absorption of
light or by collisional excitation.

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Atomic spectra

(2) The absorption and emission spectra are characteristic for

specific atoms. They allow the unambiguous determination of the
chemical element corresponding to these spectra. The spectral
analysis therefore yields the composition of chemical elements in
sample. This is particularly important in astrophysics where the
spectrum of the starlight gives information on the number and the
composition of chemical elements in the atmosphere of the star.
(3) The spectral lines are not completely narrow, even if the
spectral resolution of the spectrograph is extremely high. This
means that the atoms do not emit strictly monochromatic radiation
but show an intensity distribution I(λ ) around each wavelength λ
with a finite halfwidth ∆λ.

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Atomic spectra

• The wavelengths of these spectral lines could be determined with

great precision, and much effort went into finding and interpreting
regularities in the spectra.
• Johann Balmer, found that the lines in the visible and near
ultraviolet spectrum of hydrogen could be represented by the
empirical formula
1 1 1
= −
4
where n is a variable integer that takes on the values n = 3, 4, 5,…
The constant = 109,737 is the Rydberg constant.
This formula is called Balmer series.

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• Later on Theodore Lyman (1874–1954) and Friedrich Paschen
(1865–1947) found further series in the emission and absorption
spectrum of the Hydrogen atom, which could all be described by
the Balmer formula
1 1 1
= −

• but with = 1 (Lyman series) or = 3 (Paschen series)

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Energy levels in hydrogen atom

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Challenge with Rutherford’s planetary model of the atom.

• A major challenge for the classical treatment of the Rutherford’s

planetary model of the atom:
– The Sun and planets of the solar system are electrically neutral.
– However, atomic nucleus and orbiting electrons carry net charges.
Oscillating charges will emit electromagnetic radiation, and thus
carry away mechanical energy.
– Then according to the classical theory of the atom the electron
will spiral into the nucleus in only a matter of microseconds, all
the while continually emitting radiation.

• Clearly these are not observed - atoms are stable, do not

continually emit radiation, and do not emit a continuous
spectrum of radiation.
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The Bohr atom

• In 1911, Bohr began to put forth the idea that since light could no
long be treated as continuously propagating waves, but instead as
discrete energy packets (as articulated by Planck and Einstein), why
should the classical Newtonian mechanics on which Thomson’s
model was based hold true?
• It seemed to Bohr that the atomic model should be modified in a
similar way.
• If electromagnetic energy is quantized, i.e. restricted to take on only
integer values of hν, where ν is the frequency of light, then it seemed
reasonable that the mechanical energy associated with the energy of
atomic electrons is also quantized.

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The Bohr atom
Z = number of protons

• In 1913, the Danish physicist Niels H. D. Bohr proposed a model of

the hydrogen atom and was remarkably successful in predicting the
observed spectrum of hydrogen.
• In Bohr’s atomic model the electron (mass , charge -e) and the
nucleus (mass , charge +Ze) both move on circles with radius
or , respectively, around their center of mass.
• This movement of two bodies can be described in the center of
mass system by he movement of a single particle with reduced mass
m=( )/( + ) ≈ in the Coulomb potential
around the center r = 0, where r is the distance between electron and
nucleus.
• The balance between Coulomb force and centripetal force yields the
equation,

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The Bohr atom

v 1
=
4
• The laws of electrodynamics predict that such an accelerating charge
will radiate light of frequency f equal to that of the periodic motion,
which in this case is the frequency of revolution. Thus, classically,
/
v 1 1 1 1 1
= = = ~ /
2 4 2 4 4

• The total energy of the electron is the sum of the kinetic and the
potential energies:
1 1
= v + −
2 4

• Using equation = we can get the total energy as

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The Bohr atom

1 1 1 1
= − =− ~−
4 2 4 4 2

• Thus, classical physics predicts that, as energy is lost to radiation,

the electron’s orbit will become smaller and smaller while the
frequency of the emitted radiation will become higher and higher,
further increasing the rate at which energy is lost and ending when
the electron reaches the nucleus.
• This model predicts that the atom will radiate a continuous spectrum
(since the frequency of revolution changes continuously as the
electron spirals in) and will collapse after a very short time, a result
that fortunately does not occur.

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The Bohr atom

• Unless excited by some external means, atoms do not radiate at all,

and when excited atoms do radiate, a line spectrum is emitted, not a
continuous one.
• Bohr “solved” these formidable difficulties with two decidedly
nonclassical postulates.
– Electrons could move in certain orbits without radiating. He called
these orbits stationary states.
– The atom radiates when the electron makes a transition from one
stationary state to another and that the frequency f of the emitted
radiation is not the frequency of motion in either stable orbit but is
related to the energies of the orbits by Planck’s theory
• hf = E − E

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The Bohr atom

• where ℎ is Planck’s constant and and are the energies of the

initial and final states.

• In order to determine the energies of the allowed, nonradiating

orbits, Bohr made a third assumption, now known as the
correspondence principle, which had profound implications:
– In the limit of large orbits and large energies, quantum calculations
must agree with classical calculations.

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The Bohr atom

• Bohr proposed that the orbiting electron could only exist in certain
special states of motion - called stationary states, in which no
electromagnetic radiation was emitted. In these states, the angular
momentum of the electron L takes on integer values of Planck’s
constant divided by 2π, denoted by ℏ = ℎ/2 .
• In these stationary states, the electron angular momentum can take
on values ℏ, 2ℏ, 3ℏ, … but never non-integer values.
• This is known as quantization of angular momentum, and was one of
Bohr’s key hypotheses.
• Note that this differs from Planck’s hypothesis of energy
quantization, but as we will see it does lead to quantization of
energy.

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• For circular orbits, the position vector of the electron r is always
perpendicular to its linear momentum p.
• The angular momentum = has magnitude L = rp = vr in
this case.
• Thus Bohr’s postulate of quantized angular momentum is equivalent
to
= v = ℏ n (quantum number)
• This can be solved to give the velocity = 1, 2, 3, …
ℏ
v=
r
• The speed of the orbiting electron is given by
/
v= where 1
=
4

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• Using these equations for the circular orbits we get

ℏ ℏ /
= =
v

• Squaring r and cancelling common quantities yields

ℏ
= =

ℏ
Where = = 0.529 ̇ = 0.0529 .

• Here is known as the Bohr radius.

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• This is a significant and unexpected result when compared to the
classical behavior.
• A satellite in a circular orbit about the earth can be placed at any
altitude (radius) by providing an appropriate tangential velocity.
• However, electrons are only allowed to occupy orbits with certain
discrete radii. Furthermore, this places constraints to the allowed
velocity, momentum, and total energy of the electron in the atom.
• Total energy of the electron
1 1 1 1
= v + − = − =−
2 4 2 4 2

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• Thus,
=− =− =− =−
2 2 ℏ 2 ℏ
• This shows the energy of the electron is also quantized; i.e., the
stationary states correspond to specific values of the total energy.
• This means that energies and that appear in the frequency
condition of Bohr’s postulate must be from the allowed set .
ℎ = − =− − −

• From this
1 1
= −
ℎ

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• The possible values of the energy of the hydrogen atom(Z=1)
predicted by Bohr’s model are given by
= =−
2 ℏ
where
= = 13.6
2ℏ
is the magnitude of with = 1. (= − ) is called the ground
state.
Bohr’s postulates can be summarized as follows:
• Quantized angular momentum: = v = ℏ.
• Radiation is only emitted when an atom makes transitions
between stationary states: = − .

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 Laser
Light

Amplification by

Stimulated

Emission of Mix of Fundamental Physics

Radiation (Quantum Mechanics) and

Applied Physics (Engineering)

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Laser
Laser: merging of several discoveries

30
Applications of lasers

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Applications of lasers

32
Applications of lasers

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Light-matter interaction

• Light & light-matter interaction where quantum physics matters!

• What’s the “Matter”:
– Atom
– Electron
– Solid …
– Vacuum (really?)

Light (in) Light (out)

Emission,
Laser
Photoelectric/
photodetection Y (e.g., electron)

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Light-matter interaction

• Quantization of atomic energy levels

– Electrons of an atom have
discrete energy levels

– Thus the global energy of the

atom (or molecule, ion,…) can have only
certain discrete values as well
– Quantization of energy

– The wavelength depends on the energetic

difference between the 2 levels is
∆ =ℎ =ℎ /

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Light-matter interaction

• The term Laser stands for Light Amplification by Stimulated

Emission of Radiation.
• Laser is a device that stimulates atoms or molecules to emit light at
particular wavelengths and amplifies that light, typically producing a
very narrow beam of radiation. The emission generally covers an
extremely limited range of visible, infrared, or ultraviolet wavelengths.
• The first theoretical foundation of LASER was given by Einstein in
1917 using Plank’s law of radiation that was based on Einstein
coefficients for absorption, spontaneous emission and stimulated
emission of electromagnetic radiation.
• Light interacts with semiconductor materials with these three
mechanisms i.e., absorption, stimulated emission and spontaneous
emission.

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Absorption

• When a photon with certain energy is incident on an electron in a

semiconductor at the ground state(lower energy level E1), the
electron absorbs the energy and shifts to the higher energy level E2.
• The energy now acquired by the electron is Ee = hf = E2 – E1
(Planck’s law)
E2
E2

E1
E1 Initial state
E2

E1
Excited electron
final state
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Spontaneous emission

• E2 is unstable and the excited electron(s) will “spontaneously” return

back to the lower energy level E1.
• As they fall, they give up the energy acquired during absorption in
the form of radiation, which is known as the spontaneous emission
process.
• It gives rise to incoherent light as the photon emissions are
independent of each other, that is they are emitted at different times
from different positions within the material and in random
directions.
E2
E2
photon
E1
E1
Initial state

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Stimulated emission

• Before the occurrence of this spontaneous emission process, if

external stimulation (photon) is used to strike the excited atom then,
it will stimulate the electron to return to the lower state level.
• By doing so it releases its energy as a new photon. The generated
photons are in phase and have the same frequency, the same
direction and the same energy as the incident photon.
• The result is generation of a coherent light composed of two or
more photons.

E2 E2
Coherent light
E1 E1

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Rates of emission and absorption

• Einstein in 1917
– The Quantum Theory of Radiation
– Stimulated emission concept
• Einstein’s analysis
– Let’s consider transitions between two molecular states with
energies E1 and E2 (where E1 < E2 ).

• = number of molecules in
E2 N2 the higher state
• = number of molecules in
the lower state
E1 N1 • = ( − ) = ℎ photon
energy

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Rates of emission and absorption

• Rates at which emissions and absorption occur within a particular

material are described by Einstein relations.
• If stimulated emission occurs, the rate of those transitions must be
proportional to the number of molecules in the higher state and
the energy density of the incoming radiation .
• The rate at which stimulated transitions are made from to is
= = −

• Similarly, the probability that a molecule at will absorb a photon

can be expressed as
= =

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• The rate of spontaneous emission is
= =−

where
– , the populations of the upper and lower energy states,
respectively
– is the spectral density of photon energy
– A, , are Einstein coefficients
• 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
−
= exp(− ) = exp( − ⁄ )

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• At thermal equilibrium condition in a medium:
– Rate of emission = Rate of absorption
– Since the emission is of two types: Spontaneous and Stimulated,
the condition is rewritten as:
– Rate of spontaneous emission + rate of stimulated emission = Rate
of absorption
+ =
– In thermal equilibrium each of the are proportional to their
respective Boltzmann factors / . Thus,
• + =
/ /
• + =

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 / = / .
• In the classical limit → ∞. Then
• At high temperatures, thermal energy in the system increases. Thus
the energy density ( ) becomes very large, so the A term becomes
insignificant.
• Therefore in the classical limit
≈ ≡
• That is, the probability of stimulated emission is approximately equal
to the probability of absorption.
– Means if the absorption can occur, then we should also expect that
stimulated emission will occur.

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• The energy density can be given as

= ( )/
= /
− −
• Since ≈ ≡
1
= / −1
• Recall that the Planck’s radiation law is given by

8 ℎ 1
=
−1

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• Planck’s law of the spectral energy density of electromagnetic
radiation per volume ( , ) in the spectral range + (or ( , ) in
the spectral range + ) is given by
8 ℎ 1
, =
−1
8 ℎ 1
, =
−1
• In the SI system, multiplying by a constant factor /4 is required* to
change the energy density [ , , energy per unit volume per unit
wavelength inside the cavity ] to a spectral intensity [ ℓ( , ), power
per unit area per unit wavelength for radiation emitted from the
cavity]:
2 ℎ 1
ℓ , =
−1
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• Comparing these two equations for will result in
8 ℎ
=

• This shows that the stimulated emission probability coefficient B is

proportional to the spontaneous emission probability coefficient A
in equilibrium.
• Since we want stimulated emission to dominate in lasers, we would
compare its rate with respect to the other two processes.
• Rate of stimulated emission >> rate of absorption
– This condition is known as population inversion!
• Rate of stimulated emission >> rate of spontaneous emission
– External energy must be supplied to force the system away from
equilibrium
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Main components of laser

Pumping
• Active/gain medium

– Solid, liquid, gas

– Three-level, four-level, or more

• Optical feedback (optical cavities)

Mirror Mirror
• Pumping mechanism

– Optical pumping

– Electrical pumping: Current injection in semiconductors

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Amplification

• Stimulated emission is the fundamental physical process in the

operation of the laser.

• All the photons are identical with the original photon

– Same phase, same polarization, same direction, same amplitude

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Optical resonator

• Optical resonator reflects the amplified radiation back and forth into
the gain medium. This resonator is often also called laser cavity. An
optical cavity of a laser has mirrors located at each end of the laser
gain medium. To obtain a laser oscillator, we need to place the
amplifying medium into a cavity.
• Most of the energy gets reflected from both the mirrors, passes
through the active medium, and continues to get amplified until
steady state level of oscillation is reached. After attaining this stage,
amplification of wave amplitude within the cavity dies away and
extra energy produced by stimulated emission exits as laser output.
• The gain coefficient inside the cavity should be greater than the
threshold gain coefficient in order to start and maintain laser
oscillation inside the cavity.

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Population inversion

• Lasers require an external source of power to provide the energy

required to augment the input signal. Pumping excites the electrons
in the atoms, causing them to move from lower to higher atomic
energy levels.
• In order the achieve amplification, the pump must provide
population inversion.

Atomic system at rest Population inversion

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Population inversion

• Population inversion can be achieved by three-level or four-level

systems.
• 3-level system

• Continuous regime hard to obtain

(level 1 refills as soon as the laser is
on !)
• Population inversion is possible!
But hard!
• We need to fill level 2  OK.
• We need to empty level 1 
• Example: Ruby laser
difficult !! (fundamental level)

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Population inversion

• 4-level system
• Population Inversion easy!
– We need to fill level 2  OK
– We need to empty level 1  OK
(rapidly empty by transition to
level 0)
• Most of actual lasers work on
this 4-level scheme
• Example: solid state
– Wide variety of wavelengths
lasers with neodymium
ions
– Nd:YAG 1064 nm

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Principle of lasing

• Basically, every laser system essentially has an active (gain medium),

placed between a pair of optically parallel and highly reflecting
mirrors with one of them partially transmitting, and an energy source
to pump active medium.

• The gain media have the property to amplify the amplitude of the
light wave passing through it by stimulated emission, while pumping
may be electrical or optical.
• The gain medium used to place between pair of mirrors in such a
way that light oscillating between mirrors passes every time through
the gain medium and after attaining considerable amplification
emits through the transmitting mirror.

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Principle of lasing

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Laser modes

• The output of laser beam actually consists of a number of closely

spaced spectral lines of different frequencies in a broad frequency
range. The discrete spectral components are termed as laser modes,
and coverage range is the line-width of the atomic transition
responsible for the laser output.
• Laser modes are categorized into axial and transverse modes.

Longitudinal modes Longitudinal

of the cavity Gain modes

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Laser modes

• Axial mode: is related to the variation of the electric field with ,

where the -axis lies along the cavity axis.
• Let
2
= 2 = 2 = 2

be the phase change in the laser wave after a round-trip in the

cavity.
• In order to sustain laser oscillation inside the cavity, the phase
change should be an integral multiple of 2 , that is 2 = 2 .
In terms of frequency, this expression transforms to ν = m .
• Frequencies that can oscillate in the cavity are called Longitudinal
modes. Therefore, separation between two adjacent m and m + 1
modes is given by Δν = =

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Laser modes

• Transverse modes: Unlike the plane waves propagating along the

axis of the cavity in axial modes, there are some other waves
traveling out of the axis that are not able to repeat their own path
termed as transverse electromagnetic ( ) modes.
• These modes can be practically seen in the form of pattern when the
laser beam falls on any surface.
• These modes are assigned by two
integers p and q in the form of
, where p and q are the
number of minima in the horizontal
and vertical directions, respectively,
in the pattern of the laser beam.
• Fundamental mode:

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Laser types

• Based on their gain medium, lasers are classified into:

– Semiconductor Lasers (Laser Diodes) [GaN, InGaN, VCSELs]
– Solid-State Lasers [Ruby, Nd:YAG, Ti:sapphire …]
– Fiber Lasers (Erbium doped)
– Liquid Lasers (Dye Lasers)
– Gas Lasers
– Free electron laser, …
• Additionally, these five types of lasers can be divided into
subcategories based on their mode of operation: continuous wave
lasers and pulsed lasers.
• Furthermore, there are also multiple types of pulsed lasers.

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Laser light properties

• To greater or lesser extent laser light is:

– very intense
– highly collimated
– highly coherent
– highly polarized
– can be continuous or very short pulses
• The intensity depends on the pumping power and the efficiency of
the LASER mechanism.
• There is always a tradeoff between the intensity and the coherence
and/or collimation.

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Laser light properties

• Coherent light waves:

– same frequency
– same phase

• Incoherent light waves:

– different frequencies
– different phase

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Laser diode output power

• Typical output optical power vs. diode current characteristics and

the corresponding output spectrum of a laser diode.

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Presentation topics

1. Wave-particle duality

2. Atomic structure
• 2 persons per title
3. Lasers

4. Quantum mechanics
• 20 minutes for each

5. Semiconductors group

6. Special relativity

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Summary

• Atomic models

– J. J Thomson atom

– Rutherford atom

– Bohr atom

• Lasers

– Operating principle

– Laser components

– Types

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