Laser Lectures 1

1

Laser

Fundamentals

Dr. Mohamed Al-Fadhali

3/15/2014 1 Dr. Mohamed Al- Fadhali

3/15/2014 Dr. Mohamed Fadhali 2

Recommended texts The lectures and notes should give you a good base from which to start your

study of the subject. However, you will need to do some further reading. The

following books are at about the right level, and contain sections on almost

everything that we will cover:

1. “Principles of Lasers,” Orazio Svelto, fourth edition, Plenum

Press.

2. “Lasers and Electro-Optics: Fundamentals and

Engineering,”Christopher Davies Cambridge University Press.

3. “Laser Fundamentals,” William Silfvast, Cambridge

University Press.

4. “Lasers,” Anthony Siegman, University Science Books.

2

LASER SPECTRUM

10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 10 102

LASERS

200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 10600

Ultraviolet Visible Near Infrared Far Infrared

Gamma Rays X-Rays Ultra- Visible Infrared Micro- Radar TV Radio

violet waves waves waves waves

Wavelength (m)

Wavelength (nm)

Nd:YAG

1064

GaAs

905

HeNe

633 Ar

488/515

CO2

10600

XeCl

308 KrF

248

2w

Nd:YAG

532

Retinal Hazard Region

ArF

193 Communication

Diode

1550

Ruby

694

Laser-Professionals.com

Alexandrite

755

3/15/2014 3 Dr. Mohamed Al-Fadhali

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Introduction (Brief history of laser)

The laser is perhaps the most important optical device developed in the past 50 years. Since its

arrival in the 1960s, rather quiet and unheralded outside the scientific community, it has

provided the stimulus to make optics one of the most rapidly growing fields in science and

technology The laser is essentially an optical amplifier. The word laser is an

acronym that stands for “light amplification by the stimulated emission of

radiation”.

The theoretical background of laser action as the basis for an optical amplifier was

made possible by Albert Einstein, as early as 1917, when he first predicted the

existence of a new irradiative process called “stimulated emission”. His

theoretical work, however, remained largely unexploited until 1954, when C.H.

Townes and Co-workers developed a microwave amplifier based on stimulated

emission radiation. It was called a maser


3

3/15/2014 Dr. Mohamed Al-Fadhali 5

Following the birth of the ruby and He-Ne lasers, others devices

followed in rapid succession, each with a different laser medium and a

different wavelength emission. For the greater part of the 1960s, the laser

was viewed by the world of industry and technology as scientific curiosity.

In 1960, T.H.Maiman built the first laser device (ruby laser) which

emitted deep red light at a wavelength of 694.3 nm.

A. Javan and associates developed the first gas laser (He-Ne laser),

which emitted light in both the infrared (at 1.15mm) and visible (at

632.8 nm) spectral regions..

3/15/2014 6 Dr. Mohamed Fadhali Dr. Mohamed Al-Fadhali

4



5


3/15/2014 Dr. Mohamed Al-Fadhali

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6

A laser consists of three parts: 1. a gain medium that can amplify light by means of the basic process of

stimulated emission;

2. a pump source, which creates a population inversion in the gain medium;

3. two mirrors that form a resonator or optical cavity in which light is trapped,

traveling back and forth between the mirrors.

For examples, a ruby laser consists of a ruby rod, a flash tube with a

cylindrical reflector of elliptical cross-section, and two mirrors; a He-Ne

laser consists of a plasma tube filled with He-Ne gases, electrical

excitation and two mirrors.

11 3/15/2014 Dr. Mohamed Fadhali Dr. Mohamed Al-Fadhali

Brewster Angle Gain region


Three key elements in a laser

•Pumping process prepares amplifying medium in suitable state

•Optical power increases on each pass through amplifying medium

•If gain exceeds loss, device will oscillate, generating a coherent output

7

The active media

An active medium is made of atoms or molecules in gas, liquid or solid

states. Atoms and molecules have quantized energy levels.

Electrons with higher orbits have higher energy.

13

Without pumping, atomic and

molecular systems have more

population in the lower energy

states than in the higher energy

states. 3/15/2014 Dr. Mohamed Fadhali Dr. Mohamed Al-Fadhali

Population Inversion

• It is necessary to have a mechanism where N2 > N1 This is called POPULATION INVERSION

• Population inversion can be created by introducing a so call metastable centre where electrons can piled up to achieve a situation where more N2 than N1

• The process of attaining a population inversion is called pumping and the objective is to obtain a non-thermal equilibrium.

• It is not possible to achieve population inversion with a 2-state system.(If the radiation flux is made very large the probability of stimulated emission and absorption can be made far exceed the rate of spontaneous emission. But in 2-state system, the best we can get is N1 = N2.

• To create population inversion, a 3-level or higher system is required.

• In The 3-level system, if pumped with radiation of energy E31 the electrons in level 3 relax to level 2 non- radiatively and then from E2 will jump to E1 to give out radiation.


8

Examples of Electrical and Optical pumping



(a) Direct current (dc) is often used to pump gas lasers. The current may

be passed either along the laser axis, to give a longitudinal discharge, or

transverse to it (The latter configuration is often used for high pressure pulsed lasers, such as

the transversely excited atmospheric (TEA) CO2 lasers).

(b) Radio frequency (RF) discharge currents are also used for pumping gas lasers.

(c) Flash lamps are effective for optically pumping ruby and rare-earth solid-state lasers.

(d) A semiconductor laser diode (or an array of light emitting diodes LEDs)

can be used to optically pump Nd3+:YAG or Er3+:silica fiber lasers.

For a laser to operate, pumping must be strong enough to create population inversion and above a

threshold level. The threshold level is a function of all loss factors, the most important one of

which is the useful output from the laser.

Description of last figure pumping methods


9


Lasers may be divided into two groups:

1. continuous wave (CW) or quasi-CW,

2. Pulsed (Q-switching – Mode-locking)

A CW laser exhibits a steady flow of coherent energy and its output power

undergoes little or no change with time. Many gas lasers, such as HeNe and Ar-

ion lasers, operate CW; several solid-state lasers, such as Nd3+ and Ti3+:Al2O3

lasers, are also often operated in CW mode.

In pulsed lasers, the output beam power changes with time so as to produce a

short optical pulse, usually in a repetitive way and with pulse duration usually

ranging from nanoseconds (1 ns = 10−9 s) to femtoseconds (1 fs = 10−15 s).

Typical examples of pulsed lasers are many solid-state and liquid lasers, such as

Nd:YAG, Ti:Al2O3 , and dye lasers.

Laser Resonators

A laser resonator consists of two spherical mirrors (they might be of any other shape).

The wavefronts of the laser beam must conform to mirror curvatures.

To keep beam profile stable over time, R1, R2 and the separation d between the

mirrors must be designed such that the optical field in any cross-sectional

plane perpendicular to the optical axis will repeat itself after every round trip.

Hence, the resonator controls the properties of the optical beam emerged from

the laser (e.g. beam divergence, beam radius, etc).

The plasma tube of gas laser is terminated at the Brewster angle for

polarization selection.

Brewster Angle Gain region


10

3/15/2014 Dr. Mohamed Fadhali 19 Dr. Mohamed Al-Fadhali

Some common resonators

Light-Mater interaction

Lasers are quantum devices, requiring understanding of the gain medium.

Laser light usually generated from discrete atomic transitions

3/15/2014 20 Dr. Mohamed Fadhali

Laser: Light amplification by stimulated emission of radiation

A laser converts electricity or incoherent light to coherent light.


11

Blackbody radiation Blackbody radiation is emitted from a hot body. It's anything but black!

The name comes from the assumption that the body absorbs at every frequency

and hence would look black at low temperature . It results from a combination

of spontaneous emission, stimulated emission, and absorption occurring in a

medium at a given temperature.

It assumes that the box is filled with molecules that, together, have

transitions at every wavelength.


Hot objects radiate. Idealized object (perfect blackbody radiator): emission spectrum direct measure of energy present in system vs. frequency

(perfect blackbody: reflectivity = transmission = 0

emissivity = absorptivity = 1 )

Model: hole in large box with reflective interior walls: incident light from ~all angles will make multiple passes inside box, resulting in thermal equilibrium inside box.

Blackbody model


12

Approach:

1. calculate all possible ways EM radiation ‘fits in the box’

depending on the wavelength (density of states calculation)

2. first (wrongly) assume that each radiation mode has E=kT/2 energy

(this was the approach before the photon was known) results in paradox

3. fix this by assuming energy in field can only exist in energy quanta &

apply Maxwell-Boltzmann statistics problem solved


Light inside box reflects multiple times, depending on incidence angle

Closed paths with different length exist at different angle of incidence:

Allowed modes (2-D )


13

For three dimensional case, and taking cavity with dimensions a×a×a (V = a3), we find allowed modes with equally spaced k values

We can now calculate the density of states as a function of several parameters, e.g. number of states within a k-vector interval dk.

Each allowed k-vector occupies volume k in k-space (reciprocal space):

3

3

aaaakkkk zyx

Allowed modes and density of states


Number of k-vectors in k range of magnitude dk depends on k:

In two dimensions: number of allowed k-vectors goes up linearly with k

In three dimensions: number of allowed k-vectors goes up quadratically with k

width dk

Mode density


To Calculate number of modes in the frequency range between and +d :

1. calculate volume 1/8 sphere 2. divide by volume for 1 mode and 3. multiply by 2 (TE and TM modes allowed)

14

volume 1/8 sphere = Vs 33 2

3

4

8

1

3

4

8

1

c

nkVs

Number of modes in this volume = 2 × Vs / k and k = (/a)3

3

3

333

3

3

33

3

8

3

42

2

3

4

8

12

ac

na

c

n

a

c

nN

The mode density (modes per unit volume) in frequency range d becomes

With the group index, which we set as ngn

Mode density (2/2)


Classically (e.g. in gases), it was known that each degree of freedom had E = kT/2 (e.g. atom moving freely: three degrees of freedom: E=3/2 kT.

Applying this to the calculated mode density gives (incorrectly!) the energy density:

This gives rise to the Ultraviolet catastrophe


15

29

0 20

2x107

4x107

6x107

8x107

1x108

T = 5000 K

T = 6000 K

T = 3000 K

Spe

ctra

l Rad

ianc

e E

xita

nce

(W/m

2 - m

m)

Wavelength (mm)

M = T

Cosmic black body background

radiation, T = 3K.

Rayleigh-Jeans law

3/15/2014 Dr. Mohamed Fadhali Dr. Mohamed Al-Fadhali

From Einstein and Planck theories, we now know: light exists with energy in discrete amounts: photons

Each photon has energy : E = h

Probability of having one photon in a given mode scales with exp(-h/kT) Probability of having two photons in a given mode scales with exp(- 2 h/kT)

Average energy in a given mode given by:

one photon energy × probability of having 1 photon present in mode

two photons × probability of having 2 photons present in mode

normalization factor

The effect of energy quantization


16

Analytical solution for blackbody radiation

The equation for energy per mode can be solved analytically:

Giving the following energy density inside the cavity at a given frequency :

This is the Planck blackbody radiation formula


Blackbody Radiation Short wavelength behavior:

Result of quantum nature of light

mode density thermal population


17

Fundamental Light-Matter Interactions

• (Stimulated) Absorption

• Spontaneous Emission

• Stimulated Emission

All light-matter interactions can be described by

one of three quantum mechanical processes:

…We will now look at each. 3/15/2014 33 Dr. Mohamed Al-Fadhali

Interaction of Radiation with Atoms and Molecules: The Two-Level System

The concept of stimulated emission was first developed by Albert

Einstein from thermodynamic considerations. Consider a system

comprised of a two-level atom and a blackbody radiation field, both at

temperature T.


18

Atomic and molecular vibrations correspond to excited

energy levels in quantum mechanics.

3/15/2014 35

En

erg

y

Ground level

Excited level

E = h

The atom is at least partially in an

excited state.

The atom is vibrating at

frequency, .

Energy levels are everything in quantum mechanics.


Excited atoms emit photons

spontaneously.

3/15/2014 36

When an atom in an excited state falls to a lower energy level, it emits a

photon of light.

Molecules typically remain excited for no longer than a few nanoseconds.

This is often also called fluorescence or, when it takes longer,

phosphorescence.

En

erg

y

Ground level

Excited level


19

Atoms and molecules can also absorb

photons, making a transition from a lower

level to a more excited one.

3/15/2014 37

This is, of

course,

absorption.

Energ

y

Ground level

Excited level

Absorption lines in an

otherwise continuous

light spectrum due to a

cold atomic gas in front

of a hot source.


Decay from an excited state can occur in many steps.

Energ

y

The light that’s eventually re-emitted after absorption may occur at

other colors.

Infra-red

Visible

Microwave

Ultraviolet


20

In what energy levels do molecules reside? Boltzmann population factors

Ni is the number density of

molecules in state i (i.e.,

the number of molecules

per cm3).

T is the temperature, and

kB Boltzmann’s constant

= 1.38x10-16 erg / degree

= 1.38x10-23 j/K

exp /i i BN E k T

En

erg

y

Population density

N1

N3

N2

E3

E1

E2


Boltzmann Population Factors

In equilibrium, the ratio of the populations of two states is:

N2 / N1 = exp(–E/kBT ), where E = E2 – E1 = h

As a result, higher-energy states are always less populated than the

ground state, and absorption is stronger than stimulated emission. 3/15/2014

Dr. Mohamed Al-Fadhali 40

In the absence of collisions,

molecules tend to remain

in the lowest energy state

available.

Collisions can knock a mole-

cule into a higher-energy state.

The higher the temperature,

the more this happens.

22

1 1

exp /

exp /

B

B

E k TN

N E k T

Low T High T

En

erg

y

Molecules

En

erg

y

Molecules

3

2

1

2

1

3

21

In 1917, Einstein showed that another process, stimulated emission, can occur.


41

Before After

Absorption

Stimulated

emission

Spontaneous

emission

Calculating the gain: Einstein A and B coefficients

Recall the various processes that occur in the laser medium:

Absorption rate = B N1 r()

Spontaneous emission rate = A N2

Stimulated emission rate = B N2 r()


42

2

1

22


The processes of spontaneous emission and (stimulated) absorption

were well known. Einstein had to postulate a new process, stimulated

emission in order for thermodynamic equlibrium to be established.

2

1 Spontaneous Emission

Stimulated Absorption

Stimulated Emission

2 21N A 2 21N B r1 12N B r

3/J s mr



From thermodynamic equilibrium

3

2 21 2 21 1 12 /N A N B N B J s m r r r

Units of B must be consistent with units of r() units of A are sec-1.

Absorption calculations are best done using A to avoid confusion on units. 3/15/2014 44

1

18)(

3

33

TBkh

ec

hn

r Tk

EE

Beg

g

N

N)(

1

2

1

2

12

and

121212213

33

21 ,8

BgBgBc

hnA


23

Rate equations – spontaneous emission

Absorption, emission, amplification depend on number of atoms in various states

Define concentration of atoms in state 2 as N2 (units often cm-3)

To find N1(t) and N2(t) need to model time dependence of all processes

Process 1: spontaneous emission

Chance of spontaneous emission per unit time is A (Einstein coefficient)

If there are N2 atoms excited per volume, then t later we will have less, or

spsp

NAN

dt

dN

2

2

2

22 tNAN In differential form this becomes a rate equation of the form

where A is the rate constant for spontaneous emission and

sp is the time constant for spontaneous emission given by sp=1/A


Rate equations – spontaneous emission

Suppose you can bring atoms in the excited state by some energy input

look at time dependence of N2 after the energy input is turned off at t=0:

( ) ( ) spt

spsp

eNtNN

dt

dN

/

2222 0

Note that the N2 drops to 1/e of its original value when t=sp.

We have solved our first rate equation to calculate the time dependent

concentration of excited atoms


24

Rate equations – Stimulated emission

More complex situations, add more processes to the rate equations

Process 2: Stimulated emission

Scales with the electromagnetic spectral energy density r()

where r()d is the energy per unit volume in the frequency range {,+d}

)(2122 rBN

dt

dN

st

B21 is the Einstein coefficient for stimulated emission

Under monochromatic illumination at frequency we can write this as

h

IN

dt

dN

st

)(2122

with I / (h) the photon flux given and 21() the cross section for stimulated emission

Note that ()I /(h) is the rate constant for stimulated emission (units again s-1)


Rate equations – Absorption

Process 3: Absorption

for absorption (‘stimulated absorption’) we obtain a similar rate equation:

r

h

INBN

dt

dN

abs

)()( 1211212

with B12 the Einstein coefficient for absorption

and 12 the absorption cross section


25

Rate equations for two-level system We now have the rate equations describing the population of levels 1 and 2

Population is the ‘amount of occupation’ of the different energy levels

22121212 )()( ANBNBN

dt

dN rr

22121211 )()( ANBNBN

dt

dN rr

Since we have only two states, we find

(atoms that leave state 2 must end up in state 1)

N1 and N2 should add up to the total amount of atoms: N1+N2=N

We can now solve the time dependent population N2 under illumination

Before doing that, let’s look at the relations between A, B12, and B21

dt

dN

dt

dN 21


Einstein coefficients in thermal equilibrium

Hypothetical situation: closed system at temperature T with collection of

two-level atoms and no external illumination:

All processes together will result in a thermal equilibrium with a

population distribution described by a Boltzmann factor:

Tkh BeN

N /

1

2

In equilibrium, on average 021 dt

dN

dt

dN

This implies that in equilibrium

Tk

h

BeBA

B

N

N

r

r

)(

)(

21

12

1

2

0)()( 2212121 ANBNBN rr

Resulting in an equation relating the Einstein coefficients to the thermal distribution:


26

Einstein coefficients in thermal equilibrium

At high temperatures e-h/kT →1, and the radiation density becomes large:

BBBB

B

BA

Be

BA

B

N

N

T

Tk

h

B

2112

21

12

21

12

)(21

12

1

2 1 )(

)(

)(

)(

r

r

r

r

1

1)(

2

1

21

2

221112

2

N

NB

A

NN

N

B

A

NBNB

ANr

1

1)(

TBkh

eB

Ar

Conversely, we can derive the ‘emission spectrum’ from our two-level atom

Substituting the thermal distribution over the available energy levels we obtain

which looks very similar to the Planck blackbody radiation formula :

1

18)(

3

33

TBkh

ec

hn

r

3

3

3

33 88

hn

c

hn

B

Aimplying


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Laser Lectures 1

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