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412 PHYS

Lasers and their

Applications

Department of Physics

Faculty of Science

Jazan University

KSA

Lecture-1

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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.

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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 Alexandrite

755

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Introduction (Brief history of laser)

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

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Others devices followed in rapid succession, each with a different laser

medium and a different wavelength emission.

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

emitted deep red light at a wavelength of 694.3 nm.

- Ali 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.

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7

8

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9

10

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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.

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Brewster Angle Gain region

Examples of Electrical and Optical pumping

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Dr. Mohamed Fadhali

Dr. Mohamed Fadhali

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

Laser light usually generated from discrete atomic transitions

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Laser: Light amplification by stimulated emission of radiation

A laser converts electricity or incoherent light to coherent light.

Laser-matter interaction

Is a radiation 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.

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Blackbody radiation

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(perfect blackbody: reflectivity = trasmissivity = 0

emissivity = absorptivity = 1 )

Model: Assume a 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.

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Blackbody radiation

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=KBT/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

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

3x y zk k k ka a a a

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ALLOWED MODES AND DENSITY OF STATES

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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

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Mode density

volume 1/8 sphere = Vs 331 4 1 4 2

8 3 8 3s

k nV

c

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

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

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

This gives rise to the Ultraviolet catastrophe

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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

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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

* 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

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HW: Prove that?

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Short wavelength behavior:

Result of quantum nature of light

mode density thermal population

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Blackbody Radiation

• (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.

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Fundamentals of Light-Matter Interactions

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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.

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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.

Atoms and molecules can also absorb photons, making a

transition from a lower level to a more excited one

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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.

Energ

y

Ground level

Excited level

Excited atoms emit photons spontaneously

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

Energ

y

Population density

N1

N3

N2

E3

E1

E2

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In what energy levels do molecules reside?

Boltzmann population factors

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*

In the absence of collisions, molecules

tend to remain in the lowest energy state

available.

Collisions can knock a molecule

into a higher-energy state.

The higher the temperature,

the more this happens.

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1 1

exp /

exp /

B

B

E k TN

N E k T

Low T High T

Energ

y

En

erg

y

Molecules

3

2

1

2

1

3

Boltzmann Population Factors

2

1

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()

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

The processes of spontaneous emission and (stimulated) absorption were well

known. Einstein had to postulate a new process, stimulated emission in

order for thermodynamic equilibrium to be established.

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1 Spontaneous Emission

Stimulated Absorption

Stimulated Emission

2 21N A2 21 2 21 ( )N W N B r 1 12 1 12 ( )N W N B r

3( ) . /J s mr

From thermodynamic equilibrium

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2 21 2 21 1 12( ) ( ) ( ) . /N A N B N B J s mr 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 3

3

8 1( )

1h

k TB

hn

c e

r

2 1( )

2 2

1 1

B

E E

K TN ge

N g

and

3 3

21 21 2 21 1 123

8,

hnA B g B g B

c

2 21 2 21 1 12N A N W N W

HW: How to

reach to this

expression ???

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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), you 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 at later time t we will have less, or

2 22

spsp

dN NAN

dt

2 2N A t N

In differential form this becomes a rate equation of the form

where A is the rate constant for spontaneous emission and sp is the

life time for spontaneous emission given by sp=1/A

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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:

( ) ( )/2 2

2 2 0 spt

spsp

dN NN t N e

dt

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

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Rate equations: spontaneous emission

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22 21 ( )

st

dNN B

dtr

B21 is the Einstein coefficient for stimulated emission

Under monochromatic illumination at frequency we can write this as

22 21( )

st

IdNN

dt h

Rate equations – Stimulated emission

More complex situations, add more processes to the rate equations

Scales with the electromagnetic spectral energy density r()

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

Process 2: Stimulated emission

with I /(h ) is 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)

Process 3: Absorption

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

equation: 2

1 12 1 12( ) ( )abs

IdNN B N

dt h

r

with B12 the Einstein coefficient for absorption

and 12 the absorption cross section

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Rate equations – Absorption

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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

21 12 2 21 2( ) ( )

dNN B N B AN

dtr r

11 12 2 21 2( ) ( )

dNN B N B AN

dtr r

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

Rate equations for two-level system

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:

/2

1

Bh k TNe

N

In equilibrium, on average 021 dt

dN

dt

dN

This implies that in equilibrium

2 12

1 21

( )

( )B

h

k TN Be

N A B

r

r

1 12 2 21 2( ) ( ) 0N B N B ANr r

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

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Einstein coefficients in thermal equilibrium

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2 12 12 1212 21

( )1 21 21 21

( ) ( ) 1

( ) ( )B

h

k T

T

N B B Be B B B

N A B A B B

r r

r r

1

2

2 2

12 1 21 2 1 2

1( )

1N

N

AN NA A

B N B N B N N Br

1( )

1h

k TB

A

B er

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 : 3 3

3

8 1( )

1h

k TB

hn

c e

r

3 3 3

3 3

8 8A hn hn

B c

implying

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

Ratio between rates of spontaneous and stimulated emission

21 21

21 21

1( )

hk TB

A Ae

W B

r

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HW. 1

Compare the rates of spontaneous and stimulated emission at room temperature (T

= 300K) for an atomic transition where the frequency associated with the transition

is about 3 × 1010 Hz, which is in the microwave region (KB=1.38 × 1023 J/K)

(b) What will be the wavelength of the line spectrum resulting from the transition of

an electron from an energy level of 40 × 10−20 J to a level of 15 × 10−20 J?

HW. 2

Consider the energy levels E1 and E2 of a two-level system. Determine the

population ratio of the two levels if they are in thermal equilibrium at room

temperature, 27◦C, and the transition frequency associated with this system is

at 1015 Hz

HW. 3

The oscillating wavelengths of the He–Ne, Nd:YAG, andCO2 lasers are 0.6328,

1.06, and 10.6µ m, respectively. Determine the corresponding oscillating

frequencies. What energy is associated with each transition?