Chapter 2 Laser - UGA Physics and Astronomy

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Chapter 2

Laser

Light Amplification by Stimulated Emission

of Radiation

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How does an object emit light or radiation?

Part I

Blackbody Radiation

• Solids heated to very high temperatures emit visible light (glow)– Incandescent Lamps (tungsten filament)

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Blackbody Radiation

• The color changes with temperature – At high temperatures emission color is whitish, at lower

temperatures color is more reddish, and finally disappear

– Radiation is still present, but “invisible”

– Can be detected as heat• Heaters; Night Vision Goggles

Electromagnetic Spectrum

(m)

1000 100 10 1 0.1 0.01

visible

light

0.7 to 0.4 m

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(m)

1000 100 10 1 0.1 0.01

ultraviolet

visible

light


(m)

1000 100 10 1 0.1 0.01

ultraviolet

visible

lightinfrared


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(m)

1000 100 10 1 0.1 0.01

ultraviolet

visible

lightinfrared

Near IRMid IRFar IRTHz


(m)

1000 100 10 1 0.1 0.01

ultraviolet

visible

lightinfraredmicrowaves x-rays


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(m)

1000 100 10 1 0.1 0.01

ultraviolet

visible

lightinfraredmicrowaves x-rays

High

Energy

Low

Energy


Kirchoff’s Question (1859)

Radiant Energy and Matter in

Equilibrium

What is the thermal radiation of a bodies that emit

and absorb heat radiation, in an opaque enclosure

or cavity, in equilibrium at temperature T?

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Observation

• All object at finite temperatures radiate electromagnetic waves (emit radiation)

• Objects emit a spectrum of radiation depending on their temperature and composition

• From classical point of view, thermal radiation originates from accelerated charged particles in the atoms near surface of the object

Ideal System to Study Thermal Radiation:

Blackbody

– A blackbody is an object that absorbs

all radiation incident upon it

– Its emission is universal, i.e. independent of the nature of the object

– Blackbodies radiate, but do not reflect and so are black

Blackbody Radiation is EM radiation

emitted by blackbodies

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Blackbody Radiation

• There are no absolutely blackbodies in nature – this is idealization

• But some objects closely mimic blackbodies:– Carbon black or Soot (reflection is <<1%)

• The closest objects to the ideal blackbody is a cavity with small hole (and the universe shortly after the big bang)

– Entering radiation has little chance of escaping, and mostly absorbed by the walls. Thus the hole does not reflect incident radiation and behaves like an ideal absorber, and “looks black”

Kirchoff's Law of Thermal Radiation (1859)

• Absorptivity αλ is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength.

• The emissivity of the wall ελ is defined as the ratio of emitted energy to the amount that would be radiated if the wall were a perfect black body at that wavelength.

• At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity

αλ = ελ• If this equality were not obeyed, an object could never reach

thermal equilibrium. It would either be heating up or cooling down.

• For a blackbody ελ = 1

• Therefore, to keep your frank warm or your ice cream cold at a baseball game, wrap it in aluminum foil

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Blackbody Radiation Spectra

Blackbody Radiation Laws

• Emission is continuous

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• The total emitted energy increases with temperature, and represents, the total intensity (Itotal) – the energy per unit time per unit area or power per unit area – of the blackbody emission at given temperature, T.

– σ = 5.670×10-8 W/m2-K4

• To get the emission power, multiply Intensity Itotal by area A

4TItotal

Stefan-Boltzmann Law

• The maximum shifts to shorter wavelengths with increasing temperature– the color of heated body changes from red to orange to

yellow-white with increasing temperature


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• The wavelength of maximum intensity per unit wavelength is defined by the

– b = 2.898×10-3 m/K is a constant

• For, T ~ 6000 K,

bT max

nm 4836000

10898.2 3

max

Wien’s Displacement Law

Nobel 1911


Blackbody Radiation Spectra

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How to understand Blackbody radiation from fundamental physical principle?

The Birth of Quantum Mechanics

Classic Physics View

• Radiation is caused by EM wave radiation

• Consider a cavity at temperature Twhose walls are considered as perfect reflectors

• The cavity supports many modes of oscillation of the EM field caused by accelerated charges in the cavity walls, resulting in the emission of EM waves at all wavelength

• They are considered to be a series of standing EM wave set up within the cavity

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• Radiation is caused by EM wave radiation

• Average energy of a harmonic oscillator is <E>

• Intensity of EM radiation emitted by classical harmonic oscillators at wavelength λ per unit wavelength:

• Or per unit frequency ν:

Ec

TI3

22),(

Ec

TI3

2),(


• In classical physics, the energy of an oscillator is continuous, so the average is calculated as:

is the Boltzmann distributionTk

E

BePEP

0)(

Tk

dEeP

dEeEP

dEEP

dEEEP

E B

Tk

E

Tk

E

B

B

0

0

0

0

0

0

)(

)(

TkE B


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• This gives the Rayleigh-Jeans Law

Tkc

Ec

TITk

c

E

cTI B

B

2

2

3

2

33

22),(,

22),(

Agrees well with experiment long wavelength (low frequency) region


• Predicts infinite intensity at very short wavelengths (higher frequencies) – “Ultraviolet Catastrophe”

• Predicts diverging total emission by black bodies

No “fixes” could be found using classical physics


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Planck’s Hypothesis

Max Planck postulated that

A system undergoing simple harmonic motion with frequency ν can only have energies

where n = 1, 2, 3,…

and h is Planck’s constant

h = 6.63×10-34 J-s

nhnE

1918 Nobel

Planck’s Theory

hnhhnE

nhE

)1(

J1023000J1063.6 30134

ssE

hE

E is a quantum of energy

For = 3kHz

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• As before:

• Now energy levels are discrete,

• So

• Sum to obtain average energy:

Thus

Ec

TI3

22),(

10

0

0

0

Tk

n

Tk

n

n

Tk

n

n

BB

B

eeP

ePE

E

1

2

1

2),(

3

2

3

2

Tk

h

Tk BB e

h

ce

cTI

nnhEn

0

0

0

n

Tk

n

Tk

n

n

B

B

eP

eP)E(P

Planck’s Theory

h

Blackbody Radiation Formula

c is the speed of light, kB is Boltzmann’s constant, h is Planck’s constant, and T is

the temperature

1exp

2)(

2

2

Tk

h

h

cI

B

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Blackbody Radiation from the Sun

Plank’s curve

λmax

Stefan-Boltzmann Law

IBB T4

IBB = T4

Stefan-Boltzmann constant

=5.67×10-8 J/m2K4

More generally:

I = T4

is the emissivity

Wien's Displacement Law

peak T = 2.898×10-3 m K

At T = 5778 K:

peak = 5.015×10-7 m = 5,015 A

• Cosmic microwave background (CMBR) as perfect black body radiation

1965, cosmic microwave background was first detected by Penzias and Wilson

Nobel Prize

1976

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The Nobel Prize in Physics 2006

• "for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation"

John C.

Mather

George F.

Smoot

Stimulated Emission

Part II

How is light generated from an atomic point of view?

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Mechanisms of Light Emission

For atomic systems in thermal equilibrium with their

surrounding, the emission of light is the result of:

Absorption

E1

E2

E = hv



Absorption

And subsequently, spontaneous emission of energy

E1

E2

Phase and propagation direction of created photon is random.


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Absorption

And subsequently, spontaneous emission of energy

Stimulated emission

E1

E2Created photon has the same phase, frequency, polarization, and propagation direction as the input photon.


Stimulated Emission

• It is pointed out by Einstein that:

Atoms in an excited state can be stimulated to jump to a

lower energy level when they are struck by a photon of

incident light whose energy is the same as the energy-level

difference involved in the jump. The electron thus emits a

photon of the same wavelength as the incident photon. The

incident and emitted photons travel away from the atom in

phase.

• This process is called stimulated emission.

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Population of Energy Levels

How many atoms are in the ground states? And

how many are in the excited states?

Unexcited

electron

1E

2E Excited

electron

1E 2E

Population of Energy Levels

Maxwell-Boltzmann distribution

1E

2E Excited

electron

1E 2E

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Rate Equation of Absorption and

Emission

For absorbance, the # of E1 atoms decrease after

absorption

E

h

absorption

For emission, the # of E1 atoms increase

emission

Rate Equation of Absorption and Emission

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For stimulated, the # of E1 atoms increase after

absorption

Stimulated emission


Emission

By considering all 3 processes, the change rate of # of

E1 atoms becomes

E

h

absorption emission Stimulated emission


Emission

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Emission

If the system is under equilibrium (blackbody), then


Emission

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Emission

I needs to approach to infinite when T approaches infinite,

which implies


Emission

Compared to Plank’s formula,

For a visible light, v ~ 1014 Hz, A/B ~ 10-16

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The Laser

Part III

Population Inversion

In order to obtain the coherent light from stimulated emission:

𝐵𝑁2𝐼 ≫ 𝐵𝑁1𝐼 + 𝐴𝑁2

<<1

𝑁2 ≫ 𝑁1

Thus:

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In order to obtain the coherent light from stimulated emission, two conditions must be satisfied:

1. The atoms must be excited to the higher state. That is, an inverted population is needed, one in which more atoms are in the upper state than in the lower one, so that emission of photons will dominate over absorption.

E

2. The higher state must be a metastable state – a state in which the electrons remain longer than usual so that the transition to the lower state occurs by stimulated emission rather than spontaneously.

Metastable state

Photon of energy 12 EE

1E

2E3E

Metastable system1E

2E3E

Stimulated emission

Incident photon

Emitted photon


In order to obtain the coherent light from stimulated emission, two conditions must be satisfied:

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- Gain medium (Active medium)

- Pumping source

- Cavity (Resonator)

- Output couplerpumping laser

relaxation

relaxation

Laser light

pumping source

gain medium

cavity (resonator)

output coupler

total reflector

Four Key Elements of a LASER

iE

fE

Mirror Mirror


Lasing Process

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iE

fE

Mirror Mirror

Spontaneous emission

Lasing Process

iE

fE

MirrorMirror

Stimulated emission

Lasing Process

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iE

fE

MirrorMirror

Feed-back by the cavity

Lasing Process

iE

fE

MirrorMirror

Stimulated emission

Lasing Process

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iE

fE

MirrorMirror

Feed-back by the cavity

Lasing Process

iE

fE

MirrorMirror

Laser beam

After several round trips/many pumps…

Photons with:

- same energy : Monochromatic

- same direction of propagation : Spatial coherence

- all in synchrony: Temporal coherence

Lasing Process

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An Amplification and Cascade Process

During the entire process, the population must be kept inversed, i.e., the amplification

media should be pumped all the time, either pulsed or continuously.

An Amplification and Cascade Process

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Laser Construction

Amplifying Medium

Laser Construction

• Atoms: helium-neon (HeNe) laser; heliumcadmium (HeCd) laser,

copper vapor lasers (CVL)

• Molecules: carbon dioxide (CO2) laser, ArF and KrF excimer

lasers, N2 laser

• Liquids: organic dye molecules dilutely dissolved in various

solvent solutions

• Dielectric solids: neodymium atoms doped in YAG or glass to

make the crystalline Nd:YAG or Nd:glass lasers

• Semiconductor materials: gallium arsenide, indium phosphide

crystals.

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Homework

(1) Helium-neon (HeNe) laser

(2) Ruby laser

(3) Dye laser

(4) Semiconductor laser

Please find out the principles of the following lasers from internet or

books, and in your first lab report, i.e., “Lab #1” report, please add an

“Appendix” section to describe the principle of one of the following

lasers, with at least two figures, the construction of the laser and

energy lever diagram, you have to describe these figures and the laser

operation principle:

Resonance Cavities and Longitudinal Modes

Since the wavelengths involved with lasers spread over small

ranges, and are also absolutely small, most cavities will

achieve lengthwise resonance

Plane

parallel

resonator

Concentric

resonator

Confocal

resonator

Unstable

resonator

Hemispheric

al resonator

Hemifocal

resonator

cc

f

f

c: center of curvature, f: focal point

L = nλ/2

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Fabry-Perot boundary conditions

Multiple resonant frequencies



node

antinode

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Multi-mode laser


Resonance Cavities and Transverse Modes

Ga

us

s-H

erm

ite M

od

en

TEM 00

TEM 01

TEM 02

TEM 03

TEM 10

TEM 11

TEM 21

TEM 31

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Gaussian Beams

• Zero order mode is Gaussian

• Intensity profile: 22 /2

0

wreII

Gaussian Beams

• Gaussian beam intensity

• Beam waist: w0

• Confocal parameter (Rayleigh range): Z0

22 /2

0

wreII

𝑤 = 𝑤0 1 + (𝑧/𝑧0)2

𝑧0 =𝑘𝑤0

2

2

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Gaussian Beams

• Far from waist

• Divergence angle

0

637.0w

0z0w

I

Gaussian profile

02w

0/2/ nw

Spread angle :

0z

Near field

(~ plane wave)

Far field

(~ spherical wave)

z

𝑤 ≈2𝑧

𝑘𝑤0

𝜃 ≈ 2𝑤

𝑧=

4

𝑘𝑤0=

2𝜆

𝜋𝑛𝑤0

Power Distribution in Gaussian

• Intensity distribution:

• Experimentally to measure full width at half maximum

(FWHM) diameter

• Relation is dFWHM = w 2 ln2 ~ 1.4 w

• Define average intensity

Iavg = 4 P / (p d2FWHM)

• Overestimates peak:

I0 = Iavg/1.4

22 /2

0

wreII

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Propagation of Gaussian Beam - ABCD law

Propagation of Gaussian Beam - ABCD law

Matrix method (Ray optics)

yi

yo

ai

ao

Optical Elements

i

i

o

o y

DC

BAy

aa

DC

BA: ray-transfer matrix

Optical axis

𝑦𝑜 = 𝐴𝑦𝑖 + 𝐵𝛼𝑖

𝛼𝑜 = 𝐶𝑦𝑖 +𝐷𝛼𝑖

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Ray Transfer Matrices

Free space propagation

(paraxial ray approximation)

i

i

o

o ydy

aa 10

1

𝑦𝑜 = 𝑦𝑖 + 𝑑𝛼𝑖

𝛼𝑜 = 𝛼𝑖

𝑦𝑜𝑦𝑖 𝛼𝑖

𝛼𝑜


Propagation through curved refracting surface

i

i

o

o y

n

n

Rn

nny

aa2

1

2

21

01

n1 n2

ai ao

yi yo

R

h

s S’

𝑛1𝑠+𝑛2𝑠′

=𝑛2 − 𝑛1

𝑅𝛼𝑜 =

𝑛1𝑛2

𝛼𝑖 + (1 −𝑛1𝑛2)𝑦𝑖𝑅

𝑦𝑖 = 𝑦𝑜 = ℎ

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ABCD Law for Gaussian Beam

i

i

o

o y

DC

BAy

aa iio

iio

DCy

BAyy

aa

a

ii

ii

o

oo

DCy

BAyyR

a

a

a

DCy

BAy

ii

ii

a

a

/

/

DCR

BAR

i

i


)()( opticsGaussianqopticsrayRo

DCq

BAqq

1

12

2q1q

optical system

DC

BA

ABCD law for Gaussian beam :

0izzq

2

00

nwz

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Focusing a Gaussian beam

101w

02w

1z 2z

?

?

f/zf/

f/zzzzf/z

z

f/

z

DC

BA

1

21212

12

11

1

10

1

11

01

10

1


)/1(/

)/()/1(

11

2121122

fzfq

fzzzzqfzq

2

01

2

2

1

2

01

2

02

11

11

w

ff

z

ww

)()/()(

)(22

01

2

1

1

2

2 fwfz

fzffz

0201 ww - If a strong positive lens is used ; => 1

01

02

f

w

fw

2

1

2

01 )(/ fzw - If => fz 2

=> dfff

w

fw N

N /,2

)2(

2

01

02

: f-number

; The smaller the f# fo the lens, the smaller the beam waist at the focused spot.

Note) To satisfy this condition, the beam is expanded before being focused.


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Documents

Chapter 2 Laser - UGA Physics and Astronomy