SPH 618 Optical and Laser Physics University of Nairobi ...mitr.p.lodz.pl/raman/Lecture2-SPH 618.pdf · Optical and Laser Physics University of Nairobi, Kenya Lecture 2 LASER FUNDAMENTALS

SPH 618Optical and Laser Physics

University of Nairobi, KenyaLecture 2

LASER FUNDAMENTALSProf. Dr Halina Abramczyk

Max Born Institute for Nonlinear Optics and Ultrashort Laser Pulses, Berlin, Germany,

Technical University of Lodz, Lodz, Poland

• Overview of wave propagation in variousmedia (dielectrics, semiconductors, conductors)

• Normal and anomalous dispersion

• Emission and absorption of light

• Spontaneous and stimulated emission

• Population inversion

• Optical resonator

The stimulated transitions have several important properties:

• the probability of the stimulated transition between the states mand n is different from zero only for the external radiation field that is in the resonance with the transition, for which the photon energy ħ of the incident radiation is equal to the energy difference between these states,

• the incident electromagnetic radiation and the radiation generated by the stimulated transitions have the same frequencies, phases, plane of polarization and direction of propagation. Thus, the stimulated emission is, in fact, completely indistinguishable from the stimulating external radiation field,

• the probability of the stimulated transitions per time unit is proportional to the energy density of the external field , that is the energy per unit of the circular frequency from the range between and +d in the volume unit.

prof. Halina Abramczyk Laboratory of Laser Molecular Spectroscopy Technical

University of Łódź5

A

1

2ESpontaneous

emission

E stimulated emission

W – probability of stimulated transitions in a time unit

n – density of radiation field (spectral density in a volume unit)

n

n

2112

1221

BW

BW

Absorption

Stimulated emission



THERMODYNAMIC EQUILIBRIUM

emsamble of quantum particles

radiation field

n

n

1221

212112

12

121212

2112

BW

BAW

WWW

nWnW

NN

emwym

emsp



11212

12121

11222121

1212

exp

exp

nBkT

EEnBA

nBnBA

kT

EEnn

nn

nn

1exp

1

exp

exp

12

21

12

12

12

21

21

1221

21

kT

EE

B

B

kT

EE

kT

EE

B

A

BB

A

n

n



From the Planck equation we know that

1exp

83

2

kTh

h

c nn

nn

I

2112 BB 213

2

21

8Bh

cA n

ni

)300(1042

21

21 KTB

A

n

12

21

21 1033,1 B

A

)41500(1~ KT

1exp

1

exp

exp

12

21

12

12

12

21

21

1221

21

kT

EE

B

B

kT

EE

kT

EE

B

A

BB

A

n

n



N1

N2

21

212121

2112

nn

BnBn

BB

nn

In two level system the population inversion is not possible!!! (upper limit

N1 = N2) THREE LEVEL SYSTEM- YES

Longitudinal modes

L

c

L

cm

L

cm

Lc

mLc

m

LmLmL

c

mm

mm

mm

mm

2221

212

2122

1

1

1

1

nn

nn

nn

prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical


TOTAL NUMBER OF LONGITUDINAL MODES

2

0

2

0

0022

0

00

minmax

4

22

2

112

L

L

L

LnnN

Ln

Ln

2)(

2)(

0min

0max

0

I

I



Bandwith of a single mode

1. Resonator Quality Q

2. Degree of Population inversion N1, N23. Laser power P

1

2

12

2002mod1

g

glasNN

N

QP

h

nn

g1, g2 – degeneration degree

• Q- of resonator is usually lower than a

theoretical value

• Energy losses from diffraction

• Heating of a medium (by pumping excitation)

• Mechanical nonstabilities



RESONATOR QUALITY Q

periodoneinlossenergy

resonatortheinenergyQ 2

dumpFFma

xkxxm

A0

t

AA

A0

t

Harmonic oscillator

The model of the damped oscillator can be applied to describe phenomena occurring in the optical resonator. As a result of diffraction, reflections and other system imperfections, the optical resonator loses the accumulated energy, and the standing wave does not held the constant amplitude.

cycle1duringlostenergy

systemingainedenergy2Q

022

0

2

0

2

0

e2

TAA

AQ

The quality factor of the system Q is defined by:

so

where A0 is the amplitude at t=0.

/222

0

2

0

2

0

00 1

22

TTe

π

eAA

AπQ

.

where the relationship τ = 1 between the damping factor and the time has been employed.

The time is a time after which the amplitude A0 of the damped oscillators decays by e=2.718 with respect to the initial time

0T 02e

TAssuming that , the term can be expanded in series

0

221e 0 T

T

n

2222 00

0

0

πT

πQ

Inserting into we get

Q

Q

when

)(lossE

What is the relation to the bandwith

of a single mode ?

? 21



deftf

tnftf

ti

n

)()(

)sin()()(

A

Time domainFrequency domain

I()

t

Et

FOURIER TRANSFORM

deftf

tnftf

ti

n

)()(

)sin()()(

Fourier transform



00 )( xfdxxfxx

tdet ti

000 coscos because

Frequency domain

t

(delta Dirac)

0

I()

0 t0cos



Properties of Dirac function

00 )(

0

xfdxxfxx

0xx

0xx 0xx

0xx



t

Frequency domain

021

n

nn1Time domain

tte

0cos

nn

nn

0

0

sin

x

x

x

where



Function has a maximum for x=0, which means

=0 (n=n0), becuase

11

coslim

'

'sinlim

sinlim

000

x

x

x

x

x

xxx

1/coslim2

xx

nnnn 01221

21 nn

022;

1

21

t

Frequency domain

021

n

nn1Time domain



Q 2

1

The larger Q-of the resonator, the narrower

the bandwidth of a single mode



CONCLUSIONS

Bandwith of spantaneous emission depends

on relaxation processes characterized by the

energy relaxation T1 and phase relaxationT2

Bandwith of stimulated emission

1. Quality of resonatorQ

2. Population inversion

3. Laser power

4. Number of modes2

0

4

LN

Bandwith of a single

mode

Total bandwith

of stimulated

emission

Transverse modes

We observe an intensity distribution not only along the resonator axis, but also in the plane perpendicular to the direction

of the laser beam propagation. The longitudinal modes are responsible for the spectral characteristics of a laser such as

bandwidth and coherence length whereas the beam divergence, beam diameter, and energy distribution in the plane

perpendicular to the beam propagation are governed by the transverse modes.

Transverse modes

Transverse modes

Documents

SPH 618 Optical and Laser Physics University of Nairobi ...mitr.p.lodz.pl/raman/Lecture2-SPH 618.pdf · Optical and Laser Physics University of Nairobi, Kenya Lecture 2 LASER FUNDAMENTALS