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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
prof. Halina Abramczyk Laboratory of Laser Molecular Spectroscopy Technical
University of Łódź6
THERMODYNAMIC EQUILIBRIUM
emsamble of quantum particles
radiation field
n
n
1221
212112
12
121212
2112
BW
BAW
WWW
nWnW
NN
emwym
emsp
prof. Halina Abramczyk Laboratory of Laser Molecular Spectroscopy Technical
University of Łódź7
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
prof. Halina Abramczyk Laboratory of Laser Molecular Spectroscopy Technical
University of Łódź8
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
prof. Halina Abramczyk Laboratory of Laser Molecular Spectroscopy Technical
University of Łódź9
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
University of Łódź11
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
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź12
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
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź13
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
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź16
deftf
tnftf
ti
n
)()(
)sin()()(
A
Time domainFrequency domain
I()
t
Et
FOURIER TRANSFORM
deftf
tnftf
ti
n
)()(
)sin()()(
Fourier transform
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź19
00 )( xfdxxfxx
tdet ti
000 coscos because
Frequency domain
t
(delta Dirac)
0
I()
0 t0cos
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź20
Properties of Dirac function
00 )(
0
xfdxxfxx
0xx
0xx 0xx
0xx
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź21
t
Frequency domain
021
n
nn1Time domain
tte
0cos
nn
nn
0
0
sin
x
x
x
where
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź23
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
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź24
Q 2
1
The larger Q-of the resonator, the narrower
the bandwidth of a single mode
prof. Halina Abramczyk Laboratory of Laser Molecular spectroscopy Technical
University of Łódź25
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
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