chapter 3. classical theory of absorptionoptics.hanyang.ac.kr/~choh/degree/quantum...
TRANSCRIPT
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Nonlinear Optics Lab. Hanyang Univ.
Chapter 3. Classical Theory of Absorption
3.1 Introduction
Visible color of an object : Selective absorption, Scattering, Transmission
1) Absorption in gases : line spectrum
- Electronic resonance : UV region
ex) white daylight : N2 or O2 does not absorb at visible frequency
- Molecular vibration : IR region
- Molecular rotation : Microwave region
2) Absorption in liquids or sold : broad band spectrum
ex) Green water : absorption in the red portion
ex) Red dye : strong absorption in the blue or UV
ex) Metal (free electron, plasma freq. : UV region)
- Shine
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Nonlinear Optics Lab. Hanyang Univ.
3.2 Absorption and the Lorentz model
Absorption (?) => Frictional force that damps out dipole oscillations
fric2
2
F x),R(Ex
sktedt
dm=> Equation of motion ;
3.3 Complex Polarization and Index of Refraction
)(
0EˆEkztie
)(0
2
02
2
Eˆxdt
dx2
xd kztiem
e
dt
sol))( a x(t) kztie
where,
i2
(e/m)Eˆ-a
2
0
2
0
where,dt
dbbv
xFfric
Origin of friction : collision
(Section 3.9, 3.10)
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Nonlinear Optics Lab. Hanyang Univ.
xp e- Dipole moment,
22222
0
22
0
2
22
0
2
4)(
2
2
/)(
i
m
e
i
me
In case of many electrons ;
22222
0
22
0
0
2
22
0
0
22
4)(
21
2
/1)(
i
m
Ne
i
mNen
Complex refractive index
Ep ; - Polarizability,
Polarizability
z
j jj
jj i
m
e
122222
0
22
02
4)(
2)(
)()(
1 22
2
0
2
22
n
c
N
ck
2)]()([ IR inn
(2.3.13)
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Nonlinear Optics Lab. Hanyang Univ.
)(
0Eε̂),(Ekztietz
i) Intensity
where,
z
j jj
j
Imc
Necna
122222
0
2
0
2
4)(
2/)]([2)(
Electric field in the medium
}/)]([{/)]([
0
)/)([
0 Eε̂Eε̂cznticzncznti RI eee
zaczn eIeIzI I )(02/)]([ )()0()(
: absorption
coefficient
ii) Refractive Index
z
j jj
j
Rm
Nen
122222
0
22
0
0
22
4)(1)(
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Nonlinear Optics Lab. Hanyang Univ.
3.4 Polarizability and Index of Refraction near a Resonance
Most gain media of lasers are composed of a background material and a small
amount of resonant material.
ex) gas : background gas + active gas medium
ex) solid (ruby) : Al2O3 + Cr+3
ex) liquid (dye) : ethanol + dye
Thus we can write,
)()()( rb
b
rrb
b
rrbr
rbi
i
bi Nnn
Nn
NNn
)(1
)(1)()(1)(
2
0
2
2
00
2
2
bn (nearly const.)
bb
rrb
b
rrb
n
Nn
Nnn
2
)()(1)(
2/1
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Nonlinear Optics Lab. Hanyang Univ.
Near the resonance ( ),0
||,,|| 0 oo
)(2))((22
oooo
i
me
o
r
2/
)(2
22
0
0
0
2
)(4)(
mn
Nenn
bR
bRR
22
00
2
)(4)(
mn
Nenn
bR
bII
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Nonlinear Optics Lab. Hanyang Univ.
3.5 Lorentzian Atoms and Radiation in Cavities
Electric field in a cavity is not the form of a traveling wave but a standing wave,
ti
nn zektz sinEε̂),(E
where, ...,3,2,1,/ nLnkn
Similarly as previous sections,
zke nti sin a x(t)
where,
i2
(e/m)Eˆ-a
2
0
2
n
Maxwell wave equation for a cavity including cavity loss,
),(P1
t)E(z,1
2
2
2
0
2
2
22
0
2
2
tztctctcz
where, : conductivity
EJ
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Nonlinear Optics Lab. Hanyang Univ.
i
mNe
cccikn
2
/22
0
0
222
2
0
2
Near resonance approxiamtion ; )(2,)(22222
nnoo
cigi
m
Nein )(
2
1
)(42 220
0
0
2
0
cg
0
)2
)(22
)1 0
gccn
: cavity pulling)(22/
)2/(0
0nn
n gc
gc
gc
: threshold gain (*)
on n
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Nonlinear Optics Lab. Hanyang Univ.
(*) Relations 1), 2) are correct except the sign of g
cg
0
If , then field amplification is possible,
But, from (3.5.9),
0)(2 2200
2
mc
Neg : negative
This means that c
g0
cannot be satisfied.
That is, a classical laser theory based on the linear electron oscillator model is not possible.
And, it requires a quantum mechanical treatment of light-matter interaction to understand
how can be made positive.g
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Nonlinear Optics Lab. Hanyang Univ.
3.6 The Absorption Coefficient
[Lorentzian Lineshape]
(3.3.25) => )1when(4)(
2)(
2222
0
2
0
2
jmc
Nea
Near the resonance,
22
00
2
)(2)(
mc
Nea
2v 2
0
2
0
0
0
2
)(4)(
v
v
mc
Nea
where, 2/0
0,|| o
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Nonlinear Optics Lab. Hanyang Univ.
Lineshape Function ;
2
0
2
0
0
)(
/)(
v
vL
: Lorentzian lineshape
1)(
Ld
max
0
00 )(2
1
2
1)(
LL 02 : FWHM
00
2
04
)(
mc
Nea
In general, )()( Saa twhere, is the integrated absorption coefficient
ta
Normalization : tt aSdaad
00)()(
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Nonlinear Optics Lab. Hanyang Univ.
3.7 Oscillator Strength
Absorption Cross section :N
a )()(
Natt /
From (3.6.17) and (3.6.23), /scm1065.24
22
0
2
mc
et
: universal value
Oscillator strength,
As can be seen in Table 3.1 for the integrated cross section of hydrogen atom, actual values
of do not agree with the calculated values by classical theory. This problem of classical
theory could be patched by introducing oscillator strength assigned empirically.t
fmc
e
mc
ett
0
2
0
2
44
f
* Quantum theory removes this defect of Lorentz’s model.
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Nonlinear Optics Lab. Hanyang Univ.
3.11 Doppler Broadening
- 1842, C. J. Doppler, Predicted
- 1854, C.H.D. Buys Ballot, Demonstrated (trumpet in a moving train)
To an atom moving with velocity away from a source of radiation of frequency ,
the frequency of the radiation appears to be shifted :
cv
v
'
)1('c
v
Atomic velocity distribution :
dvekT
mvdf
kTvmx x 2/
2/12
2)(
: Maxwell-Boltzman distribution
Thus, the atom with resonance frequency
will absorb radiation near to the frequency :0
)1(0c
v
dc
dvc
v0
0
0
,)(
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Nonlinear Optics Lab. Hanyang Univ.
dv
ce
kT
mvdf
kTcmx x
0
2/)(
2/12
02
02
2)(
Since the absorption rate must be proportional to , we may write
the Doppler line shape function as
)(vdf
20
20
2 2/)(
2/1
2
0
2
2)(
kTcmx xekT
cmvS
: Doppler line shape function
2ln
000 )()(2
1)
2
1( eSSS D
: FWHM
2/1
0
2/1
0 2ln22
2ln2
2
xx
DM
RT
m
kT
c
2/1
0
2ln41)(
D
S
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Nonlinear Optics Lab. Hanyang Univ.
3.12 The Voigt Profile
Collisional broadening : Homogeneous => Lorentzian
Doppler Broadening : Inhomogeneous => Gaussian
In general, we cannot characterize an absorption lineshape of a gas
as a pure Lorentzian or a pure Gaussian.
Voigt profile : Described the absorption lineshape when both collision broadening
and Doppler broadening must be taken into account.
RTvMx xeRT
MvSdvvS
2/
2/12
2),()(
where,2
0
200
0
)(
)/1(),(
cv
vS
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Nonlinear Optics Lab. Hanyang Univ.
2
0
2
00
2/
0
2/1
)/(2)(
2
cv
edv
RT
MvS
RTvM
xx
22
0
2
2/3 )(
12
bxy
edyb y
: Voigt profile
where, ,)2ln4(0
02/1
x
D
b
02/1)2ln4(
)(erfc)(2
0
2/3
2
0 beb
bS b
where,
2
2/1
2erfc uedu
: complementary error function
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Nonlinear Optics Lab. Hanyang Univ.
Limit case of Voigt profile
10 bDb
beb2/1
1)(erfc
2
0
0
1)(
S : nearly pure Lorentzian
10 bD
2/1
0
2ln41)(
D
S : nearly pure Gaussian
ii) 1)(erfc2
beb
General case : numerical calculation
i)
)(ReRe)(
22
22ibxw
bibyx
edyi
bbxy
edy yy
where, w : error function of complex argument
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Nonlinear Optics Lab. Hanyang Univ.
Example) Na vapor
K 300T,line)-(D A5890
MHz1300D
MHzP17000 )torr(P2.2b : If P < 0.1 torr => Doppler regime
Absorption coefficient, 2/1
0
2
0
0
2
0
2ln41
4)(
4)(
D
Nmc
feSN
mc
fea
A5890,1 f
TN
)torr(P1065.9 18
-150 cm P(torr)102.2)( a