Chapter 3. Classical Theory of Absorptionoptics.hanyang.ac.kr/~choh/degree/Quantum Optics/quantum... · 2016. 8. 29. · Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Classical Theory

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


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)


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)


)(

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)(


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


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


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


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


(*) 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


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


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)()(


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.


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

,)(


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


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


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


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


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

Chapter 3. Classical Theory of Absorptionoptics.hanyang.ac.kr/~choh/degree/Quantum Optics/quantum... · 2016. 8. 29. · Nonlinear Optics Lab.Hanyang Univ. Chapter 3. Classical Theory

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