chapter 7. emission and absorption and rate equations

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.

Chapter 7. Emission and Absorption and Rate Equations

7.1 IntroductionFor most considerations (total relaxation

rate is much faster than the rate at which external forces cause electron to jump between atomicenergy levels.

)A(2

112121

(6.5.18)

The result of the external force,

F=-eE is only to produce a gradual

increase or decrease in probability.

)A(2

112121

(6.5.19)

Therefore, such a fast phenomena can often be

treated with sufficient accuracy in an average sense. Absorption rate / Stimulated emission rate

(Rates of increase and decrease in probability)


7.2 Stimulated Absorption and Emission Rates

When )A(2

12121

quasisteady-state approximation, 02112

)(2

A 21*

12222111111 i

)(2

)A( 21*

122221222 i

)(2

)( 1122

*

1212 ii

)(2

)( 11222121 ii

(6.5.14)

(6.5.17)

In Chapter 6, density matrix equations considering the relaxation effects are given by

is possible ;

(6.5.17) => )(2/

1122

*

12

i

i

)(2/

112221

i

i: Adiabatic solution(7.2.1)

(condition for adiabatic following to occur),

2112 , adiabatically follow the inversion 1122,

※


)( 112222

2

21*

12

i

(6.5.14) =>)(

2/112222

2

222111111

A

)(2/

)( 112222

2

2221222

A

: Population rate equation(7.2.2)

1122 , are coupled only to each other

※

<Stimulated Absorption and Emission Rates>

Stimulated absorption or emission rates

2221

2

0

)(

2//ˆrateAbsorption

Ee 21r

2221

2

0

)(

2//ˆrateemissionStimulated

Eεe 21r

1) For nondegenerated transitions,

(7.2.3)


Calculation of ||2

In many cases (unpolarized radiation, rotational or collisional disorientation, etc), orientational average of |is simpler and useful

2

022

012

2

0norientatio

2

0

3

1

3

1

ˆ

EDEe

EEe

221

21

rr

r

DεrD 21 ˆand e

)(2

12

2

12

2

122

1222 zyxeeD 221 rrD

(7.2.4)

where, : Complex dipole moment and its projection on ε̂

In terms of cartesian components,

Homework : Problem 7.1

22

2

02

2221

2

012 6)(

2//

EDE

R

Induced transition rates (abbreviation), : 12R

(orientation-averaged)(7.2.7)


2) For degenerated transitions (Homework : Refer to Appen. 7.A)

1

2 g2=5

g1=3

Absorption cross section, : [Refer to (7.4.2)]

12

200

12abs

2/fluxenergy radiation incident

1 levelin atomper rate absorptionenergy R

Ec

R

abs12 R

where,

(7.2.8)

: photon flux

<Example of degenerated transition>

21 ,

212

21 ),(1

mm

mmAg

A

21 ,

211

12 ),(1

mm

mmRg

B

21 ,

212

21 ),(1

mm

mmRg

B

: In the case of natural excitation (the # of atoms in each of the different degenerated states of the same level are equal) ;

22221111 /,/ gPgP


7.3 Population Rate Equations

Densities of atoms in levels 1 and 2 ;

222111 , NNNN (7.3.1)

where, N : total density of atoms

(7.2.2) => )( 12221111 NNNANN

)( 12221222 NNNANN (7.3.2)

※ ??0221121 NNNN

: This indicates that ineleastic collisions will takes all of the atoms out of levels 1 and 2 into other atomic levels. Nevertheless, are practically small relative to , and the intermediate time behavior is of the most interest. => We can ignore the

21, ,21A

21,

2

1

2

1


constant21 NNN

NNA

NNNAN

221

22212

)2(

)2(

22)0()(

21

)2(

2122

21

A

Ne

A

NNtN tA

(No inelastic collision), then 21 NNN

Sol)

(7.3.7)

)( 122211 NNNAN

)( 122212 NNNAN (7.3.4)


Examples)

1) No radiation field ; 0

)decaysspontaneou()0()( 2122

tAeNtN

2) Weak radiation field ;21A

, 0)0(2 N

, 0)0(2 N

)excitaionweak(1)( 21

212

tAeA

NtN

3) Strong radiation field ;21A , 0)0(2 N

NA

NNt

212,

: Lorentz classical theory is valid.※

)statesteadysaturated(222

)0()( 222

NNe

NNtN t


NN )0(20)0(2 N

[Fig. 7.1]


Power Broadening

In the limit, 1)2( 21 tA

(7.3.7) =>

2)(

212 A

NN

)statesteady(/

2/

/)(21

/)()(

21222

212

21

212

A

A

A

A

N

N

2/1

21

2/1

21

22

2/1

)0(21

A

A

Half width ;


7.4 Absorption Cross Section and the Einstein B Coefficient

abs12 R (7.2.8), )(2/200 IEc

(7.2.7) => 22210

2

)(3

c

D

)(6

)()2()()2(

2

3

0

2

221

2221

221

0

2

Lc

D

c

D

212Put,

221

221

21

)()(

/)(

Lwhere, : Lorentzian line shape function

)(S : generaliztion for arbitrary line shape function


Examples)

1) For descrete radiation frequencies

)(S

21

0

hI

h

IS

c

DR

)(

6 0

2

12

For a single frequency, :00

00

0

2

12

)(

6

h

IS

c

DR

(narrow band limit)2) For continuous band radiation

)(S

)(I

dh

Sc

DR

00

2

12

)()(

6

cI

21

If 021 )()(, 21 S

)()(6 21212

0

2

12

B

DR

: Einstein’s empirical definition

20

2

6 D

B (broad band limit)

0012 )(1

IBSc

R


7.5 Strong Fields and Saturation

What is the criterion for “strong” field ? => Saturation the population ; !2/2 NN

This criterion is satisfied in (7.3.13), if 2221

221 /or 2/ AA

Define,

intensity saturation : 2/ I

flux (photon) saturation : 2/

21sat

sat

21sat

A

A

ex) 1821

210 s10~ ,resonance)(on cm10~ A2

sat218sat W/cm0.3I and s,photons/cm 10


7.6 Spontaneous Emission and the Einstein’s A Coefficient

An atom in an excited state will eventually drop to a state of lower energy, even in the absence of any field or other atoms. => Spontaneous emission(※ Spontaneous emission would occur even for a single excited atom in a perfect vacuum !)

ex) Luminescence, Fluorescence, Phosphorescence

tAeNtN 21)0()( 22

(7.3.8) =>

212 /1 A : characteristic time constant (excited state “life time’)

- In the case that there are multi-channels for radiative transition,

,22 m

mAAmm AA 22

2

11


Quantum mechanical expression ;

222

31

3

32

0

2

pol3

32

0

4224

1

sinˆ24

1

nmnmnmnm

kkkknmnm

nm

zyxc

e

ddc

eA

εr

30

32

3

32

0 33

4

4

1

c

D

c

DA nmnmnmnmnm

(7.6.4)

(7.6.5)

2222nmmnnmnm eeD rrr where,

Line shape for the spontaneous emission : Lorentzian

2rad

20

rad

)(

/)(

S

where, )(4

112rad AA

: Natural linewidth

(7.6.7)

Homework : Problem 7.3

chapter 7. emission and absorption and rate equations

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