chapter 8. semiclassical radiation theoryoptics.hanyang.ac.kr/~choh/degree/quantum...

Nonlinear Optics Lab. Hanyang Univ.

Chapter 8. Semiclassical Radiation Theory

8.1 Introduction

Semiclassical theory of light-matter interaction (Ch. 6-7)

- Ignores the quantum-mechanical nature (ex : quantum fluctuations) of the EM field

- Treats the matter quantum-mechanically through the Schrodinger equation

=> Semiclassical theory is not perfect to describe fully the light-matter interactions

(ex : spontaneous emission), but successful to describe all of atomic radiation

when the number of photons are much larger than unity.

In this chapter, we will complete our development of the semiclassical theory ;

Maxwell-Bloch equation.


8.2 Optical Bloch Equation

: An equivalent set of vector equations of the density-matrix equations ;

)(2

A 21

*

12222111111 i

)(2

)A( 21

*

122221222 i

)(2

)( 1122

*

1212

ii

)(2

)( 11222121

ii

(6.5.14)

(6.5.17)

)(2

1122

*

1212

ii

)(2

11222121

ii

)(2

21

*

1211 i

)(2

21

*

1222 i

(6.5.2) : without relaxation

: with relaxation


For two-level atomic system, 12211

Define,

1221 u

)( 1221 iv

1122 w

(8.2.1)

1) In the case of no relaxation

(6.5.2) => Δvdt

du

χwΔudt

dv

χvdt

dw

(8.2.2)


Consider a fictitious space with unit vectors,

and define a “coherence vector” (Pseudo spin, Bloch vector),

and a “torque vector” (Axis vector),

3̂,2̂,1̂

wvu 321S ˆˆˆ

S

Q

31Q ˆˆ

(8.2.2) =>

SQS

dt

d(8.2.5)


※ QThe effect of is only to rotates about the direction of

It cannot lengthen or shortenS Q

S

※ 0)(222

SQSS

Sdt

d

dt

dS

2

111122

2

221221

2

1122

2

1221

2

1221

2222

24

)()()(

wvuS

11)(

)(2)(4

22*

11

*

22

2*

11

*

11

*

22

2*

22

*

21

*

12

cccc

cccccccccccc

: Conservation of probability in the two-level atom.

(in the absence of collision or relaxation)

※ 1122 w : Degree of inversion

)1,0(1

)0,1(1

1122

1122

w

w : population is entirely in the upper level

: population is entirely in the lower level


Ex) const. ,resonance)(at 0

1Q ˆ : Bloch vector rotates about axis 1̂

1̂2̂

3̂

cos

sin

w

v

(8.2.2c) => dt

dθ

sinsin

dt

dt

tw

tv

cos

sin

1122

2211 1

w

2

1,

2

12211

ww

)cos1(2

1

)cos1(2

1

22

11

t

t

: Same form to (6.3.18)

when 0


Ex) const. ,resonance)(at 0

t

dttt0

'')()( : “area” of the pulse

)()()ˆ()(

00 tEtEet

εr21

where,

[ pulse]

If , external wave inverts the atomic population from the lower to the upper level.


If E0 is time-dependent, ※

1) The rotating wave approximation in going from (6.3.13) to (6.3.14) fails

if itself contributes rapid temporal variation.

=> We must assume that E0(t) is a slow varying time function.

2) If not just the amplitude but the phase of E0 changes in time, we can no longer

assume that an adjustment of the wave function phase will make (t) real.

=> , E should be complex.

[Remarks]


2) In the case considering the relaxation processes

(6.5.14), (6.5.17) =>

wiivui

wii

iiviu

))((

)(2

)()(22

21

1122212112211221

}{222 21

*

1222212221221122 iAw

)( 21 ivu

)put,( 212

ivuivuw

211222 ,,2

1 By definitions,


)]()([2

)1(1 *

1

ivuivui

wT

w

where,

12

2121

1

2

111

1

TT

AT

: longitudinal lifetime of spin

: transverse lifetime of spin

)2( 12 TT

2̂

3̂

1̂

1T

2T


8.3 Maxwell-Bloch Equations

Atomic state under the influence of an external perturbation (EM wave or light)

can be described by Schrodinger equation. But, in order to describe the atomic state

exactly, we need additional equation to express the behavior of the light by the

Interaction with the atoms. = Maxwell equation !

Assumptions :)(),(ˆ),(1) kztietzt εrE : monochromatic, plane wave propagating along the z-direction

; r,both in field varyingSlow 2) t

,2

2

zk

z

,

kz

t

; varyingslow also ison Polarizati 3) )(

2112 ),(2),( kztietzNetz rP

,2121

t tt

21

2

21

2


Maxwell wave equation :

),(),( 21

*

0

tzNik

tzctz

),(

1),(

12

2

2

0

2

2

22

2

tztc

tztcz

PE

)ˆ( *

12

*εr ewhere,

)]()([2

)1(1

))((

)(2

),(

*

1

*

0

2

ivuivui

wT

w

wiivuiviu

ivuNik

tzctz

ivu21

(8.2.18) =>Maxwell-Bloch Equations


8.4 Linear Absorption and Amplification

In practice, there may be background atoms in laser active medium,

and we should add the background atom effect to the Maxwell equation, (8.3.6).

),(21

*

0

tzNik

(8.3.1) : background atom effect

Background-atoms are far from resonance and come to steady state extremely quickly,

so we can use the adiabatic result (7.2.1) for 21. And, these atoms are at most only slightly

excited, so that 1,0 1122

)(2/

112221

i

i(7.2.1)

2221

))(2/(2/

ii

i

i/εwhere,

ia

i

c

NN

ik

2

1

2 22

0

*

21

*

0 where,

22

0

2

c

Na

22

0

2

)/(

c

Na


(8.3.6) =>

),(),(22

21

*

0

tzNik

tzct

ia

z

[Quasi-steady solution]

0)(2

z

i

z

0,0 1212

ct

))((2

1

22211

iaia

iwhere,

022

**

i

z

i

z0

2

zzzz

**

2||

I|| 2 II

z

ze I(0)I(z)

where, )( 2211 aa


1) If the resonant atoms are all in their ground state, 11~1, 22~0

field theofn attenuatio :0 aa

2) If the resonant atoms are all in their excited state, 11~0, 22~1

)0,1(& aaaa

field theofon Aplificati negative large a becan :

* Threshold condition for amplification ; 22 - 11 aa /


8.5 Semiclassical Laser Theory

Electric field in a laser cavity ;

ti

mmmm zekt sin)(ε̂E

Lmkm /

),(),( tzEtzm

mE where,

Polarization of m-th mode ;

ti

m

m

m zektzNe sin),(2 )(

2112rP

Maxwell wave equation considering cavity loss ;

zktzNi

zktt

i

m

m

mmm

sin),(

sin)(2

)(

)(

21

*

0

0

Actually, different cavity modes are coupled through the z-dependence of m21z,t,

but in many lasers this coupling is not important.

=> Take the average value of 21t) instead of individual m21z,t).


)()(2

)( 21

*

00

tNi

tt

i mm

For quasisteady-state lase operation, and021

)(2/

112221

i

i(7.2.1) : adiabatic approximation

)]()()[(2

)(2/

2)(

112222

0

2

1122

*

00

tttiN

i

iN

i

ti

m

mmm


mmm igcit

)()(2

2

1

0

,))(()( 121222

0

2

NNNNc

g

where, g

21

)()(),()( 222111 tNtNtNtN

: Fundamental equations of semiclassical laser theory

)()( 12221111 NNNANN m

)()()( 1222122 NNNAN m


In steady state, 0)( tm

)(22

)(0

ig

ci m

gain) (threshold/ 0cg

)(2

21

gc

m

)(2

21

2

21

21

mm

m

gc

gcgc

pulling) (frequency

Cf) section 3.5 : a positive sign of g is now possible !※


[Einstein laser model]

mmmmmmc

gc

*

0

** )(1

2

0

2

12

2

))(( mm

mNNc

dt

d

(8.5.13) =>

(8.5.12) => Add a pumping term, K,

and Assume N2>>N1 condition is maintained by the pumping :

KNNAdt

dN 22212

2 )()(

Define atom number : , photon number :VNn 22 Vc

I

c

VVq

20 ||

2/

qqnV

c

dt

dq

0

2

)(

KVnAqnV

c

dt

dn

22122

2 )()(

Results of (1.5.1), (1.5.2)

Einstein laser model

chapter 8. semiclassical radiation theoryoptics.hanyang.ac.kr/~choh/degree/quantum...

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