Chapter 8. Semiclassical Radiation Theory

Nonlinear Optics Lab

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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.

Nonlinear Optics Lab

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8.2 Optical Bloch Equation

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

) 2(

A_{21 22 12} ^* ₂₁

11 1

11     

    i 



) 2(

) A

( _{2 21 22 12} ^* ₂₁

22    

      i 



) 2 (

)

( _{22 11}

* 12

12     

   i i  ) 2 (

)

( _{21 22 11}

21     

  i i 

(6.5.14)

(6.5.17)

) 2 ( ^{22 11}

* 12

12    

 i i 

) 2 ( ^{22 11}

21

21    

  i i  )

2( ²¹

* 12

11   

  i 



)

2( ²¹

* 12

22   

  i 



(6.5.2) : without relaxation

: with relaxation

Nonlinear Optics Lab

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For two-level atomic system, 1122 ¹

Define,

12

21 

  u 

) (₂₁ ₁₂

 i v

11

22 

 

 w

(8.2.1)

1) In the case of no relaxation

(6.5.2) => Δv dt

du 

χw dt Δu

dv  

dt χv dw  

(8.2.2)

Nonlinear Optics Lab

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Consider a fictitious space with unit vectors,

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

and a “torque vector” (Axis vector),

3ˆ , 2ˆ , 1ˆ

w v

u 2 3 1

S  ˆ  ˆ  ˆ

S

Q







 1 3 Q ˆ ˆ

(8.2.2) =>

S S Q

 dt 

d

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※ The effect of is only to rotates about the direction ofQ It cannot lengthen or shorten

S ^Q

S

※ 2 2 ( ) 0

2   S  S QS 

S dt d dt

dS

2 11 11 22 2

22 12

21

2 11 22 2

12 21 2

12 21 2

2 2 2

2 4

) (

) (

) (

















































 u v w S

1 1

) (

) (

2 )

( 4

2 2

* 1 1

* 2 2

2

* 1 1

* 1 1

* 2 2 2

* 2 2

* 2 1

* 1 2

















c c c

c

c c c

c c c c

c c

c c c

: Conservation of probability in the two-level atom.

(in the absence of collision or relaxation)

※

w  

 



















) 1 ,

0 (

1

) 0 ,

1 (

1

11 22

11 22







 w

w : population is entirely in the upper level

: population is entirely in the lower level

Nonlinear Optics Lab

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Ex)   0 (at resonance) ,   const.

 1 Q   ˆ

 : Bloch vector rotates about axis 1ˆ

1ˆ 2ˆ

3ˆ





 cos sin







 w v

(8.2.2c) =>   

dt

 dθ

 

 sin

sin 

dt

d     t

t w

t v



 cos sin









11 22

22

11 1

















w      

2 , 1

2 1

22 11

w w 



) cos 1

2 ( 1

) cos 1

2 ( 1

22 11

t t

















: Same form to (6.3.18) when  0

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Ex) 0(at resonance) , const.



 ^t t dt

t 0

' ') ( )

( 

 : “area” of the pulse





) ( )

( ˆ) ) (

( e E⁰ t E⁰ t

t 

  r²¹ε  where,

[ pulse]



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

Nonlinear Optics Lab

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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]

Nonlinear Optics Lab

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2) In the case considering the relaxation processes

(6.5.14), (6.5.17) =>

w i iv u i

w i i

i i

v i u













































































) )(

(

) (

2

) (

) (

2 2

21

11 22

21 21

12 21

12

21    







} {

2 2

2 _{22 21 22 21 22 12} ^* ₂₁

11

22       

        

 A i

w   

) ( ₂₁ u iv

) put,

( ₂ ₂₁

iv u iv

w u







 

 _{12 21}

22 , ,

2

1  

 By definitions,

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)]

( )

( 2[ ) 1

1 ( _*

1

iv u iv

i u T w

w       

 ^  

where,

1 2

21 21

1

2 1 1

1 1

T T

T A













 

: longitudinal lifetime of spin

: transverse lifetime of spin

) 2 (T²  T₁

2ˆ 3ˆ

1ˆ

T1

T2

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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)^{E r} t ^ε



; r, both in

field varying

Slow

2) t

2 ,

2

k z

z 

 





 















 t

; varying slow

also is

on Polarizati

3) ^{( )}

21

12 ( , )

2 )

,

(z t  Ner  z t e^{i tkz} P

21 ,

21  

 _





t t t

 



 ₂₁

2 21

2  

Nonlinear Optics Lab

.

Maxwell wave equation :

) , ( )

,

( ^* ₂₁

0

t z ik N

t ct z

z  





 







 



  )

, 1 (

) , 1 (

2 2 2 0 2

2 2 2 2

t t z

t c t z

c

z E P



 



 







 







ˆ ) ( ₁₂ ^*

*  e r ε



where,

















































 



 







 





)]

( )

( 2[ ) 1 1(

) )(

(

) 2 (

) , (

* 1

* 0

2

iv u iv

i u T w

w

w i iv u i

v i u

iv u ik N

t ct z

z









 













u iv

21

(8.2.18) =>

Maxwell-Bloch Equations

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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

t z ik N  

 (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 _220, _111

) 2(

/

11 22

21  



  





 

i (7.2.1) i

2 21 2

) )(

2 / ( 2

/









 







 



 

 i i

i

i _where, ε/

  















 

 

i a

i c

N N ik













 



2 1

2 ₀ ^{2 2}

* 21

*

0  ^where, ₀ ^{2 2}

2













 

c a N



2 2 0

2

) /

(  



 

  

 



 c

a N



Nonlinear Optics Lab

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(8.3.6) =>

) , ( )

, 2 (

2 ²¹

* 0

t z ik N

t ct z

i a

z  



 _





 







 



 



[Quasi-steady solution]

0 )

2  ( 



 



 

 

  i z

z ^{ }



0 ,

0 ₁₂ ₁₂

 





ct





2 1

2    ¹¹ ²²

  _{    }

i a i

i a where,

2 0 2

*

*  



 



 

 

 



 



 

 

 

 



z i

z 0

2



 

 



  ^



z

z z

z 

 



 



||² _*   ^*

I

|

| 



^



  z

e



 I(z) I(0)

where, aa(₁₁₂₂)

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

the of n attenuatio

:

0





  a a

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

) 0 ,

1 (

&   





  a a a a 

field the

of on Aplificati negative

large a

be can

:  

* Threshold condition for amplification ; ₂₂ - ₁₁  a /a

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8.5 Semiclassical Laser Theory

Electric field in a laser cavity ;

t i m m

m

m^ˆε



L m

k_m   /

) , ( )

,

(z t E z t

m





E where,

Polarization of m-th mode ;

t i m m

m 2Ner₁₂₂₁^{( )}(z,t)sin k ze^ P

Maxwell wave equation considering cavity loss ;

z k t

z i N

z k t t

i

m m

m m

m

sin ) , ( sin ) 2 (

) (

) ( 21

* 0

0



 





 









 







 







Actually, different cavity modes are coupled through the z-dependence of ^m₂₁z,t, but in many lasers this coupling is not important.

=> Take the average value of ₂₁t) instead of individual ^m₂₁z,t).

Nonlinear Optics Lab

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) ( )

2 ( )

( ^* ₂₁

0 0

t i N

t t

i _{m m}  



 



 

  



 

















For quasisteady-state lase operation, and^₂₁0

) 2(

/

11 22

21  



  





 

i

i (7.2.1) : adiabatic approximation

)]

( )

( )[

2 (

) 2(

/ ) 2

(

11 2 22

2 0

2

11 22

* 0

0

t t

i t N

i N i

i i t

m m m

m





 











 

 







 





 





 

 



 



 







 











Nonlinear Optics Lab

.

m m

m i c g i

t







 



    

 

  2 ( ) ( )

2 1

0

, ) )(

( )

( _{2 1 2 1}

2 2 0

2

N N N

c N

g    



  











where,  g





^21

) ( )

( ), ( )

( _{11 2 22}

1 t N t N t N t

N    

: Fundamental equations of semiclassical laser theory )

( )

( _{2 1}

2 21 1

1

1 N A N N N

N^   



) ( )

( _{2 21 2 2 1}

2 A N N N

N^      _m 

Nonlinear Optics Lab

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In steady state,



) 2(

) 2 (

0

 

 

 c g i

i  _m   





gain) (threshold

/ ₀c g 



) 2 ( ²¹

 

   gc 

m

) 2 (

1 2 2

21 21

m m

m

gc

gc gc



 



 

 











 



 



 



 





pulling) (frequency

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

※

Nonlinear Optics Lab

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[Einstein laser model]

m m m

m m

m g c

c

     



 _*

0

*

* )

1(



 



 









2

0 2 1

2 2

) )(

( _{m m}

m c N N

dt

d

  



   



 (8.5.13) =>

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

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

N N

dt A

dN² (₂  ₂₁) ₂ () ₂ 

Define atom number : n₂N₂V, photon number : V c I c

V V

q  







  



 ⁰/2| |²

q q

V n c dt

dq

0 2

) (









KV n

A V qn

c dt

dn²  () ₂ (₂  ₂₁) ₂ 

 Results of (1.5.1), (1.5.2)

Einstein laser model