Laser spectroscopy

Nonlinear Optics Lab . Hanyang Univ.

Laser spectroscopy

- Basic concepts and instrumentation -

2

enlarged edition

Wolfgang Demtröder

Nonlinear Optics Lab . Hanyang Univ.

Spectroscopy(분광학) :

To analyze the characteristics of EM radiation (light) interacting with matters  Absorption/Emission spectra

Spectroscopic infomations :

 Wavelength : energy levels of atomic system

 Line intensity : transition probability

 Natural linewidth : mean lifetime of excited states

 Doppler width : velocity distribution (temperature)

 Pressure broadening/shift : collision/interatomic potential

 Zeeman/Stark splitting : magnetic/electric moments

Spectroscopic technique : Sensitivity/Resolution

 Optical instrumentation : light source, detector, optics

 Technique : Linear/Nonlinear, Molecular beam, Time resolved, Fourier, ...

1. Introduction

Nonlinear Optics Lab . Hanyang Univ.

2. Absorption and Emission of Light

2.1 Cavity Modes

Cubic cavity (L, T)

Stationary radiation field : superposition of plane waves

c.c )]

r k t i(ω [ exp A

E

p

p

  

  ^{  }



Boundary condition :

standing waves (cavity modes) ), n

, n , n L (n

k   π

: positive integers

2 3 2

2 2

1

n n

L n

k

π

, π/ k

λ 2

ω

c k λ n n n ,

L

2  



n

n

n

L

ω

πc

(2.1)

(2.2) (2.3)

(2.4)

Nonlinear Optics Lab . Hanyang Univ.

) k eˆ , eˆ δ ;

eˆ eˆ ( eˆ a eˆ a

A









 _{1 1 2 2 1 2 12 1 2}

(and taking into account that EM wave has two possible polarization components)

(2.5)

Number of modes with frequency :

Number of integer set satisfying the condition of



  )

( n

, n

, n

c

k

 ω

 ω

If (radius of sphere, is large enough compared to ),

the number of integer set is roughly given by the volume of the octant of sphere : L  

2  / c  ^{/ L}

3 2

3 3 3

3 1 3

4 8 2 1 )

( π c

L ω πc

π Lω ω

N

 



 



 

Nonlinear Optics Lab . Hanyang Univ.

Spectral mode density : n(  )  N(  )/L

c dν dν πν

ν n or c dω

π dω ω ω

n

2 3

2

2

8

) ( )

(  

examples)

a) visible (=500 nm), : b) microwave (=1 cm), : c) X-ray ((=1 nm), :

GHz 1

d   n(  )d   3  10

m

Hz

0 1

d  

ⁿ⁽  ^)d  ^{ 10}

^m

Hz

0 1

d  

n(  )d   8 . 4  10

m

Nonlinear Optics Lab . Hanyang Univ.

2.2 Thermal radiation and Planck’s Law

In classical thermodynamics, each mode would represent a classical oscillator with mean energy kT.

Therefore from (2.7), spectral energy density of the radiation field is given by

c Tdν πν k n(ν(ν)kT

ρ(ν)dν

8





Rayleigh-Jeans law (IR region)

(2.8)

Planck’s radiation law :

 Each mode can only emit or absorb energy in discrete amounts , which are integer multiples of a minimum energy quantum (photon).

 In thermal equilibrium, the partition of the total energy into the different modes is governed by the Maxwell-Boltzmann distribution :

hν

qhν q

qhνhν/

e /Z q

p ( )  ( 1 )

where, (partition function)

^ 

q

kT /

e

Z

(2.9)

(2.10)

Nonlinear Optics Lab . Hanyang Univ.

Mean energy per mode

1 ) 1

(   

  

q

kT / qh

q

e

e h Z qh

qh q p

W  



Spectral energy density : mode density mean energy per mode 

 

 









d

e h W c

d ) ( n d

)

(

1 8

3 2

 





(Planck’s radiation law)

(2.13)

kT e

 1 h





(Rayleigh-Jeans) (Planck)

Nonlinear Optics Lab . Hanyang Univ.

2.3 Absorption, Induced and Spontaneous Emission

In a two-level system,

transition probabilities per second for (induced) absorption, induced emission, and spontaneous emission

21 21

21 21

12 12

) (

) (

A dt p

d

B ν dt p

d

B ν dt p

d

spont











(2.16)

(2.17)

(induced) absorption

stimulated(induced) emission

spontaneous emission

where, A₁₂, B₁₂, B₂₁ : Einstein A, B coefficients

Nonlinear Optics Lab . Hanyang Univ.

Relations between the coefficients

At thermal equilibrium, the population distribution is given by the Boltzmann distribution

kT / i E i

e

Z N g

N 

where, N : total number density, g_i = 2J_i+1 : degeneracy, and : partition function ^{E /kT}

i i

e i

g Z ^



Steady state solution

) ( ]

) (

[ B

   A

N

 B

N

 

using the relation,

N

/ N

 ( g

/ g

) e

 ( g

/ g

) e

1

21 12 2 1

21 21





kT /

e

B B g

g

B / ) A

( 

 1

8

3 2



 e

h ) c

(  





3 21 3 21

21 1 2 12

8 B

c A h

, g B

B _ g _ 

(2.20)

(2.19)

(2.21, 22)

Nonlinear Optics Lab . Hanyang Univ.

Analysis

 If g₁=g₂, B₁₂=B₂₁

 ^{B h}  n

A

21 21

)

( 

 , where : number density of mode

n (  )  8 

/ c

 If

: The spontaneous emission per mode equals to the induced emission.

: The ratio of the induced- to the spontaneous-emission rate in an arbitrary mode is equal to the number of photons in this mode.

, h c n

h  

 

 8 ( )

)

( ₃

3





3 21 3 21

21

) 8

( A

c B h

B      

: Stimulated emission > spontaneous emission ]

[ ₂₁

1 2 12 1 2

21 B

N B N

N N

A   ( ) [1 ]

1 2 2 1 12

1 2

21 N

N g B g

N N

A    

: If N₂g₁>N₁g₂ (population inversion)  Lasing!!

Nonlinear Optics Lab . Hanyang Univ.

2.6 Transition Probabilities

The intensities of spectral lines depend on the population density of in the absorbing or emitting level and also on the transition probabilities of the corresponding transitions.

 If the probabilities are known,

the population density can be obtained from line intensity measurements.

2.6.1 Lifetimes. Spontaneous and Radiationless Transitions

<Spontaneous emission>

 Spontaneous emission probability :

p

A

dt

d 

 Total Spontaneous emission probability :

^ 

k ik

i

A

A

 The decrease of the population density :

t A i i

i i i

e

N t

N , dt N A

dN   ( ) 

 Mean lifetime :



 1 / A

Nonlinear Optics Lab . Hanyang Univ.

<collision-induced radiationless transition>

coll ik B coll

ik

v N

dt p

d  

<induced(stimulated) emission>

] ) (

[ )

(

ind

ik

B N g / g N

dt p

d    

<Effective lifetime>

] }

) (

{ ) ( 1 [

ik B

σ v N

N g / g N

B

A

k eff ik

i







  ^{ }



(2.43)

Nonlinear Optics Lab . Hanyang Univ.

2.6.2 Semiclassical Description; Basic Equations

 Semiclassical description :

EM wave  classically, Atom  quantum mechanically

<EM wave>

)

0

cos( t kz E

E ^  ^  

Spatial variation of the EM field can be neglected when

  d

(e.g. In the visible region ~500 nm, d~0.5 nm)

0 0

0

0

2

) 1 (

cos t A e e , A E

E

E   

 







 

Nonlinear Optics Lab . Hanyang Univ.

<Atom>

Hamiltonian operator of the atom interacting with light field can be written as a sum of the unperturbed Hamiltonian of the free atom plus the perturbation operator :

V H

H 



Dipole approximation for the perturbation operator :

t E

p E p

V    

cos 









Time-dependent Schrodinger equation :

t H i



  

 

General solution :   E t/^ n

n n

e

r u t c t

,

r





 ( ) ( ) )

(

1



For our two-level system, ^{ }



b /

t iE a

b

a

b t u e

e u t a t

,

r )  ( )

 ( )

 (

b(t) ), (t a

where, : time-dependent probability amplitudes

E

E



,

k 

Nonlinear Optics Lab . Hanyang Univ.

Substituting (2.52), (2.45) into (2.47), and using the relation, gives (2.53)







 



a

t u

e

i b t u

e

aVu

e

bVu

e

i

n n

n E u

u H₀ 

Multiplication with u^*n ^{(n=a, b)},and spatial integration ;

] )

( )

( )[

/ -(

) (

] )

( )

( )[

/ -(

) (





 

 

/ t ) E E ( i ba bb

/ t ) E E ( i ab aa

b a

b a

e V t a V

t b i

t b

e V t b V

t a i

t a













 (2.54a)

(2.54b) where,

V

u

Vu

d e E

u

r

u

d E

D













 ^  ^









D e u r u d  D

(dipole matrix element)

(2.55a) (2.55b)

cf) Expectation value of the dipole moment under the influence of the EM field,

ab t

i

* t

i

* ab

*

r d D ( a be ab e ) D

e

D













 ^{  }

has odd parity ! r

ab a

b

ba (E E )/ 

    

Nonlinear Optics Lab . Hanyang Univ.

Put,

ba ab

ab

ab

D E / D A / R

R        

 



0

0

2

) ( ) e

(e /2)

( ) (

) ( ) e

(e /2)

( ) (

ab ab

ab ab

( - (

ab

( (

ab

t a R

i t

b

t b R

i t

a

t ) i

t ) i

t ) i

t ) i









































(2.57)

(2.54) 

(Rabi frequency)

(2.58a) (2.58b)

Nonlinear Optics Lab . Hanyang Univ.

2.6.3 Weak-Field Approximation

 Assume the field amplitude to be sufficiently small so that for time t<T(maximum interaction time) the population of upper level remains small compared with that of lower level, i.e., ^{b(t)2 1}

From (2.58) with the initial condition of a(0)=1, b(0)=0,

) e

(e /2)

( ) (

0 ) (

ab

ab -i(

i(

ab

t ) t

R )

i t b

t a







   







 (2.59a)

(2.59b)



1) e

1 )(e

( 2 ) (

1 ) 0 ( ) (

ab

ab -i(

i(



















 



 









ba t )

ba t )

Rab

t b

a t

a _(2.60a)

(2.60b)

can be neglected (rotating-wave approximation)

ba

ba  

  

, and second term oscillates fast enough to be vanished out in time averaged measurement.

Nonlinear Optics Lab . Hanyang Univ.

Probability that the system is at time t in the upper level :

2 2 2

2 ) (

2 ) sin(

) 2

( 



 







 



 



 

/ t

/ t t R

b

ba ba

ab

 





At the resonance,   _ba

2 2 2

) 2

( R t

t

b ^ab

ba 



 









valid in a time,

0 2

2 1 or

E T D

t t

R

ab ab

 





(2.62)

(2.63)

Nonlinear Optics Lab . Hanyang Univ.

2.6.4 Transition Probabilities with Broad-Band Excitation







 







 

 







 

 

 d

/ t

/ t d D

t b t

p

ba ba ab

ab

2

2 0 2 2

2 ) (

2 ) ) sin(

2 ( ) ) (

( )

( 

t / d

t / t

ab ba

ba    







 

 ( )2

2 ) (

2 ) ) sin(

(

2

 



 









For broad-band light source,

because (_ab) is slowly varying over the absorption-line profile ;

t D

t

p

0



 



 



(2.64)

(2.65)

, E //

D_ab ₀ If

See, Fig. 2.18(c)

Nonlinear Optics Lab . Hanyang Univ.

In general, when the level k and i are degenerated with the degeneracies g_k, g_i, respectively

ik i g

m g

n

k i i

ik

S

D g g

B e

i k

n

m 2

1 1 0

2 2

0 2

3 1

3   





 _

 

 

Einstein B coefficient

From (2.65), with considering that

the averaged component of the square of dipole moment for isotropic radiation: p_z²  p² cos²  p² /3 )

3 ( )

( ₂ ²

0

ab ab

ab t D

dt p

d  





 

) ab

( )

(t B

dt p d

ab

ab   

From (2.16)



0 2

3 







d u r e u

B

^ _ 

^

(2.66)

S

Nonlinear Optics Lab . Hanyang Univ.

2.6.5 Phenomenological Inclusion of Decay Phenomena

Decay phenomena (spontaneous emission, collision-induced relaxation) can be treated by adding phenomenological decay terms to (2.58) as follows ;

) ( e

/2) ( b(t) )

2 1 ( )

(

) ( e

/2) ( a(t) )

2 1 ( )

(

ab ab

( ab

( - ab

t a R

i /

t b

t b R

i /

t a

t ) i

b

t ) i

a



































(2.70b)

Weak-field Approximation :









2 2

2 2 2

2 ) (1 ) 2 (

) 1 ( )

( ) (

ab ba

t ab ab ab

t R b e ,

t b

p ^ab





 



 ^ _(2.70c)

Nonlinear Optics Lab . Hanyang Univ.

Dispersion and Absorption

After taking the second time derivative of (2.56b) and using (2.70), the equation of motion for the dipole moment D under the influence of a radiation field

) cos(

)

( ^  ^  ₀  



 D a be^ ab e D t

D _ab ^{* i bat * i bat} _ba (2.56b)

] )sin 2 (

)cos )[(

( ) 4

(

/ R

t / t

D

D    

  

 



     



Assume a close-to-resonance condition and weak damping : (_ba )2, _ab _ba

t

D t D

D 

cos  

sin 

2 2

ab 2

2 2

ab 1

) 2 (

) (

(1/2)R

) , 2 (

) (

) (

R

D / D /

ab ba

ab ab ba

ba





















 





 

(Absorption)

(2.73) (2.74a)

(2.74b)

Nonlinear Optics Lab . Hanyang Univ.

2.6.6 Interaction with Strong Fields

From (2.58), in the rotating wave approximation :

) ( e

/2) ( ) (

) ( e

/2) ( ) (

) (

ab

) ( ab

ab ab

t a R

i t

b

t b R

i t

a

t i

t i

























(2.75b)

) e e

( )e

/ 2 ( ) (

e e

) (

2 ab 1

2 1

2 2 1

1 ) (

1 1

t i t

i t

i ab

t i t

i

C C

R t

b

C C

t a















 









General solutions:

(2.77a) (2.77b)

where, ₁₂ ( )^{2 2}

2 ) 1 2(

1

ab ba

ba

,       R



2 1

1 2

2 1

2

1  









 

 

 ,C

C

(2.76)

Nonlinear Optics Lab . Hanyang Univ.

Probability amplitude :

t/2) sin(

)e / ( )

( t  i R





b

(2.80)

where, ^{2 2}

2

1

  (

 )  R



    

(2.78)

Probabilities :

t/2) (

sin ) / ( - 1 )

( 1 )

(

t/2) (

sin ) / ( )

(

2 2 2

2

2 2 2

















ab ab

R t

b t

a

R t

b

(2.81)

cf) Rabi frequency : Rabi-flopping frequency at the resonance,

  

See, Fig. 2.18(b)

Nonlinear Optics Lab . Hanyang Univ.

At the resonance,

  

) /2 (

sin )

(

) /2 (

cos )

(

0 2 2

0 2 2

 



 



t E D t

b

t E D t

a

ab ab







 (2.82a)

(2.82b)

After the time,

T   / ( D 

 E 

)   / R



the population probability of the initial system 1

) ( 0 ) 0 ( 0

) ( 1 ) 0

( ²  , b t ²   a ²  , b t ²  a

(-pulse)

has been inverted ;

Nonlinear Optics Lab . Hanyang Univ.

Now include the damping terms,

2 2

2

2 )

2 ( 2 2

) 2 ( ) (

] ) 2 (

) [ (

ab ab

t / ab

R /

t / sin e

t R b

ab







 ^ 









(2.84)

(2.85)

2 2 2

1 2 2

1 2

2 1

ab ba

ab ba

, i i  R



 



  





 



  



      



where, _ab _a _b,  _a _b

where, Rabi-flopping frequency :

2 2

2

1

2

1

ab

ba

  R



 



  







     

See, Fig. 2.19

Nonlinear Optics Lab . Hanyang Univ.

For the case of a Closed two-level system

) ( 2 e

) 2 (

) 1 2 (

) 1 (

), ( 2 e

) 2 (

) 1 2 (

) 1 (

) ( ab

) ( ab

ab ab

t a i R

t a t

b t

b

t b i R

t b t

a t

a

t i

a b

t i

b a











































 _(2.87a)

(2.87b)

ab ab

ab i , R

R 2

1 2

2 1

2

1     

 , and ₂ ^{a b} ₂

2 2

) 2 (

2 ) 1

(

R /

t R b

ab ab

ab









 



 For the resonance case,

  

(2.88)

Special case, a b

2 ) 1

(

t

b

: impossible the population inversion!!