Chapter 10. Laser Oscillation : Gain and Threshold

Nonlinear Optics Lab . Hanyang Univ.

Chapter 10. Laser Oscillation : Gain and Threshold

Detailed description of laser oscillation

10.2 Gain

<Plane wave propagating in vacuum>

Consider a quasi-monochromatic plane wave of frequency n propagating in the +z direction ;

z A

z z+Dz

u_n

The rate at which electromagnetic energy passes through a plane of cross-sectional area A at z is _In(z)A

The flux difference ;

z A z I

A I z z

I D



 

 D

 ) ] ( ) (

[ _{n n n}

Nonlinear Optics Lab . Hanyang Univ.

This difference gives the rate at which electromagnetic energy leaves the volume ADz,

z A z I

z A

t u D



 

 D

 

 ( _n ) ( _n )

1  0



 





n

n I

I z t

 c

I c

u_{n n}/ : Equation of continuity

<Plane wave propagating in medium>

The change of electromagnetic energy due to the medium should be considered.

=> The rate of change of upper(lower)-state population density due to both absorption and stimulated emission.

(7.3.4a) => energy flux added to the field : (n)(N₂N₁)hn(n)I_n(N₂N₁) )

( ) 1 (

1

2 N

N I z I

t

c   



 







 



  _n

 n

Nonlinear Optics Lab . Hanyang Univ.

(7.4.19) : (n)(hn/c)BS(n)

n n

n n n

n



 n

 n n n n

I g N

N g

I S g N

N g A

I S g N

N g c B

h

I S N N c B

I h z t c



 



 





 



 





 



 





 



 







 





1 1 2 2

1 1 2 2 2

1 1 2 2

1 2

) (

) 8 (

) ( ) ( ) 1 (

(no degeneracies)

(degeneracies included)

Define, gain coefficient, g(n)



 



 





 



 



1 1 2 2

1 1 2 2 2

) (

) 8 (

) (

g N N g

S g N

N g g A

n



 n n 

Nonlinear Optics Lab . Hanyang Univ.

n n

n g n I

t I c z

I 1 ( )

 

 



 

Temporal steady state :  0





t ⁿ

n g

n

dz

dI  ( )



z

e

I z

I

( ) 

( 0 )



* g>0 when ₁

2 1

2 N

g

N g : population inversion



* If 1^,

2 1

2 N

g N g

) ( )

(

) 8 (

) (

1 2

1 2 2

1 2 1

1 2 1

1 2 2

n n



 n n 

g a Ng

g NS g g A

g N N g

g N g

g N g























The gain coefficient is identical, except for its sign, to the absorption coefficient.

Nonlinear Optics Lab . Hanyang Univ.

10.3 Feedback

In practice, g~0.01 cm^-1 if the length of active medium is 1 m.

=> A spontaneously emitted photon at one end of the active medium leads to a total of e^{0.01 x 100}=e¹=2.72 photons emerging at the other end.

=> The output of such a laser is obviously not very impressive.

=> Reflective mirrors at the ends of the active medium : Feedback.

10.4 Threshold

In a laser there is not only an increase in the number of cavity photons because of stimulated emission, but also a decrease because of loss effects.

(Loss effects : scattering, absorption, diffraction, output coupling)

In order to sustain laser oscillation the stimulated amplification must be sufficient to overcome the losses.

Nonlinear Optics Lab . Hanyang Univ.

<Threshold condition>

Absorption and scattering within the gain medium is quite small compared with the loss occurring at the mirrors of the laser. => Consider only the losses associated with the mirrors.

 1



 t s r

where, r : reflection coefficient t : transmission coefficient s : fractional loss

At the mirror at z=L and 0 ; ) ( )

( ₂ ^{( )}

)

( L r I L

I_n^  _n^ ) 0 ( )

0

( ₁ ^{( )}

)

(  _n

n r I

I

Nonlinear Optics Lab . Hanyang Univ.

In steady state (or CW operation), g(n) : constant

) ( )

(

)

( ^

  _n

n g n I

dz

dI ( )

) (

)

( ^

   _n

n g n I

dz dI

z

eg

I z

I_n^()( ) _n^()(0) ⁽ⁿ⁾



) )(

( )

( )

( (z) I (L)e^{g L z} I_n^  _n^{ n }

Amplification condition : after one round trip must be higher than the initialI_n^()(0) I_n^()(0)

) 0 ( )

0 ( ]

[

] )

0 ( [

)]

( [

)]

( [

) 0 ( )

0 (

) ( )

( ) ( 2 2 1

) ( )

( ) ( 2 1

) ( 2 ) ( 1

) ( ) ( 1

) ( 1 )

(





























n n n

n n n

n n n n n n

I I

e r r

e I

e r r

L I

r e

r

L I

e r

I r I

L g

L g L

g L g

L g

2

1

2

1



 r r e

Nonlinear Optics Lab . Hanyang Univ.

Threshold gain,

) 2 ln(

1 ln 1

2 1

2 1 2

1

r L r

r r

g_t L  

 



 

gt

* In the case of high reflectivity, ^r₁^r₂^{1, and ln(1-x)-x} 0.9)

ties, reflectivi (high

) 1

2 ( 1

2 1 2

1 





 rr rr

g_t L

* If the “distributed losses” (losses not associated with the mirrors) are included, a

r L r

g_t ln( ) 2

1

2 1

where, : effective loss per unit lengtha

Nonlinear Optics Lab . Hanyang Univ.

Example)

1 4

1

10 2 . 2

)]

980 . 0 )(

998 . 0 ) ln[(

50 ( 2

1















cm

cm g_t

Threshold population inversion ; ) 8 (

)

( ₁

1 2 2 2

2 n



n  N S

g N g n

g A 

 



 

 ( ) ( )

8

2 2 1

1 2

2  n  n

 _{t t}

t

g AS

g N n

g N g

N  



 



 

 D



nm 632.8 at

sec 10 4 .

1  ^{6 1} 

 ^ 

A

MHz 1500 M MHz

T 10 1

15 . 2 amu

20 M

, K 400

~

2 / 1 6

D 











 



 



 





 n 

T Ne

3 9

10 1

6 2

8

1 4

cm / atoms 10

6 . 1

sec) 10

3 . 6 )(

sec 10 4 . 1 ( ) cm 10 6328 (

) cm 10 2 . 2 )(

8 (











 

D _  ^ _ ^ _

Nt

6 3

1 15

9

10 10 8 . 4

10 6 .

1   _



  D

N N_t

(very small !)

Nonlinear Optics Lab . Hanyang Univ.

10.5 Rate Equations for Photons and Populations

Time-dependent phenomena ?

1) Gain term (stimulated emission or absorption)

) ( )

( )

(

)

1 ^ ( _

 



 





n n

n g n I

t I c z I

) ( )

( )

(

)

1 ^ ( _

 



 



  ^{n n} g n I_n t

I c z

I

) )(

( ) 1 (

)

( ^{() () ()} ^()  ^() ^()



 

 

  _{n n} I_n I_n g n I_n I_n t

I c z I

In many lasers there is very little gross variation of either or with z. =>I_n^{() I}_n^() _^{  0}

z

* l<L (l : gain medium length, L : cavity length) )

)(

( )

(I_n^()  I_n^() cg n I_n^() I_n^() dt

d

) )(

( )

( _n^()  _n^()  g n I_n^()  I_n^() L

I cl dt I

d (10.5.4b)

Nonlinear Optics Lab . Hanyang Univ.

2) Loss term (output coupling, absorption/scattering at the mirrors)

By the output coupling, a fraction 1-r₁r₂ of intensity is lost per round trip time 2L/c.

) )(

1 2 ( )

( _n^() _n^()  r₁r₂ I_n^()I_n^() L

I c dt I

d (10.5.8)

From (10.5.4b), (10.5.8), and put : total intensityI_n I_n^()I_n^()

n

n

n

r r I

L I c

L g cl dt

dI ( 1 )

) 2

(  



or photon number,

q

 I

n n

n n n

n n

q L g q cl L g

cl

q r L r

q c L g

cl dt

dq











) (

) 1

2 ( )

(

Nonlinear Optics Lab . Hanyang Univ.

If g₁=g₂,

n n

n n n

n



 n



I r L r

I c N L N

cl

I r L r

I c S N A N

L cl dt

dI

) 1

2 ( )

)(

(

) 1

2 ( )

( ) 8 (

2 1 1

2

2 1 1

2 2

















Coupled equations for the light and the atoms in the laser cavity including pumping effect :

n n n

n n

n

n n









 



























) 1

2 ( )

(

) ( )

(

) (

2 1 2

2 2

2 1

1 1

r L r

g c L cl dt

d

K g

N dt A

dN

g AN

dt N

dN

Nonlinear Optics Lab . Hanyang Univ.

10.7 Three-Level Laser Scheme

1) Two-level laser scheme is not possible.

2 ) 2

(

21 2

N A

N N 





 

 

Neglecting ₁, ₂ in (7.3.2) =>  : can’t achieve the population inversion

2) Three-level laser scheme

P : pumping rate

1 pumping

1 PN

dt

dN  



 





1 pumping

1 pumping

3 pumping

2 PN

dt dN dt

dN dt

dN  



 





 



 







 





(10.5.14) =>

) (

) (

1 2

2 21 1

2

1 2

2 21 1

1

N N

N dt PN

dN

N N

N dt PN

dN



























n n





n

n 

n

 (1 )

) 2

( r₁r₂

L g c

L cl dt

d

Nonlinear Optics Lab . Hanyang Univ.

<Threshold pumping>

i) Near threshold the number of cavity photons is small enough that stimulated emission may be omitted from Eq. (10.7.4)

ii) Steady-state : N₁N₂0

1 21

2 P N

N 



N

N N





NT

N P

21 21

1 

 

 NT

P N P

21

2  

21

 P : total population is conserved And,

# Positive (steady-state) population inversion :

# Threshold pumping power : ^Pwr _{h 31PN1} V  n

NT

P N P

N

21 21 1

2 



 



NT

P P h

21 21 31







n



 P

31 2 21

1 N_Thn



Nonlinear Optics Lab . Hanyang Univ.

10.8 Four-Level Laser Scheme

Lower laser level is not the ground level : The depletion of the lower laser level obviously enhances the population inversion.

Population rate equations :

n n

n



n







































) )(

(

) )(

(

1 2

2 21 0

2

1 2

2 21 1

10 1

1 10 0

0

N N

N dt PN

dN

N N

N dt N

dN

N dt PN

dN

Nonlinear Optics Lab . Hanyang Univ.

NT

N N

N₀  ₁ ₂ constant  Steady-state solutions :

T T

T

P N P

N P

P N P

N P

P N N P

21 10

21 10

10 2

21 10

21 10

21 1

21 10

21 10

21 10 0













 













 















  Population inversion :

P P

N N P

N ^T

21 10

21 10

21 10

1 2

) (

















 



# Positive inversion :

21 10

 



lower level decays more rapidly than the upper level.

If



 

, P

N

P N P

N N

21 2

1

2

    

Nonlinear Optics Lab . Hanyang Univ.

10.9 Comparison of Pumping Requirements for Three and Four-Level Lasers

1) Threshold pumping rate, P

21 laser

level three

)

( 

D

 D

 



t T

t T

t N N

N P N

21 laser

level

)four

( 

D



 D



t T

t

t N N

P N (10.7.9) =>

(10.8.7), (10.8.6) =>

) 1 (

) (

laser level three

laser level

four



D



 D







t T

t t

t

N N

N P

P

2) Threshold pumping power, (Pwr)

21 31

laser level

three 2

1 )

Pwr

(   



 







T

t h N

V n

(10.7.15) =>

21 30

laser level four

) Pwr

(   D 



 







t

t h N

V n ^T

t t

t

N N V

V

31 30 laser

level three

laser level

four

2

) / ) Pwr ((

) / ) Pwr ((

n n D









Nonlinear Optics Lab . Hanyang Univ.

10.11 Small-Signal Gain and Saturation

For the three-level laser scheme, steady-state solution including the stimulated emission term ;

n









 

 2

) (

21 21 1

2 P

N N P

N ^T

Gain coefficient (assuming g

=g

),



) (

) fn ( 2

) )(

) (

( ¹

21

21   











  _n

n

 n

n 

N

g P ^T

: A large photon number, and therefore a large stimulated emission rate, tends to equalize the populations and . In this case, the gain is said to be

“saturated.” ¹

N N₂

<Microscopic view of the gain saturation>

As the cavity photon number increased, the stimulated absorption as well as the stimulated emission increased. => The lower level’s absorption rate is

exactly equal to the upper level’s emission rate in the extreme limit.  The gain is zero.

Nonlinear Optics Lab . Hanyang Univ.

<Small signal gain / Saturation flux>

sat 0

21 21

21

/ 1

) (

)]

/(

) ( 2 [ 1

1 )

)(

) ( (

n n

n

n

n

 n

n 





 















 

g

P P

N

g P ^T

Where we define, Small signal gain as

21 21 0

) )(

) (

( 



 

P

N

g n  n P ^T

Saturation flux as

) ( 2

sat 21

n

n 



 

 P

* The larger the decay rates, the larger the saturation flux.

* The saturation flux  Stimulated emission rate :

The average of the upper- and lower-level decay rates.

Nonlinear Optics Lab . Hanyang Univ.

<Gain width>

When the absorption lineshape is Lorentzian, 1

) /(

) (

) 1 ( )

( ₂

21 2

21

21  

 n n n n n



* Small signal gain width : Dn_gn₂₁ (10.11.3) =>

) /

( 1 ) /(

) (

) 1 ( )

( _{2 sat}

21 2

21 21

0

n21

n n

n n n

n  g     

g

where,

21 21 21

21 0

) )(

) (

( 



 

P

N

g n  n P ^T : Line-center small signal gain

* Power-broadened gain width :

2 / 1

21 sat

21

1 













 

 D

n

n n

n_g

Nonlinear Optics Lab . Hanyang Univ.

21 2 21

21 2

21 sat 21

) 4 (

) ( 2

21

n



n

 n

n 



 P

A P

: Line-center saturation flux is directly proportional to the transition linewidth.

Nonlinear Optics Lab . Hanyang Univ.

10.12 Spatial Hole Burning

In most laser, we have standing waves rather than traveling waves.

=> Cavity standing wave field is the sum of two oppositely propagating traveling wave fields ;

) , ( )

, (

)]

sin(

) [sin(

sin cos

) , (

2 0 1

0

t z E t

z E

t kz t

kz E

kz t

E t

z E



 



















) sin(

) ,

(z t ₂¹ E₀ kz t

E_  

where,

The time-averaged square of the electric field gives a field energy density : 0² ) 0

(

8 E c

hn 

n 

 ^

=> Total photon flux,

2 kz

) ( )

( ]sin

[

2  ^  ^



_{n n n}





Nonlinear Optics Lab . Hanyang Univ.

(10.11.3) =>

kz

g _{( )} g⁰_{( ) sat 2}

sin ] /

) [(

2 1

) ) (

(

n n

n

n n









  _{ }