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 ;
A
un
z
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
z+Dz
[ In ( z Dz ) In ] A
Nonlinear Optics Lab.

( In A)Dz
z
Hanyang Univ.
This difference gives the rate at which electromagnetic energy leaves the volume ADz,


  (un ADz ) ( In A)Dz
t
z
1 

In  In  0 : Equation of continuity
un  In /c 
c t
z
<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 )( N 2  N1 )hn  (n ) In ( N 2  N1 )

1   
  In  (n ) In ( N 2  N1 )

 c t z 
Nonlinear Optics Lab.
Hanyang Univ.
(7.4.19) :  (n )(hn /c) BS (n )
hn
1   
  In  B ( N 2  N1 ) S (n ) In

c
 c t z 

(no degeneracies)

hn 
g
B N 2  2 N1  S (n ) In
c 
g1 
(degeneracies included)

2 A
g
 N 2  2 N1  S (n ) In

8 
g1 


g
 (n )  N 2  2 N1  In
g1 

Define, gain coefficient, g(n)

2 A 
g
 N 2  2 N1  S (n )
g (n ) 
8 
g1 


g
  (n ) N 2  2 N1 
g1 

Nonlinear Optics Lab.
Hanyang Univ.

In 1 In

 g (n ) In
z c t
Temporal steady state :

0
t
 In ( z ) In (0)e g (n ) z
* g>0 when N 2 
* If N 2 
g1
N1 ,
g2

dIn
 g (n ) In
dz
: valid only for low intensity ( saturation effect, g=g(I))
g1
N1 : population inversion
g2
g
g
g
N 2  2 N 1   2 N1   2 N
g1
g1
g1
2 A g 2
g (n )
NS (n )
8 g1
 N
g2
 (n )a (n ) The gain coefficient is identical, except
g1
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
e0.01 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.
r  t  s 1
where, r : reflection coefficient
t : transmission coefficient
s : fractional loss
At the mirror at z=L and 0 ;
In(  ) ( L)  r2 In(  ) ( L)
In(  ) (0)  r1 In(  ) (0)
Nonlinear Optics Lab.
Hanyang Univ.
In steady state (or CW operation), g(n) : constant
dIn(  )
 g (n ) In(  )
dz
dIn(  )
  g (n ) In(  )
dz
 In(  ) ( z )  In(  ) (0)e g (n ) z
In(  ) ( z )  In(  ) ( L)e g (n )( Lz )
Amplification condition : In( ) (0) after one round trip must be higher than the initial In( ) (0)
 In(  ) (0) r1 In(  ) (0)
 r1[e g (n ) L In(  ) ( L)]
 r1e g (n ) L [r2 In(  ) ( L)]
 r1r2 e g (n ) L [ In(  ) (0)e g (n ) L ]
 r1r2e 2 gL 1
[r1r2 e 2 g (n ) L ]In(  ) (0)  In(  ) (0)
Nonlinear Optics Lab.
Hanyang Univ.
Threshold gain, gt
gt 
1  1 
1
  
ln 
ln( r1r2 )
2L  r1r2 
2L
* In the case of high reflectivity, r1r2 1, and ln(1-x)-x
 gt 
1
(1r1r2 ) (high reflectivi ties, r1r2 0.9)
2L
* If the “distributed losses” (losses not associated with the mirrors) are included,
g t 
1
ln( r1r2 )  a
2L
where, a : effective loss per unit length
Nonlinear Optics Lab.
Hanyang Univ.
Example) He-Ne laser, L=50 cm, r1=0.998, r2=0.980
gt  
1
ln[( 0.998)(0.980)]cm 1
2(50)
 2.2 10 4 cm 1
Threshold population inversion ;
2 A 
g2 

g (n ) 
N

N1 S (n )
2
8n 2 
g1 

g
g 2  8n 2 gt
 DN t  N 2  N1  2
 t
g1   AS (n )  (n )

A1.4106 sec 1 at  632.8 nm
 1  T 1/2 
T ~400K, M Ne 20 amu  n D 2.1510     MHz 1500 MHz
   M  
4
1
(8 )( 2.210 cm )
DN t 
(632810 8 cm) 2 (1.4106 sec 1 )(6.310 10sec)
6
1.6109 atoms /cm3
DN t 1.6 109 1

 3 10 6 (very small !)
15
N
4.8 10
Nonlinear Optics Lab.
Hanyang Univ.
10.5 Rate Equations for Photons and Populations
Time-dependent phenomena ?
1) Gain term (stimulated emission or absorption)
In(  ) 1 In(  )

 g (n ) In(  )
z
c t
In(  ) 1 In(  )


 g (n ) In(  )
z
c t
 () () 1  () ()

( In  In )
( In  In ) g (n )( In(  )  In(  ) )
z
c t
()
In many lasers there is very little gross variation of either In(  ) or In with z. =>
d ()
( In  In(  ) )  cg (n )( In(  )  In(  ) )
dt
* l<L (l : gain medium length, L : cavity length)
d ()
cl
(10.5.4b)
( In  In(  ) )  g (n )( In(  )  In(  ) )
dt
L
Nonlinear Optics Lab.
Hanyang Univ.

0
z
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.
d ( ) ()
c
( In  In ) (1r1r2 )( In(  )  In(  ) )
dt
2L
(10.5.8)
From (10.5.4b), (10.5.8), and put In  In(  )  In(  ) : total intensity
dIn cl
c
 g (n ) In 
(1  r1r2 ) In
dt
L
2L
or photon number, qn  In
dqn cl
c
 g (n )qn  (1r1r2 )qn
dt L
2L
cl
cl
 g (n )qn  g t qn
L
L
Nonlinear Optics Lab.
Hanyang Univ.
dIn cl 2 A
c
If g1=g2,

( N 2  N1 ) S (n ) In 
(1  r1r2 ) In
dt
L 8
2L
cl
c
  (n )( N 2  N1 ) In 
(1  r1r2 ) In
L
2L
Coupled equations for the light and the atoms in the laser cavity including pumping effect :
dN1
 1 N1  AN 2  g (n )n
dt
dN 2
 (2  A) N 2  g (n )n  K
dt
dn cl
c
 g (n )n 
(1  r1r2 )n
dt
L
2L
Nonlinear Optics Lab.
Hanyang Univ.
10.7 Three-Level Laser Scheme
1) Two-level laser scheme is not possible.
Neglecting 1, 2 in (7.3.2) => N 2 ()
N
N
: can’t achieve the population inversion

A21 2 2
2) Three-level laser scheme
(10.5.14) =>
P : pumping rate
 dN1 
  PN1


dt

 pumping
dN1
  PN1  21 N 2  n ( N 2  N1 )
dt
dN 2
 PN1  21 N 2  n ( N 2  N1 )
dt
dn cl
c
 g (n )n  (1r1r2 )n
dt L
2L
 dN 
 dN 2 
 dN 
 3
  1 
 PN1


 dt  pumping  dt  pumping
 dt  pumping
Nonlinear Optics Lab.
Hanyang Univ.
<Threshold pumping> for the steady-state operation
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
P
 N 2  N1
21
And, N1  N 2  NT : total population is conserved
 N1 
21
NT
P21
P
N2 
NT
P  21
N 2  N1 
# Positive (steady-state) population inversion : P21
# Threshold pumping power :
P  21
NT
P  21
Pmin  21
h n P
Pwr
 hn 31PN1  31 21 N T  12 21N T hn 31
V
P 21
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 :
dN 0
  PN 0  10 N1
dt
dN1
 10 N1  21N 2   (n )( N 2  N1 )n
dt
dN 2
 PN 0  21N 2   (n )( N 2  N1 )n
dt
Nonlinear Optics Lab.
Hanyang Univ.
N 0  N1  N 2  constant  NT
Steady-state solutions :
1021
N0 
NT
1021  10 P  21P
Population inversion :
21P
N1 
NT
1021  10 P  21P
N2 
N 2  N1 
10 P
NT
1021  10 P  21P
P(10  21 ) N T
1021  10 P  21P
# Positive inversion :
10  21
lower level decays
more rapidly than
the upper level.
If 10 21, P
P
N 2  N1  N 2 
NT
P  21
Nonlinear Optics Lab.
Hanyang Univ.
10.9 Comparison of Pumping Requirements for Three and
Four-Level Lasers
1) Threshold pumping rate, Pt
(10.7.9) => ( Pt ) threelevel laser 
NT  DN t
21
NT  DN t
(10.8.7), (10.8.6) => ( Pt ) four level laser 
DN t


1
( Pt ) threelevellaser NT DN t
( Pt ) four levellaser
DN t
21
NT  DN t
2) Threshold pumping power, (Pwr)t
1
 (Pwr )t 
 hn 31NT 21

 V  threelevellaser 2
(10.7.15) => 
 (Pwr )t 
 hn 30DN t 21


V

four level laser
(( Pwr ) t /V ) four levellaser 2n 30DN t


(( Pwr ) t /V ) threelevellaser n 31 NT
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  N1 
( P  21 ) NT
P  21  2 n
 Gain coefficient (assuming g1=g2),
g (n )
 (n )( P21 ) NT
1
fn (n )
P21 2 (n )n
: A large photon number, and therefore a large stimulated emission rate, tends
to equalize the populations N1 and N 2 . In this case, the gain is said to be
“saturated.”
<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>
g (n ) 

 (n )( P  21 ) NT
P  21
1
1  [2 (n )n /( P  21 )]
g 0 (n )
1  n / nsat
Where we define, Small signal gain as
 (n )( P  21 ) NT
g 0 (n ) 
P  21
Saturation flux as
nsat 
P  21
2 (n )
* 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
 (n ) (n 21 )
(n 21 n ) 2 /(n 21 ) 2 1
* Small signal gain width : Dn g n 21
(10.11.3) =>
g (n )  g 0 (n 21 )
1
(n 21 n ) 2 /(n 21 ) 2  1  (n / nsat21 )
where, g 0 (n 21 ) 
 (n 21 )( P  21 ) NT
P  21
: Line-center small signal gain
* Power-broadened gain width :
1/ 2

n 

Dn g  n 21 1  sat
 n 
21 

Nonlinear Optics Lab.
Hanyang Univ.
P21 4 2n 21
n 21 
 2
( P21 ) n 21
2 (n 21 )
A
sat
: 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 ;
E ( z, t )  E0 cos t sin kz
 12 E0 [sin( kz  t )  sin( kz  t )]
 E ( z , t )  E ( z , t )
where, E ( z, t )  12 E0 sin( kz  t )
The time-averaged square of the electric field gives a field energy density :
hn (  )  0 2
n  E0
c
8
=> Total photon flux,
n  2[n(  )  n(  ) ] sin 2 kz
In hnn 

In  2[ In(  )  In(  ) ] sin 2 kz
Nonlinear Optics Lab.
Hanyang Univ.
(10.11.3) =>
g (n ) 
g 0 (n )
1  2[(n(  )  n(  ) ) / nsat ] sin 2 kz
Nonlinear Optics Lab.
Hanyang Univ.