Chapter 11. Laser Oscillation : Power and Frequency
Power & Frequency of Single mode continuous wave (cw) laser ?
11.2 Output Intensity : Uniform-Field Approximation
Assumptions :
(1) Homogeneous broadened gain medium
(2) Mean-field (uniform-field) approximation
(3) All loss processes are independent of the cavity intensity
(4) Steady-state (temporal) or CW operation
r1 r2 1
1) Gain
(10.5.8) ,
I  0
=>
cl
L
g ( ) I  
c
2L
(1  r1 r2 ) I 
 g ( ) 
1
2l
(1 r1 r2 )  g t
: The gain is clamped on the threshold gain
for cw(steady-state) oscillation.
Nonlinear Optics Lab.
Hanyang Univ.
2) Output intensity
(10.12.10) or (10.11.14) =>
( )

I
g ( ) 

g 0 ( )
()
1  I
()
 I
/ I
sat
 gt

 g ( ) 
()
sat
 I   I   0
1 
g
t


Output intensity : I out  t1 I (  )
If r1 r2 1 =>

gt 
1
2l
(1  r ) 
1
()
()
I  
1 sat  g ( ) 
out
I   t I   0
1 
2
 gt

: A given medium, laser intensity depends on
how the laser cavity (t or s) is chosen.
=>
(t  s )
2l
out
I


sat  2 g 0 ( ) l
t I 
 1
2
 ts

1
Nonlinear Optics Lab.
Hanyang Univ.
11.3 Optimal Output Coupling
1) Optimal mirror transmission coefficient
  I  out

 t


 0

 t op t
1

2
I
 2 g 0 l  1 sat 2 g 0 l
1   tI 

2
t

s
(t  s )

 2
sat
0
t  t opt
 t opt  2 g 0 ( )ls  s
2) Maximum output intensity
I 
ou t

max

 I
ou t

 I
sat
t  t opt

g 0 ( )l  s / 2
: Lasing is possible when g 0 ( ) 
( g t ) opt 
If
g 0 ( ) 
s
2l
,
1
2l
(1 r ) opt 
I 
out

max
1
2l
( t opt  s ) 
s

2
: scattering loss coefficient
2l
g 0 ( ) s
2l
 g 0 ( 21 ) I  l
sat
: theoretical upper limit of laser intensity
Nonlinear Optics Lab.
Hanyang Univ.
2) Maximum output intensity – another approach
 dN 2 
h 

 dt  stimulated
   ( ) ( N 2  N 1 ) I 
emission
(10.12.8) =>

()
 dN 2 
 h 

 dt  stimulated
(11.2.5) =>
If
()
I  2 I  I
sin
2

()
()
kz ,  I   I   I 

()
()
  g ( ) I   I 
 : uniform field approx.

emission
g ( )I   I 
()

()
 g ( ) I
sat

 g 0 ( ) 
 g ( )  sat

  g t I sat  0


1
 g

 g 1   I   g 0 ( )  g t 
t
t





g 0 ( )  g t , g ( ) I (  )  I (  )  g ( ) I sat


0

: maximum intensity per unit volume
Maximum intensity extracted from the medium of length l ;
I
out
 21

max
 g 0 ( 21 ) I  21 l
sat
: (11.3.5a)
Nonlinear Optics Lab.
Hanyang Univ.
3) Input-to-output power conversion efficiency
e max 
3-level system, (10.7.12) =>
(10.11.12) =>
(10.11.7) =>
g 0 ( 21 ) 
I  21
sat

g 0 ( 21 ) I
sat
h  31 P N 1
 ( 21 )( P  21 ) N T
P   21

e max 
( P   21 ) N T h  21
h  21 ( P   21 )
2 h  31 P N 1
2 ( 21 )
1
In the case of strongly saturated case, N 1  N 2  N T
2

e max 
P   21  21
P
 31

 21
 31
: quantum efficiency
Nonlinear Optics Lab.
Hanyang Univ.
11.4 Effect of Spatial Hole Burning
Standing wave inside the cavity, (10.2.9) =>
 dN 2 

dt

 stimulated
(11.3.6) =>  h  
g ( ) 

g 0 ( )
()
1  2 I
sat
/ I
 sin
2
kz
 2 g ( ) I  sin kz
2
emission
2 g 0 ( ) I  sin kz
2

1 ( 2 I  / I 
sat
: Power per unit volume, at the point z,
)sin kz
extracted from the medium
by stimulated emission.
2
The rate at which the field gains energy should equal the rate at which it losses energy ;

l
0
For
gI dz  2 g 0 I
kl  1,

l
0

sin
l
0
sin
2
1  (2 I / I
2
1  (2 I / I
kz dz
sat
kz dz
sat
) sin
2
) sin

kz
2
kz

1 
2I 

lI
sat

(t  s ) I
1
1 2I / I
sat




()
 1
Nonlinear Optics Lab.

1
2
(t  s ) I  g t l I
1
1 2 I / I

sat
gtI
g0I
sat
Hanyang Univ.
Put,
x 1
2 I
I

sat
 1
x
2g0
x
gt
2I
I
sat
I  I
sat


2g0
1
 
gt 2
 g0 1

 
 g
4
 t
2I
x 1
I
2g0
gt

sat

2g0
1
 
gt 2
2g0
gt

1
4
: Disired solution is the one with minus
sign since x should be equal to 1 when
g0/gt=1.
1
4
1 


2 g t 16 
g0
Output intensity : I  out  ( t / 2 ) I 
out
I

t
2
sat
I
 g 0 ( ) 1

 
 g
4
t

g 0 ( )
2gt
1 


16 
: The effect of spatial hole burning
is to reduce the output intensity
Nonlinear Optics Lab.
Hanyang Univ.
11.5 Large Output Coupling
Our analysis of output power thus far has assume that the output coupling is small( r1 r2 1 ),
and we have also assumed time averaged intensity I(+) and I(-) are independent of z.
We will now allow arbitrary output coupling and therefore allow the possibility I(+) and I(-)
may vary with z.
(10.4.3) =>
dI
1  [I
 g (z) I
()
(z)
d

()
  g (z) I
()
dz
I
()
I
()
 I
()
dI
()
I
1 dI
()
()
( z )] I
sat
()
dz
1
I
()
dI
()
0
(z)
g0

()
dz
i ,e ; I
I
(z)  I
dz
dz

()
()
dz
dI
g0
g (z) 
Ignoring the spatial hole burning, (11.2.4) =>
C /I
I
sat
()
1
,
I
()
dI
()
dz
()
( z )I
()
( z )  constant  C
 g0

1
I
()
C /I
I
()
sat
Nonlinear Optics Lab.
Hanyang Univ.
1 
1  ()
C
g 0 dz  (  ) 1  sat  I
 ( )
I
I 
I



0
g 0 dz 
I
()
()
()
()

sat
I
I
()
I
()
(0)
I
()
()
I
 r2 I
()
()
1

I
sat
sat

I
I
()
()
g 0 l  ln
()
()
I
()

2
=>
(L)
dI
I
(0)
()
(L)

(0)

 C  r2 I
()
(L)

()
(L)
1

(0)
I
sat
I
()
(L)  I
()
(0)
I
I
()
()
(L)
1

(0)
I
sat
I
()
(0)  I
()
(L)

C 
1
1



sat 
()
()
I  I (0) I ( L ) 
2
I
()
(0) 
r2 r1 I
( )
(L)
=>
I
()


C 
1
1



sat
()
()
I  I ( L ) I ( 0 ) 
g 0 l  ln
()
()
I
()2
(0)
 r1 I
C dI

()
dI
(L)
()
I
dI
(0)
()
(L)
C
(0) 
(L)
dI
sat
I
C
(L) 
I
I
I
C

I

I
1

()
I
L
()
dI

  dI

(0) 
r1 r2 I
Nonlinear Optics Lab.
()
(L)
Hanyang Univ.

g 0 l  ln
=>

I
1

I
()
I
r1 r2
( L )
 r2 
sat

I

(L)
sat
1
r1 r2


I
()
( L )
r2 

r1 
()

I
sat
g
0
l  ln
r1 r2

1 r2 r1  r2  r1 r2
r1 I
sat
( r1  r2 )(1 r1 r2 )
 g l  ln
0
Output intensity :
I
out
 t1 I
()
( 0 ) t 2 I

 I sat  t 2 


()
(L)
r1
r2 
t
 g 0 l  ln

r1  ( r1  r2 )(1 r1 r2 )

r1 r2

When r1=1, t1=s1=0, r=r2, t=t2, s=s2, and t+s<<1
I
out

1
2
tI
sat
 2 g 0l

 1

 ts

 (11.2.11) : small output coupling
Nonlinear Optics Lab.
Hanyang Univ.
r1 r2

<Total two-way intensity>
(11.5.10) =>
I
(11.5.12) =>
I
(11.5.13) =>
I
()
C
(L) 
I
()
( 0 )  r1 I
()
(0) 
()
(L)
()
 r2 I
()
(L)
( 0 )  r1 r2 I
r2 r1 I
( )
()
(L)
(L)
r1 1, r2  r 
I
I
()
()
( L ) I
()
(0) I
()
(L)
(0)

1 r
2 r
Total intensities are comparable at the two
mirrors for reflectivity as low as 50%.
Nonlinear Optics Lab.
Hanyang Univ.
11.6 Measuring Small-Signal Gain and Optimal Output Coupling
Eq. (11.2.11) for output intensity or its generalization (11.5. 18) has been shown
experimentally to be quite accurate, because the spatial hole burning effect is usually
negligible in gas lasers.
In general the small signal gain and the saturation intensity are
difficult to calculate accurately, because the puping and decay
rates of the relavant atomic levels may not be well known.
=> Experimentally measured !
Nonlinear Optics Lab.
Hanyang Univ.
<Maximal loss method to measure g0>
- The cavity loss is varied by inserting a reflecting knife-edge into the cavity
- The cavity loss is increased until the laser oscillation ceases.
- (11.2.4) => g0(=gt) : loss just when the laser oscillation ceases.
<Simultaneous measurement method>
- Scattering coefficient, s=Pin/P+
- Effective output coupling : t+s
- t=Pout/P+ : known => s=tPin/Pout
- Ptotal=Pin+Pout : total output power
- Determine s-value for which Ptotal is maximum
=> topt=sopt + t., Ptotal=(Pin+Pout)topt
- Small signal gain : s-value at which laser oscillation stops.
Nonlinear Optics Lab.
Hanyang Univ.
11.7 Inhomogeneously Broadened Laser Media
In an inhomogeneously broadened gain medium the different active atoms have
different central transition frequencies 21.
# Small signal gain : Doppler broadened lineshape
 A
2
g 0 ( ) 
8
  N 0 S ( )
 A
2
1  4 ln 2 
  N 0



8
 D   
1/ 2

exp  4   21   ln 2   D
2
2

 g 0 ( 21 )exp[  4 (  21 ) ln 2 / D ]
2
2
The gain coefficient is obtained by integrating the contributions from the different
frequency components, each of which saturates to a differnet degree depending on its
detuning from the cavity mode frequency .

g ( )   g 0 ( )

1
( 21  ) /(  21 ) 1 (   /   21
2
2
sat
)
d  21
Nonlinear Optics Lab.
Hanyang Univ.
* 2  k ( v z  dv z )  ( 21   ) 2

*
x

dx
2
a
2



g ( ) 
a
g 0 ( )
1 I I
sat
where, I sat  h   
sat
21
The gain saturation set in more slowly as the intensity I is
increased in the case of inhomogeneous broadening medium.
Output intensity :
out
I

t
2
I
sat
  g (
 0

  g t

)
  1



2
cf) homogeneous medium, (11.2.9)
t sat  g ( ) 
out
I  I  0
1
2
g
t


Nonlinear Optics Lab.
Hanyang Univ.
11.8 Spectral Hole Burning and the Lamb Dip
<Spectral hole burning>
Spectral packets : The atom group with the central transition frequency of  21   21  c
The gain for spectral packets with frequency 21~(field frequency) is saturated more
strongly than others : spectral packets with frequency detuned from  by much more
than the homogeneous linewidth, i.e., |21-|>>21, are hardly saturated at all.
Spectral hole burning
(Bennet hole)
Nonlinear Optics Lab.
Hanyang Univ.
<Lamb dip>
Suppose the cavity mode frequency   the center frequency of the Doppler gain profile
i) The traveling-wave field propagating in the +z direction will strongly saturate the
spectral packet of atoms with Doppler-shifted frequencies ’21=.
: The Doppler effect has brought these atoms into resonance with the wave. Therefore,
those atoms have the z component of velocity given by

   1 

v

c
or
v  

c

ii) Similarly, the traveling-wave field propagating in the -z direction will strongly saturate
those atoms with the z component of velocity given by
v
c

 

Nonlinear Optics Lab.
Hanyang Univ.
=> The standing wave cavity field will burn two holes in the Doppler line profile.
When the mode frequency is exactly at the center of the Doppler
line, the two holes merge together. => The field can now strongly
saturate only those atoms having no z component of velocity. =>
The output power exactly at resonance will be lower than slightly
off resonance. : Lamb Dip.
Nonlinear Optics Lab.
Hanyang Univ.
11.9 Cavity Frequency and Frequency Pulling
 
Cavity mode frequency :
In general, 
or,
l
L

mc
2L
 m
mc 2
n ( ) l  ( L  l )
n    1
  m 
where, l : gain medium length, n: refractive index.
where,
m
mc
2L
: bare(passive) cavity mode frequency
(3.3.22), (3.3.25) =>
n    1  

put,

 c 
 21  21  
4
 21
g 

for homogeneous broadening medium
 21 c g ( ) l 4 L  m  21
c g ( ) l
cg ( ) l
4 L
4 L   21
: cavity bandwidth
 
 21 c   m  21
 c   21
or  c v  v 21  v 21  m  
Nonlinear Optics Lab.
: frequency pulling
Hanyang Univ.
<Frequency pulling and gain>
In most lasers,

 21   c
 21 c  21  m 
v
   21 c  m
 v 21

1  c  21
 m   21  m 
vc
 21    D
 m 1 . 88  21  m 
*   m   c
and




(homogeneous broadening)
 v 21

  m   21  m  c
 D
  v c
  1
  v
21

4 ln 2 
 vc
(inhomogeneous broadening)
vD
 c 
cl
4 L
gt
: The larger the threshold gain gt, the greater the frequency pulling for fixed gain linewidth (21 or D).
Nonlinear Optics Lab.
Hanyang Univ.
<Mode spacing>


(m )
( m 1)

 21 c   m  21

 c   21
(m )


 m 1   m  21
 c   21
c
1
2 L 1   c /  21
: The effect of frequency pulling is to reduce the mode spacing from c/2L.
Nonlinear Optics Lab.
Hanyang Univ.
11.11 Laser Power at Threshold
Laser power near the threshold ? => spontaneous emission.
(10.5.7)
dq 
dt
(10.11.12)
P   21 

cl
L
N 2  N1
g ( ) q 
g t q
L
(Mean-field approx.)
( P   21 ) N T
P   21  2 ( ) 
N 2  N1 N 2 

cl
PN T
P  2 ( ) 
NT
1 q q
sat
Including the spontaneous emission :
dq 
dt
 c  ( )
l
L
N 2  q  1  
l
L
cg t q
Nonlinear Optics Lab.
Hanyang Univ.
Steady-state solution :
q 
define,
q 


(11.11.1) => N 2  N T
 ( ) N 2
 y g0 gt
g t   ( ) N 2
N2
x
and
N t
x
x 
1 x
q  q
sat
q
1
2
q
sat
y
<1 : below the threshold
>1 : above the threshold
NT
N t
N T N t
1  q q
(1 y ) q  q
1
sat
sat

y  1 (far above threshold)
y
1  q q
sat
q  q
y 0
sat
 g0 


 g 1 
 t

: (11.2.5), (11.2.9)
: (11.2.5), (11.2.9)
4y
 ( y 1) 
( y 1)  sat
2
2
q
2
y 1 (near threshold)
(10.11.8), (10.11.10), P>>21
q
Nonlinear Optics Lab.
sat
 m  1 

  2 0 2     21  PV
e  f 

Hanyang Univ.

In many lasers, PV~103s-1, f~1, 21~10GHz => qsat~1010
y 1   q threshold 
1
2
q
sat
4 q
e
cf)  q thermal 
h  kT
 ( q ) t   q

sat
1

1
q
sat
 10
5
(very low power)
(ex) 6328 A He-Ne laser,  q thermal  0 . 023
thermal
<The rate of change of q with y>
d q
dy
 d q

 dy

1
2
q
sat

1 





 threshold
1
2
y 1 2 / q
sat
( y  1)  4 y / q
2
q
sat

1
2
 10
sat




Extremely rapid
rise in the cavity
photon number
at the point y=1
(threshold).
10
Nonlinear Optics Lab.
Hanyang Univ.
11.12 Obtaining Single-Mode Oscillation
1) Short cavity length
 
c
2L
 g

 g
Ex) g~1500 MHz (He-Ne laser) => c/2L>1500MHz => L<10 cm (low power)
2) Homogeneous broadening medium
Nonlinear Optics Lab.
Hanyang Univ.
3) Fabry-Perot etalon / Grating / Prism
=> Selective transmission
Ex) Fabry-Perot etalon

- resonance frequency :  m  m 

,
 2 nd cos  
    m 1   m 
c
m  1, 2 , 3 , . . .
c
2 nd cos 
Nonlinear Optics Lab.
Hanyang Univ.