Laser Physics EAL 501
Lecture 6
Power & Frequency
• We said that there are two types of cavities
• 1- Ring cavities
• Standing wave cavities
• 1- Longitudenal modes
due to standing waves
c
 
2nL
Do ring lasers have longitudenal modes?
2-Transversal modes [Gaussian beams]
The electric field distribution in the x-y plane
Satisfy the wave eq.
The Fresnel-Kirchoff’s theory states that
“if we know the field distribution at a plan 1 (x1,y1,z1) then the
distribution at any plane (x,y,z) is given by
We can write
Where
The solution is
Higher order modes are given by
Gaussian Beams
The solution of wave equation can be plane waves
But it is not real
q(z) is the complex beam parameter of the beam
Amplitude factor
Transverse phase factor
Rayleigh range
Beam waist
Radius of curvature
Longitudenal phase factor
Gain Saturation
We drive the rate equations
g ( )   ( )(N 2 
g1
N1 )
g2
r1r2 e
2 gt L
1
1
1
1
gt 
ln( ) 
(1  r1r2 )
2 L r1r2
2L
1
cavity..losses 
(1  r1r2 )
2L
• The steady state solution of the rate equation in the
case I=0 gives us the threshold condition
• The steady state solution of the rate equation in
the case after the threshold can be shown to be
Steady state under threshold solution is
In case of 3 level laser
In case of 4 level laser
Solution above threshold in the case of two levels without pumping
Solution above threshold in the case of 3levels with
pumping
Since g(ν)=σ(v)(N2-N1) then we can write
g o ( )
g ( ) 
I ( )
1
I sat ( )
g o ( )   ( )
P  21
NT
P  21
P  21
I sat ( )  h
2 ( )
Problem : find small signal gain and saturation intensity for 4 level laser
Hole burning
1. Spatial hole burning
2. Spectral hole burning
1- Spatial hole burning
in the case of standing wave cavity there are points of zero intensity and
points of max. intensity which depletes the gain much more rapidly
2- Spectral hole burning
in the case of inhomogeneous broadening the atoms that feed the laser
modes will be depleted much more rapidly
Output power
Usually only one mirror produces output and
if the mirror reflection is high
t
 I
2
1
 g ( )  (1  r1r2 )  g t
2l
g o ( )
g ( ) 
 gt
1  I / I sat
The condition of steady state lasing
Where s is the
scattering losses
It is easy to find the value of transmission t to get the maximum output
So the optimum output intensity is