2 a solid state laser material of refractive index

2. A solid state laser material of refractive index
1.45 n 
has a spontaneous emission lifetime of
150 nsec, a spectral emission bandwidth of 10 THz, and a central wavelength of 1.5  m. The
lineshape is Gaussian. In other words the lineshape is described as
 
 
2
0
2
2
1
( )
E E
h f
E e
f







,
where E
0
and
f 
are the energies corresponding to the central wavelength and the spectral
bandwidth of the lineshape.
2
(a) What are the Einstein A and B coefficients for this system?
(b) What is the gain cross section for this system? The gain at a given energy E is expressed as
 
2 1
( ) ( ) g E E n n    , where
 
E 
is the gain cross section, and
2
n
and
1
n
are the populations
in the upper and lower energy states respectively.
(c) How big a population inversion is required to amplify a weak 1.5  m signal by a factor of 2 in a 1
cm length? What are the gain and amplification factor at 1.55  m under these conditions?

(d) If there are 10
17
absorbing ions per cm
-3
, what is the transmission through 1 cm of this material at
1.5  m if all the ions are in the ground state?
3. A distributed amplifier at 1.5 m    is built using a doped fiber with the following
characteristics: Spontaneous emission lifetime of
1
500 sec
A
    , length 2 m L