Lasers
To achieve laser emission we must create a situation where stimulated emission can occur. To do this we create a population inversion N2 > N1. (Note: this is not possible in a two level system considered and we shall see how to do this in three or four level system.) However, even if we create a population inversion we must remember that spontaneous emission is always present and must be separated from the stimulated emission that gives rise to the laser effect.
Absorption and stimulated emission
The absorption coefficient
g is
where n is the real part of the index of refraction and
ce'' is the imaginary part of the electric susceptibility. The absorption coefficient g is positive for attenuation (absorption) and negative when a population inversion is created and light amplification of stimulated emission or the laser effect is observed. To see this we begin with Beer's law:dI = -
gIdx or dI/dx = -gIThe power per unit volume is
The intrinsic rate is w12 for an individual molecule. This rate is equal to B12
r(n). Note that W12 = N1w12.We can express the transition rate as
Note that we used the relations
and the effective speed of light c/n instead of c. We can also replace the coefficient of spontaneous emission by
Thus,
The radiation energy density is
There is a general lineshape function g(
n) that represents all of the broadening mechanisms in the spectra. This function is normalized so
Thus we have the relation
From this we can infer that
Hence,
Substituting this individual rate constant into the power density equation above we have
From a comparison with our initial expression for the absorption coefficient we see that
As we stated at the outset
g > 0 if N1 > N2. This is the normal situation where the ground state population N1 is greater (usually much greater) than the excited state population. However, g < 0 if N2 > N1, i.e. if there is a population inversion.