1. Summary Blackbody radiation Einstein Coefficients
Indistinguishable interaction to atoms
Limited lifetime in the upper level
Homogeneous Broadening:
Lorentzian lineshape
Linewidth usually in MHz range
Distinguishable interaction to atoms
Doppler broadening
Inhomogeneous Broadening:
Gaussian lineshape
Linewidth usually in GHz range
Population Inversion

2. §4.5 Gain Saturation in Homogeneous Laser Media
Gain Constant
In practice, the population inversion is caused by a “Pumping” process, such as electric current in injection lasers, flashlamp light in pulsed ruby lasers, or the energetic electrons in plasma-discharge gas lasers …
Now, we are going to derive gain constant in presence of an optical field 2 1
Upper level
Lower level
Ground level R1 R2
1/t20
1/t1
1/tsp
Wi(n)
R1, R2 : Pump rate
t20 : nonradiative relaxation
t2, t1 : lifetime
Wi(n) : induced emission rate

3. §4.5 Gain Saturation in Homogeneous Laser Media
I. Rate Equations
At steady state:
If the optical field is absent, i.e.:

4. §4.5 Gain Saturation in Homogeneous Laser Media
In an efficient laser system:
and
Saturation intensity :
Gain constant :
Small signal gain constant :

5. §4.5 Gain Saturation in Homogeneous Laser Media
II. Population Inversion Saturation :
Population inversion has no relation with intensity : 1
0.5
At saturation intensity point, induced radiation rate is comparable with other relaxation processes, and causes the inversion to drop to one half of its nonsaturated value.

6. §4.5 Gain Saturation in Homogeneous Laser Media
and
and
That is, the more closer to resonant, the smaller the saturation intensity
While at the same incident intensity, the smaller saturation intensity means the population inversion drops more, the saturation is more serious. So when incident laser frequency is close to the resonant, the saturation effect gets stronger; while laser frequency is far away from the resonant, the saturation effect gets weaker.

7. §4.5 Gain Saturation in Homogeneous Laser Media
Lorentzian lineshape population inversion
Only in this region, saturation effect is obvious n n0 DN

8. §4.5 Gain Saturation in Homogeneous Laser Media
III. Gain Saturation
Small signal gain constant :
Gain constant in Lorentzian lineshape homogeneous laser media

9. §4.5 Gain Saturation in Homogeneous Laser Media
Now we consider a higher intensity light with frequency n1, how does another weaker light with frequency n2 gain constant change?
The higher intensity light not only changes itself gain constant, but also changes other frequency weaker light gain constant in same degree. n n0 g n1
Gain constant uniformly drops in homogeneous broadening case

10. §4.6 Gain Saturation in Inhomogeneous Laser Media
Assume the inhomogeneous medium is made of classes of atoms each designated by a continuous variable x. Moreover, define a function f(x) so that a priori probability that an atom has its parameter between x and x+dx is p(x)dx, and satisfy:
The atoms within a given class x are such considered as homogeneously broadening, having a lineshape function that is normalized as:
So,

11. §4.6 Gain Saturation in Inhomogeneous Laser Media
I. Rate Equations
Steady state solution:

12. §4.6 Gain Saturation in Inhomogeneous Laser Media
II. Gain Constant
Total power emitted by induced transitions per unit volume by atoms in class x:
Summing over all the classes: Or

13. §4.6 Gain Saturation in Inhomogeneous Laser Media
Small signal gain constant:
In the absence of saturation the gain constant of a homogenous and an inhomogeneous atomic system are identical
Other cases:
In each class, all the atoms are identical and are homogeneous broadening
Substitute above equation to the gain expression for inhomogeneous media

14. §4.6 Gain Saturation in Inhomogeneous Laser Media
In the extreme inhomogeneous case, the width of p (n) is much larger than that of g x (n).
Gain constant in inhomogeneous laser media

15. §4.6 Gain Saturation in Inhomogeneous Laser Media
III. Gain Saturation
Small signal gain constant :
If the inhomogeneous broadening is Doppler Broadening, then
Gain constant in a Doppler broadening laser media
Saturation intensity

16. §4.6 Gain Saturation in Inhomogeneous Laser Media
IV. Spectral Hole Burning Effect
对于非均匀加宽，表现中心频率为n的粒子发射一条中心频率为v，线宽为Dv的均匀加宽谱线。因而这部分粒子的饱和行为可以用均匀加宽情况下的增益饱和来描述。
When
and
When ,
saturation effect gets weaker
When
and ,
saturation effect can be ignored

17. §4.6 Gain Saturation in Inhomogeneous Laser Media
因此在频率为v1，光强为Iv1的强光作用下，非均匀加宽介质中的反转布居数密度在v1 处产生局部的饱和，即在以下频率范围有显著的饱和效应，并形成一个以v1为中心的孔，而在此范围以外的粒子没有影响。以上现象称为反转布居数得“烧孔效应”。
孔的宽度：
孔的深度：
Spectral hole burning effect in the laser cavity