Laser output below, just above and way above threshold?

[unscientific]

I've been reading up on lasers recently and learned that within the cavity while the laser light is amplified by the gain coefficient, it loses power due to transmission and absorption.

Suppose the pumping radiation comes from a monochromatic source whose energy is not exactly equal to the energy difference between the two levels.

For laser operation below threshold, due to broadening of the levels it would still be possible to stimulate gain?

But if it the gain way above threshold, there is no overlap between the pumping radiation and energy difference between the two levels as the peaks become very sharp. Would the gain be 0 then?

 

[blue_leaf77]

What actually happens when you keep pumping the gain medium is the so-called gain saturation, i.e. the peak of the gain profile will gradually decrease over time. Saturation takes place when the laser photons causes more and more stimulated emission from the upper laser level so that the pumping rate cannot keep up with stimulated emission rate anymore. At saturation condition the populations in the upper and lower laser level are equal, obviously in this condition no amplification can occur hence the gain coefficient is zero since it's proportional to the population difference between the two levels, but the gain (different from gain coefficient) is just unity. However, for simplicity assuming homogeneously broadened medium, the gain profile remains unchanged, it does not shrink nor widen as you said above. Only its peak will drop when saturation is taking place.

Would the gain be 0 then?

You are right only in the sense that the gain will be zero due to saturation but not because the overlap diminishes between the gain and pump spectrum. Again, it becomes zero because the populations of the upper and lower laser levels equal in saturated condition.

 

[unscientific]

What actually happens when you keep pumping the gain medium is the so-called gain saturation, i.e. the peak of the gain profile will gradually decrease over time. Saturation takes place when the laser photons causes more and more stimulated emission from the upper laser level so that the pumping rate cannot keep up with stimulated emission rate anymore. At saturation condition the populations in the upper and lower laser level are equal, obviously in this condition no amplification can occur hence the gain coefficient is zero since it's proportional to the population difference between the two levels, but the gain (different from gain coefficient) is just unity. However, for simplicity assuming homogeneously broadened medium, the gain profile remains unchanged, it does not shrink nor widen as you said above. Only its peak will drop when saturation is taking place.

You are right only in the sense that the gain will be zero due to saturation but not because the overlap diminishes between the gain and pump spectrum. Again, it becomes zero because the populations of the upper and lower laser levels equal in saturated condition.

I read that way above the operation threshold, the gain saturates and the upper population threshold stays fixed too. But the intensity of the output laser increases. I got these graphs from a textbook, not sure what to make of them:

 

[blue_leaf77]

Personally I don't know how to relate your original question about the narrowing of the gain profile (which is not the case in reality) with the above picture.

I got these graphs from a textbook, not sure what to make of them:

Are you looking for the explanation of this picture? Since you got it from a textbook I'm sure the author must have explained what is shown in that picture in the related paragraphs. I will try to give a brief picture of what's happening there. In picture (a) since the gain peak lies just above the threshold, the lasing must have been already in saturation. In principle if there is absolutely zero loss in the resonator, the resonator can accommodate infinite number of longitudinal modes. However if there are losses, only modes which experience more gain than tehy do loss will survive as the time flies. In picture (a), there is only a small part in the frequency axis where the gain is higher than the threshold (or loss, threshold is another way of saying cavity loss anyway), which is that around the gain peak. Consequently, only longitudinal modes which lie within this small frequency region will experience a net amplification and hence contribute to the output photons, in that picture there is only one longitudinal mode gets to survive, the other die out.
As you increase the pump power, nothing will change on the gain profile. It won't become higher nor lower. Indeed by increasing the pumping power/rate, more and more atoms are brought up to the upper laser level per unit of time. But at the same time, the increasing population inversion will make stimulated emission more frequent to take place hence reducing the population inversion. These two effects balance each other in such a way that stable operation is maintained, this is represented by the unchanging gain profile (figure (a)). On the other hand, as stated before, more faster pumping will result in more stimulated emission events, thus increasing the output photon number per unit time (figure(b)).

At saturation condition the populations in the upper and lower laser level are equal, obviously in this condition no amplification can occur hence the gain coefficient is zero since it's proportional to the population difference between the two levels, but the gain (different from gain coefficient) is just unity.

I apologize, this statement of mine is not true realistically. The gain coefficient will never drop to zero, it will only drop until it's equal to the threhold value because the cavity loss is never exactly zero - one always need some amount of population inversion left in order to overcome the loss. Therefore any following statement of mine in the same comment saying that the gain is zero is not correct either.