Gain (laser)

Gain in quasi two-level system

In the simple quasi two-level system, the gain can be expressed in terms of populations ${\displaystyle N_1}$ and ${\displaystyle N_2}$ of lower and excited states:

${\displaystyle G={\sigma }_{\mathrm{e}}{N}_2-{\sigma }_{\mathrm{a}}{N}_1}$

where ${\displaystyle {\sigma }_{\mathrm{e}}}$ and ${\displaystyle {\sigma }_{\mathrm{a}}}$ are effective emission and absorption cross-sections. In the case of non-pumped medium, the gain is negative.

Round-trip gain means gain multiplied by the length of propagation of the laser emission during a single round-trip. In the case of gain varying along the length, the round-trip gain can be expressed with integral ${\displaystyle g=\int G\mathrm{d}z}$. This definition assumes either flat-top profile of the laser beam inside the laser, or some effective gain, averaged across the beam cross-section.

The amplification coefficient ${\displaystyle K}$ can be defined as ratio of the output power ${\displaystyle P_{\mathrm{o}\mathrm{u}\mathrm{t}}}$ to the input power ${\displaystyle P_{\mathrm{i}\mathrm{n}}}$:

${\displaystyle K=P_{\mathrm{o}\mathrm{u}\mathrm{t}}/P_{\mathrm{i}\mathrm{n}}}$.

It is related with gain; ${\displaystyle K=\exp \left(\int G\mathrm{d}z\right)}$.

The gain and the amplification coefficient should not be confused with the magnification coefficient. The magnification characterizes the scale of enlarging of an image; such enlargement can be realized with passive elements, without gain medium.

Alternative terminology and notations

There is no established terminology about gain and absorption. Everyone is free to use own notations, and it is not possible to cover all the systems of notations in this article.

In radiophysics, gain may mean logarithm of the amplification coefficient.

In many articles on laser physics, which do not use the amplification coefficient ${\displaystyle K}$ defined above, the gain is called Amplification coefficient, in analogy with Absorption coefficient, which is actually not a coefficient at all; one has to multiply it to the length of propagation (thickness), change the signum, take inverse of the exponential, and only then get the coefficient of attenuation of the sample.

Some publications use term increment instead of gain and decrement instead of absorption coefficient to avoid the ambiguity, exploiting the analogy between paraxial propagation of quasi-monochromatic waves and time evolution of a dynamic system.