**L**ight

**A**mplification by

**S**timulated

**E**mission of

**R**adiation.

A population inversion occurs when a system (such as a group of atoms or molecules) exists in state with more members in an excited state than in lower energy states. The concept is of fundamental importance in laser science because the production of a population inversion is a necessary step in the workings of a standard laser.

To understand the concept of a population inversion, it is necessary to understand some thermodynamics and the way that light interacts with matter. To do so, it is useful to consider a very simple assembly of atoms forming a active laser medium.

Assume there are a group of N atoms, each of which is capable of being in one of two energy states, either

# The ground state, with energy E

_{1}; or

# The excited state, with energy E

_{2}, with E

_{2}> E

_{1}.

The number of these atoms which are in the ground state is given by N

_{1}, and the number in the excited state N

_{2}. Since there are N atoms in total,

The energy difference between the two states, given by

,

determines the characteristic frequency ν

_{12}of light which will interact with the atoms; This is given by the relation

,

h being Planck's constant.

If the group of atoms is in thermal equilibrium, it can be shown from thermodynamics that the ratio of the number of atoms in each state is given by a Boltzmann distribution:

,

where T is the thermodynamic temperature of the group of atoms, and k is Boltzmanns constant.

We may calculate the ratio of the populations of the two states at room temperature (T ≈ 300 K) for an energy difference ΔE that corresponds to light of a frequency corresponding to visible light (ν ≈ 5×10

^{14}Hz). In this case ΔE = E

_{2}- E

_{1}≈ 2.07 eV, and kT ≈ 0.026 eV. Since E

_{2}- E

_{1}≫ kT, it follows that the argument of the exponential in the equation above is a large negative number, and as such N

_{2}/N

_{1}is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As T increases, the number of electrons in the high-energy state (N

_{2}) increases, but N

_{2}never exceeds N

_{1}for a system at thermal equilibrium; rather, at infinite temperature, the populations N

_{2}and N

_{1}become equal. In other words, a population inversion (N

_{2}/N

_{1}> 1) can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state. --- from wikipedia:Population_inversion

This following a simulation for laser pumping.

There are three energy levels: E0=0,E1,E2 (E2>E1). You can use slider to change E1 and E2.

Use checkbox to set allowed energy between energy level.

chekc 0-1 means allow transition between energy level E0 and E1.

The probability to jump from E0 to E1 is proportional to

Pumping rate for ground state can be adjusted with slider (default value=0.5).

Laser pumping is the act of energy transfer from an external source into the gain medium of a laser. The energy is absorbed in the medium, producing excited states in its atoms. When the number of particles in one excited state exceeds the number of particles in the ground state or a less-excited state, population inversion is achieved.

You can uncheck 0-1 and check 0-2,1-2 (3 levels) to reach laser pumping condition.

Enjoy!

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