Nonlinear Fiber Optics – 1. Introduction-2

Hi everyone again, in our previous article, we talked about the spatial and temporal coherence of the laser, the components required for laser gain, and the interactions between the atoms (absorption, spontaneous and stimulated emission) that Einstein promised about laser formation [1]. (https://steemit.com/steemstem/@onderakcaalan/nonlinear-fiber-optics-1-introduction-1).

PS: Unfortunately, I did not know some of the writing rules in Steemit (the bibliography should be given full adresses as a link), but there is a little misfortune, but in this case, more careful writing will be presented.

Today, you will be presented with an article about the levels of the laser systems. Although there are some mathematical operations, I am convinced that it is easy to understand because there are quite simple operations.

1.2 Laser Gain

As described in Introduction-1, there are three interactions for laser formation. The first is the absorption, the second is the spontaneous emission and the other is the stimulated emission [2] (figure-1).

Figure-1: Absorption, spontaneous and stimulated emission.
(http://www.equestionanswers.com/notes/laser.php)

Absorption Rate: BN₁I
Spontaneous Emission Rate: AN₂
Stimulated Emission Rate: BN₂I

N_i is number of molecules in the i^th level, I is the power of light (irradiance) and A, B is the Einstein’s variables. The gain in the laser is due to the interactions given above. Fig.2, shows a basic laser cavity.

Figure-2: A basic laser cavity.

If spontaneous emission is neglected,

dI/dt = c(dI/dt)= BN₂I - BN₁I = B[N₂ - N₁]I

If you solve this first order differential equation then,

I(z) = I(0)exp(σ[N₂ - N₁]z)

According to the value of N₂ and N₁, there will be two options:

If N₂ > N₁, then there is gain,
If N₂ < N₁, then there is loss.

For the differential equation, you do not need to solve and you can easily understand that stimulated emission is more than absorption. This phenomenon is called “population inversion” [3]. This condition is not valid for steady states. Therefore, to get a gain you need to make the system, unsteady state. This unsteady state can be obtained by pumping the gain medium.

In Fig.3, you can see a scheme of a pumping the gain medium. The energy levels of the laser gain environment determine whether the intensity of the light is sufficient. The simplest system in terms of energy levels is a 2-level system, which is in 3 and 4 level systems.

Figure-3: An example of pumping the gain medium to get population inversion.

1.2.1 2-Level Systems

The simplest system is 2-level system, Fig.4. In this system you excite the low level electrons to another level and those excited electrons falls to low level again and creates some photons (spontaneous emission). When you continue to pump this low level there the photons will be amplified (stimulated emission). This is the main idea of the 2-level system.

Figure-4: 2-level system.

Let’s calculate whether there is a population inversion or not [4].

The number of molecules in the higher level in terms of time: dN₂/dt = BI[N₂ - N₁] - AN₂

The number of molecules in the lower level in terms of time: dN₁/dt = BI[N₁ - N₂] + AN₂

Let’s explain what these equations mean. For higher and lower level: BI[N₂ - N₁] is the difference between absorption and stimulated emission. What about spontaneous emission. Spontaneous emission is just creating photon and the electrons turns to lower level again. It means that for low level you can you that electron again so “+AN₂” but for higher level you just miss it means “-AN₂”.

Let’s make some conversions to improve our equations. If we say the total number of the molecules is N then,

N = N₁ + N₂ , ΔN = N₁ - N₂ => 2N = N-ΔN

So is we use this conversions into the lower and higher level eqns. in terms of time,

d(ΔN)/dt = -2BI(ΔN) + 2AN₂
d(ΔN)/dt = -2BI(ΔN) + AN – A(ΔN)

When there is a steady state for this system, which means that there is no changes for ΔN, so it means that the derivative of the ΔN = 0,

0 = -2BI(ΔN) + AN – A(ΔN) =>
(A + 2BI)(ΔN) = AN

If you solve this equation for ΔN,

ΔN = AN/(A + 2BI)
=> ΔN = N/(1+2BI/A)
=> ΔN = N/(1+2I/I_sat) where I_sat = A/B

When we examine the solution, we see that each value for I, ΔN becomes positive (N₂ < N₁). So there will be no population inversion. It means that there will no lasing. As a conclusion, if you have a 2-level system you cannot get a light.

I want to end up this article. Because you need to understand clearly this part to continue for the 3 and 4-level systems solution.

In this article, we tried to find some answers:
1- Why Einstein interactions are needed,
2- How laser gain becomes,
3- What population inversion is
4- Whether 2-level system is enough to get light.

From now on, we will try to explain 3 and 4 level systems. These are the main levels for all laser systems such as Yb, Er, Tm, etc.

Please do not hesitate to ask and make a comment. Have nice days.