[Laser] Inversion number above and below threshold

[ApexOfDE]

Homework Statement

Consider a 3-level laser. Give the occupation numbers N1(R), N2(R), the inversion $$\Delta$$N(R)/N = [N2(R) - N1(R)]/N and the flux below and above the threshold as a function of pump rate R.

F - the photon flux

Homework Equations

The Attempt at a Solution

I have written down the total rate equation for N1, N2 and F:

$$\frac{dN1}{dt} = -RN1 + T21N2 + \sigma F(N2 - N1)$$
$$\frac{dN2}{dt} = RN1 - T21N2 - \sigma F(N2 - N1)$$
$$\frac{dF}{dt} = \sigma F(N2 - N1) + T21N2 - F/ \tau$$

$$\sigma$$ = the absorption cross section
T21 = the spontaneous emission rate from 2 to 1
$$\tau$$ = the resonator decay rate

In steady state, all rhs of these eqs are equal to zero. Above the threshold the amplification factor remains constant. I get stucked here. Can someone give me a hint or an idea so that I can solve it?

Thx in advance

 

[mark726]

Hello there,

I would first start by defining all the variables and parameters in the given equations. This will help us understand the physical meaning of each term and how they relate to the system.

N1 and N2 are the population numbers in levels 1 and 2, respectively. These are related to the total population N by the equation N = N1 + N2.

R is the pump rate, which represents the rate at which energy is being pumped into the system.

T21 is the spontaneous emission rate from level 2 to level 1. This is the rate at which photons are emitted without any external stimulation.

σ is the absorption cross section, which represents the probability of a photon being absorbed by the system.

F is the photon flux, which is the rate at which photons are passing through a unit area.

τ is the resonator decay rate, which represents the rate at which photons are lost from the system due to leakage or absorption.

Now, in steady state, as you correctly mentioned, all the right-hand sides of the equations are equal to zero. This means that the rates of change of N1, N2, and F are all zero, and their values remain constant.

Above the threshold, the amplification factor remains constant, meaning that the rate at which photons are being emitted from level 2 to level 1 is equal to the rate at which they are being absorbed by the system. This can be expressed as T21N2 = σF(N2 - N1).

Using this relationship, we can solve for N1 and N2 in terms of the other parameters:

N1 = N2 - Fσ/2T21
N2 = N - N1 = N - N2 + Fσ/2T21

Substituting these expressions into the equation for the inversion ΔN/N, we get:

ΔN/N = [N2(R) - N1(R)]/N = [N - N2 + Fσ/2T21 - (N2 - Fσ/2T21)]/N = Fσ/(NT21)

Similarly, we can solve for the photon flux F as a function of the pump rate R:

F = 2RT21/(σ + 2T21τ)

Below the threshold, the amplification factor is less than 1, meaning that the rate of photon emission from level 2 to level 1 is less than the rate