Laser

A laser is a medium that stores energy, surroudned by two mirrors. A partially reflecting output mirror lets some light out.

Gain

G \equiv exp (σ (N_{2} - N_{1}) L)

$g, α$ are gain and absorption coefficients

N_{2} < N_{1} \to α \equiv (N_{1} - N_{2}) σ N_{2} > N_{1} \to g \equiv (N_{2} - N_{1}) σ

N_{2} > N_{1} \equiv Δ N < 0

$Δ N = N_{1} - N_{2}$ , so $Δ N < 0 ⟺$ inversion

Δ N = \frac{N}{1 + 2 I _{pump} / I _{sat}}

$Δ N$ is always positive and it is impossible to achieve an inversion.

Δ N = N \frac{1 - I _{pump} / I _{sat}}{1 + I _{pump} / I _{sat}}

when $I_{pump} > I_{sat}$ , $Δ N < 0$ and inversion occurs!

Δ N = - N \frac{I _{pump} / I _{sat}}{1 + I _{pump} / I _{sat}}

$Δ N$ is always negative, so inversion is always occuring!

I_{saturation} = \frac{ℏ ω}{σ τ} = \frac{A}{B}

N_{i} \propto exp (- \frac{E _{i}}{k _{B} T})

$N_{i}$ is the number/population density of molecules in state $i$ , i.e. the number of molecules per cubic centimeter
$T$ is the tempurature
$k_{B} = 1.38 \times 1 0^{- 23}$ is Boltzmann’s constant

\frac{N _{2}}{N _{1}} = exp (- \frac{E _{2}}{k _{B} T}) / exp (- \frac{E _{1}}{k _{B} T}) = exp [- \frac{Δ E}{k _{B} T}]

where $Δ E = E_{2} - E_{1} = h ν$
- $h ν$ is the frequency of a photon for the transition
In the absence of light and collisions (low T), molecules tend to fall to and remain in the lowest energy state available
Light an collision can nexcite molecules into higher-energy states. The higher T is, the more this happens