Lasers Intro

Absorption #

Stimulated #

atom: E₁→E₂ (E₂>E₁) Rate ∝

• number of atoms in ground state
• density of photons

Emission #

Spontaneous #

De-excitation: E₂→E₁ Rate ∝

• number of atoms in excited state

Stimulated #

Photon of energy E₁-E₂=hν interacts with atom in E₂, De-excitation takes place: E₂→E₁, two photons are released
Rate ∝

• number of atoms in excited state
• density of photons

Difference #

Coherence #

It is of two types

1. Temporal coherence: A point on a wave is coherent with another point on the same wave
2. Spatial coherence: A point on a wave is coherent with a point on another wave.

Population Inversion #

When N₂ atoms in E₂ > N₁ atoms in E₁ where E₂>E₁ Usually more atoms stay in lower energy levels so if the inverse happens, it’s called a population inversion.

Distribution #

$$ N_i=N_0\cdot e^\left(-\frac{E_i}{kT}\right)$$

Einstein coefficients #

N₁,N₂ atoms in E₁,E₂ energy levels number of photons per unit volume: n Energy density of photons p(v) = nhv At equilibrium upward transition = downward transition

Upward Transition #

• Stimulated absorption rate = $B_{12}N₁p(v)$
  • $B_{12}$: Einstein coefficient of stimulated absorption

Downward Transition #

• Stimulated emission rate = $B_{21}N_2p(v)$
  • $B_{21}$: Einstein coefficient of stimulated emission.
• Spontaneous emission rate = $A_{21}N_2$
  • $A_{21}$: Einstein coefficient of spontaneous emission.

Deriving p(v) #

$$ \begin{align} B_{12}N₁p(v)=B_{21}N_2p(v)+A_{21}N_2\\ p(v)=\frac{A_{21}N_2}{B_{12}N_1-B_{21}N_2}\\ Dividing\\ numerator\\ and\\ denominator\\ by\\ B_{21}N_2\\ p(v)=\frac{\frac{A_{21}N_2}{B_{21}N_2}}{\frac{B_{12}}{B_{21}}\cdot e^{-\frac{E_1-E_2}{kT}}-1} \end{align} $$ From Plank’s Hypothesis we know that u(v) or p(v)=$\frac{8\pi v}{c^3}\cdot \frac{hv}{e^{\frac{hv}{kT}}-1}=\frac{8\pi hv^3/c^3}{e^{\frac{E_2-E_1}{kT}}-1}$ So, $\frac{B_{12}}{B_{21}}=1$ and $\frac{A_{21}}{B_{21}}=8\pi hv^3/c^3$