🗒️ Note: in this section we care bout interaction with matter, photon nature of light
💼 Case: consider transitions between two energy levels $1$ and $2$ with electron densities $N_1$ and $N_2$ All photons emitted and abosrbed are of frequency $\nu$ where $h\nu =E_2-E_1$
Spontaneous emissions
$$ \begin{aligned} \frac{\text dN_2}{\text dt}&=-N_2 A_{21} \\ \Rightarrow \quad N_2(t)&=N_2(0)e^{-A_{21}t} \end{aligned} $$
$A_{21}$ is known as the Einstein $A$ coef
Stimulated emission
$$ \frac{\text dN_2}{\text dt}=-N_2 B_{21}\rho $$
where $\rho=I/c$ is the energy density in the radiation field and $B_{21}$ is the Einstein $B$ coef for stimulated emission
Absorption
$$ \frac{\text dN_1}{\text dt}=-N_1 B_{12}\rho $$
where $B_{12}$ is the Eintein $B$ coef for absorption
In general all three process can be combined into one expression
$$ \boxed{\frac{\text d N_2}{\text dt}=-N_2 B_{21} \rho - A_{21} N_2+N_1 B_{12}\rho} $$
🗒️ Notes: In thermal equilibrium we have
$\frac{\text d N_2}{\text dt}=0$
$\rho=\frac{8\pi h \nu^3}{c^3}\frac{1}{\exp(h\nu /kT)-1}$ (blackbody radiation)
$N_2/N_1=\exp(-h\nu /kT)$ (Boltzmann)
Hence by substituting and comparing coefficients
$$ \boxed{B_{12}=B_{21} \qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h \nu^3}{c^3}} $$
💼 Case: Consider a medium where the molecules are in an excited stated called a gain medium
In a length $\text dz$ the gain in intensity $\text dI$ is
$$ \begin{aligned} \text dI=s(N_2 B_{21}-N_1 B_{12})\frac Ic h\nu \,\text dz \end{aligned} $$
Using $B_{12}=B_{21}$ we get
$$ I(z)=I(0) e^{\gamma z} $$
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/f6b82d47-71ad-426d-929d-8e79f0001774/Stimulated_emission_cross_section.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/f6b82d47-71ad-426d-929d-8e79f0001774/Stimulated_emission_cross_section.png" width="40px" /> Stimulated emission cross section: $\sigma \equiv sB_{21}h \nu/c$.
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<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3d2e8622-9061-4ca5-84e3-39932e2a6567/gain_coefficient.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/3d2e8622-9061-4ca5-84e3-39932e2a6567/gain_coefficient.png" width="40px" /> gain coefficient: is $\gamma$ where is defined as follows and has dimensions $\rm m^{-1}$
$$ \gamma =s(N_2-N_1)B_{21}\frac{h\nu}{c}\equiv (N_2- N_1)\sigma $$
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🗒️ Notes:
