### Photonics I

The Einstein description and Einstein coefficients: consider two energy levels of an atom, E1 and E2. There are three processes by which radiation can interact with atoms in these states.

1. Spontaneous emission: an atom in the upper level decays to the lower level by the emission of a photon with energy $\bar{h}\omega_{21}=E_2-E_1$ The spontaneously emitted photon can be emitted in any direction.

Einstein’s postulate: the rate per unit volume at which atoms in the upper level (2) decay to the lower level (1) is equal to N2A21. N2 is the number of atoms per unit volume in level 2; A21 is a constant characteristic of the transition (units s-1).

1. Absorption: an atom in the lower level is excited to the upper level by the absorption of a photon of energy $\bar{h}\omega_{21}$

Einstein’s postulate: the rate per unit volume at which atoms in the lower level are excited to the upper level by absorption of photons of energy $\bar{h}\omega_{21}$ is equal to N1B12ρ(ω21).

N1 is the number of atoms per unit volume in level 1; ρ(ω21) is the energy density of radiation of angular frequency ω21; B12 is a constant characteristic of the transition (units m3 J-1 rad s-1).

1. Stimulated emission: an incident photon of energy $\bar{h}\omega_{21}$ stimulates an atom in the upper level to decay to the lower level by the emission of a second photon of energy $\bar{h}\omega_{21}$. The stimulated photon is emitted into the same mode as the incident photon, and hence has the same frequency, direction and polarisation as the incident photon.

Einstein’s postulate: the rate per unit volume at which atoms in the upper level decay to the lower level by the stimulated emission of photons of energy $\bar{h}\omega_{21}$ is equal to N1B21ρ(ω21).

B21 is a constant characteristic of the transition (units m3 J-1 rad s-1)

A12, B12 and B21 are known as the Einstein A and B coefficients.

At this point we are assuming that the energy levels are perfectly well-defined. We will deal with broadening later.

Density of states $g(1)dw=\frac{\omega^2}{\pi^2c^3}dw$ (we’ve included a factor of 2 because we have 2 polarisations).

Radiation modes are harmonic oscillators $E_n=(n+\frac{1}{2})\bar{h}\omega$

Photons are excitations of the different modes.

Mean number of photons

$\bar{n}_{\omega}=\frac{\sum_n ne^{\frac{-(n+\frac{1}{2})\bar{h}\omega}{kT}}}{\sum_ne^{\frac{-(n+\frac{1}{2})\bar{h}\omega}{kT}}}$

$\bar{n}_{\omega}=\frac{\sum_n ne^{\frac{-n\bar{h}\omega}{kT}}}{\sum_ne^{\frac{-n\bar{h}\omega}{kT}}}$

The numerator might look hard to work out, but if we put $x=\frac{\bar{h}\omega}{kT}$ and we see that the numerator is $\frac{-d}{dx}$ of the denominator. So if we sum over all n, we get

$\bar{n}_{\omega}=\frac{1}{e^{\frac{\bar{h}\omega}{kT}}-1}$ Planck distribution.

Energy density of radiation field in range ω -> ω+dω

$\rho(\omega)d\omega=\bar{n}_{\omega}g(\omega)d\omega \times \bar{h}{\omega}$

$\rho(\omega)d\omega=\frac{\bar{h}\omega^3}{\pi^2c^3}\frac{d\omega}{e^{\frac{\bar{h}\omega}{kT}}-1}$

Relations between the Einstein coefficients: for a given transition the Einstein coefficients are postulated to be constant, i.e. they depend upon the atomic energy levels but not the radiation field.

Take stationary atoms immersed in black body radiation at temperature T and energy ρ(ω). These atoms will be in dynamic equilibrium; i.e. a given atom will be making transitions but on average the total number of atoms in a given level will be constant.

The Principle of Detailed Balance states that in thermal equilibrium the transitions between any pair of levels are in dynamic equilibrium (each frequency ω must be in equilibrium and as each Ei -> Ej has a unique frequency, each transition must be in equilibrium).

So rate of transitions from 1->2 = rate of transitions from 2->1.

$N_1B_{12}\rho_B(\omega_{21}= N_2B_{21}\rho_B(\omega_{21}+N_2A_{21}$

Simple rearrangement gives

$\rho_B(\omega_{21})=\frac{N_2A_{21}}{N_1B_{12}-N_2B_{21}}$

$\rho_B(\omega_{21})=\frac{\frac{A_{21}}{B_{21}}}{\frac{N_1}{N_2}\frac{B_{12}}{B_{21}}-1}$

But in thermal equilibrium the population ratio is given by the Boltzmann factor

$\frac{N_2}{N_1}=\frac{g_2}{g_1}e^{-\frac{\bar{h}\omega_{21}}{kT}}$ where g1, g2 are degeneracies.

So $\rho_B(\omega_{21})=\frac{\frac{A_{21}}{B_{21}}}{\frac{g_2}{g_1}e^{-\frac{\bar{h}\omega_{21}}{kT}}\frac{B_{12}}{B_{21}}-1}$

But this must be consistent with Planck’s Law

$\frac{\frac{A_{21}}{B_{21}}}{\frac{g_2}{g_1}e^{-\frac{\bar{h}\omega_{21}}{kT}}\frac{B_{12}}{B_{21}}-1}=\frac{\bar{h}\omega^3}{\pi^2c^3}\frac{1}{e^{\frac{\bar{h}\omega}{kT}}-1}$

So

$g_1B_{12}=g_2B_{21}$

$A_{21}=\frac{\bar{h}\omega^3}{\pi^2c^3}B_{21}$