Supplement 2.1: Transition Probabilities

The Einstein coefficients

Which conditions must be present in order to achieve an amplification of photons in a laser resonator?

Imagine a chamber in which atoms in ground state as well as atoms in an excited state are present; we consider one excited state only. The number of atoms in ground state are indicated as $N_{1}$ , and the number of atoms in an excited state are indicated as $N_{2}$ . In addition, photons having energy equivalent to the laser wavelength are also present and these can come to an interaction with the atoms. The energy states that are involved in the process of producing laser light are denoted as as laser niveaus.

A photon can interfer either with an atom in ground state or an atom in excited state. Should it interfer with an atom in ground state, there is a certain probability that the photon shall be absorbed by the atom, and the atom moves from ground state into excited state. The probability of absorption is described with the first Einstein coefficient $B_{12}$ , where index 1 stands for ground state and index 2 stands for excited state.

The time rate of transition for the number of atoms in ground state absorbing a photon, ${(\frac{d N_{1}}{d t})}_{a b s}$ , is proportional to the number of atoms in ground state, $N_{1}$ , and to the density of photons having a frequency suited for absorption, $ρ (f)$ . Using the Einstein coefficient $B_{12}$ as proportionality factor, we obtain the equation

{(\frac{d N_{1}}{d t})}_{a b s} =- B_{12} \cdot ρ (f) \cdot N_{1}

Similarly, we derive the equation for the time rate of transition for the number of atoms in an excited state by using the second Einstein coefficient $B_{21}$ :

{(\frac{d N_{2}}{d t})}_{i n d . e m .} = - B_{21} \cdot ρ (f) \cdot N_{2}

It holds $B_{21} = B_{12}$ , i.e., the probability of absorption when a photon strikes an atom in ground state is therefore the same as the probabiltiy of induced emission when a photon strikes an excited atom.

Derivation ↓

In addition to the equations for absorption and induced emission, we also have an equation for spontaneous emission. Since it is spontaneous, it does not depend on the intensity of the radiation field $ρ (f)$ :

{(\frac{d N_{2}}{d t})}_{s p . e m .} = - A_{21} \cdot N_{2}

Thermodynamic equilibrium requires that the rates of transition between ground state and excited state should be identical because the system is not changeable. Hence:

{(\frac{d N_{1}}{d t})}_{a b s} = {(\frac{d N_{2}}{d t})}_{s p . e m .} + {(\frac{d N_{2}}{d t})}_{i n d . e m .}

\Rightarrow B_{12} \cdot ρ (f) \cdot N_{1} = (A_{12} + B_{12} \cdot ρ (f)) \cdot N_{2}

Now we can convert to the population ratio applying the Boltzmann distribution (which is discussed on another page in this chapter). The energy difference of the levels is given by $h \cdot f$ :

\frac{N_{1}}{N_{2}} = \frac{A_{21} + B_{21} \cdot ρ (f)}{B_{12} \cdot ρ (f)} = e^{\frac{h \cdot f}{k \cdot T}}

Solving for the energy density yields:

ρ (f) = \frac{A_{21}}{e^{\frac{h \cdot f}{k \cdot T}} \cdot B_{12} - B_{21}}

Assuming that the temperature becomes infinitively high (where the equation must hold as well), it follows $ρ (f) \to \infty$ and since $A_{21} < \infty$ it follows that the denominator must be zero, and hence $B_{12} = B_{21}$ .

The change in the number of atoms in the ground state is exactly the same as the change in the number of photons with the corresponding energy. This is because for every atom that transits from ground state to excited state, a photon disappears.

The change in the number of atoms in the excited state is exactly the same as the negative change in the number of photons with the corresponding energy. This is because for every atom that transits from excited state into ground state during an induced emission, i.e., an additional photon is released.

Therefore, in order to achieve an increasing number of photons, we derive:

{(\frac{d N_{2}}{d t})}_{i n d . e m .} - {(\frac{d N_{1}}{d t})}_{a b s} > 0

\Rightarrow B_{21} \cdot ρ (f) \cdot N_{2} - B_{12} \cdot ρ (f) \cdot N_{1} > 0

\Rightarrow N_{2} > N_{1}

Hence, in order to obtain laser light, there must be more atoms in an excited state than atoms in the ground state. This condition is known as population inversion.

Population inversion is necessary in order to obtain laser light.

Remote Sensing Using Lasers

Supplement 2.1: Transition Probabilities

The Einstein coefficients