# Absorption and Emission of Light

The classical description of light absorption is done by the complex dielectricity describing the phenomena of absorption and refraction with the help of the Maxwell equations. In contrast, a quantum mechanical approach formulates the interaction between the electric field and the dipole strength of atoms and molecules. The spectra for light absorption and light emission can only be understood in a full quantum mechanical picture. Here we use rate equations to model the dynamics of an electron that moves from one state to another. The coupling of photons transferring the system between an electronic ground state *N _{0}* and an electronically excited state

*N*is shown in Figure 7.

_{1}**Figure 7. **Absorption (left panel) spontaneous emission (middle panel) and induced emission (right panel) of light due to electronic transitions between the electronic ground state N_{0} and the excited state **N _{1}.**

In full accordance with the concept of rate equations such systems are described by linear differential equations containing the ground state *N*_{0}, the excited state *N*_{1}, probability coefficients which are called the “Einstein coefficients” and the frequency dependent energy density *u(oa,t*) = *n(t)-%-(o* where n(t) describes the frequency dependent photon density according to eq. 21 for the monochromatic beam.

If the laser exclusively excites the first excited state ( *N*_{1} ) then the change of the ground state population denotes to:

in full accordance with the general master equation formulation presented in eq. 27. If we assume a continuous laser beam the time dependency of the energy density vanishes.

In eq. 28, the transfer matrix elements are T_{10} = B_{10}(©) *•u(cai)* for the absorption process and T_{01} = A_{01} + B_{01}(©) *-u(co)* for the spontaneous emission (A_{01}) and the induced emission (B_{01}(©) -n(®)). A_{01} and

*B*_{01}*, B** _{10}* are called the Einstein coefficients for spontaneous emission, induced emission and absorption, respectively.

The probability for absorption and induced emission is strongly dependent on the wavelength distribution of the photon flux with the maxi-

*E _{l} N_{l} E_{0} N*

_{0}E,

mum probability for *co = co _{0} =* — - о о •

^{-}, when the ground

state energy is defined as zero: (E_{0}(N_{0}) := 0).

To analyse the relation between the Einstein coefficients and the population densities a general system of the two states N. and *N** _{k}* is investigated in the equilibrium, i.e.

*N.=N*0. Then eq. 28 claims (here the indices i and k stand exemplarily for the two states

_{k}=*N*

*and*

_{t}*N*

*):*

_{k}*B _{ik} (со)* •

*u(co)- N*) +

_{k}(t*A*) =

_{ik}? N_{k}(t*B*). (29)

_{ki}(со) ? u(co)- N_{t}(tWe can look at eq. 29 for the special case of a system of quantisized niveaus coupled to a thermal radiation field (i.e. the absorption, the emission and the induced emission is just due to the thermal radiation that couples to the system). In that case we can express the equilibrium popu-

* _{N} *(

^{E}<~

^{E}k)

lation of *N** _{k}* /

*N*. according to eq. 10 (see chapter 1.3):

^{k}-*e*

*and*

^{kBT}' *N** _{t}* g

eq. 29 writes as:

Eq. 30 can be solved for the energy density of the radiation field *u(oo)* to

which uses the fact that the energy of the radiation *Tioo* must be equal to the energetic difference *h oo = E _{k}-E_{t}*. A more accurate derivation would use the spectral line shape functions for

*B*and

_{k}(со)*B*in eq. 31.

_{k}. (со)Now the Einstein coefficients in eq. 31 can be identified for special cases. For example in the thermal radiation field one expects that the energy density distribution is in accordance with Max Planck’s derivation of the black body radiation field:

If eq. 32 is compared with eq. 31 it is found: *B _{kl}g_{l} = B_{lk}g*

_{k}, i.e. the probability per time unit for the absorption equals the induced emission

with respect to the degeneration of the states and *—— =* regarding

^{B}ik ^{c}

the relation between spontaneous emission *— _{ik}* and induced emission

*B*

_{k}, i.e. the induced emission becomes more probable for longer wavelengths with the third power of the photon wavelength.

Most experiments presented in this study were performed with very low radiation intensities and therefore the induced emission is not relevant. All calculations were evaluated taking into account only the absorption and the spontaneous emission. Induced emission is the dominant process for all laser effects. For a monograph regarding lasers see (Eichler and Eichler, 1991).