Editing Stimulated emission (section)

== Mathematical model ==

Stimulated emission can be modelled mathematically by considering an atom that may be in one of two electronic energy states, a lower level state (possibly the ground state) (1) and an ''excited state'' (2), with energies ''E''<sub>1</sub> and ''E''<sub>2</sub> respectively.

If the atom is in the excited state, it may decay into the lower state by the process of [[spontaneous emission]], releasing the difference in energies between the two states as a photon. The photon will have [[frequency]] ''ν''<sub>0</sub> and energy ''hν''<sub>0</sub>, given by:
<math display="block">E_2 - E_1 = h \, \nu_0</math>
where ''h'' is the [[Planck constant]].

Alternatively, if the excited-state atom is perturbed by an electric field of frequency ''ν''<sub>0</sub>, it may emit an additional photon of the same frequency and in phase, thus augmenting the external field, leaving the atom in the lower energy state. This process is known as stimulated emission.

In a group of such atoms, if the number of atoms in the excited state is given by ''N''<sub>2</sub>, the rate at which stimulated emission occurs is given by
<math display="block">\frac{\partial N_2}{\partial t} = 
-\frac{\partial N_1}{\partial t} =
 - B_{21} \, \rho (\nu) \, N_2 </math>
where the [[proportionality constant]] ''B''<sub>21</sub> is known as the ''[[Einstein coefficients|Einstein B coefficient]]'' for that particular transition, and ''ρ''(''ν'') is the radiation density of the incident field at frequency ''ν''. The rate of emission is thus proportional to the number of atoms in the excited state ''N''<sub>2</sub>, and to the density of incident photons.

At the same time, there will be a process of atomic absorption which ''removes'' energy from the field while raising electrons from the lower state to the upper state. Its rate is precisely the negative of the stimulated emission rate,
<math display="block">\frac{\partial N_2}{\partial t} = 
-\frac{\partial N_1}{\partial t} =
  B_{12} \, \rho (\nu) \, N_1 .</math>

The rate of absorption is thus proportional to the number of atoms in the lower state, ''N''<sub>1</sub>. The B coefficients can be calculated using dipole approximation and time dependent perturbation theory in quantum mechanics as:<ref name="Hilborn2002">{{Cite web |last=Hilborn |first=Robert |date=2002 |title=Einstein coefficients, cross sections, f values, dipole moments, and all that |url=https://arxiv.org/ftp/physics/papers/0202/0202029.pdf |access-date=}}</ref><ref name=":0">{{Cite web |last=Segre |first=Carlo |title=The Einstein coefficients - Fundamentals of Quantum Theory II (PHYS 406) |url=http://phys.iit.edu/~segre/phys406/21S/modules_16.pdf |page=32}}</ref>
<math display="block">B_{ab}=\frac{ e^2}{6 \epsilon_0 \hbar^2} |\langle a|\vec{r}| b\rangle|^2 </math>
where ''B'' corresponds to energy distribution in terms of frequency ''ν''. The B coefficient may vary based on choice of energy distribution function used, however, the product of energy distribution function and its respective ''B'' coefficient remains same.

Einstein showed from the form of Planck's law,{{citation needed|date=April 2021}} that the coefficient for this transition must be identical to that for stimulated emission:
<math display="block"> B_{12} =B_{21} . </math>

Thus absorption and stimulated emission are reverse processes proceeding at somewhat different rates. Another way of viewing this is to look at the ''net'' stimulated emission or absorption viewing it as a single process. The net rate of transitions from ''E''<sub>2</sub> to ''E''<sub>1</sub> due to this combined process can be found by adding their respective rates, given above:
<math display="block">\frac{\partial N_1^\text{net}}{\partial t} = 
- \frac{\partial N_2^\text{net}}{\partial t} = 
  B_{21} \, \rho(\nu) \, (N_2-N_1)  =
  B_{21} \, \rho(\nu) \, \Delta N .</math>

Thus a net power is released into the electric field equal to the photon energy ''hν'' times this net transition rate. In order for this to be a positive number, indicating net stimulated emission, there must be more atoms in the excited state than in the lower level: <math>\Delta N > 0</math>. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition <math>N_2 > N_1</math> is known as a [[population inversion]], a rather unusual condition that must be effected in the [[gain medium]] of a laser.

The notable characteristic of stimulated emission compared to everyday light sources (which depend on spontaneous emission) is that the emitted photons have the same frequency, phase, polarization, and direction of propagation as the incident photons. The photons involved are thus mutually [[coherence (physics)|coherent]]. When a population inversion (<math>\Delta N > 0</math>) is present, therefore, [[optical amplification]] of incident radiation will take place.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency <math>\nu_0</math> corresponding to the energy of the transition. At frequencies offset from <math>\nu_0</math> the strength of stimulated (or spontaneous) emission will be decreased according to the so-called [[spectroscopic line shape|line shape]].
Considering only [[homogeneous broadening]] affecting an atomic or molecular resonance, the [[Atomic spectral line|spectral line shape function]] is described as a [[Cauchy distribution|Lorentzian distribution]]
<math display="block"> g'(\nu) = {1 \over \pi } { (\Gamma / 2) \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 }</math>
where <math> \Gamma </math> is the [[full width at half maximum]] or FWHM bandwidth.

The peak value of the Lorentzian line shape occurs at the line center, <math> \nu = \nu_0</math>. A line shape function can be normalized so that its value at <math>\nu_0</math> is unity; in the case of a Lorentzian we obtain
<math display="block">g(\nu) = { g'(\nu) \over g'(\nu_0) } = { (\Gamma / 2)^2 \over (\nu - \nu_0)^2 + (\Gamma /2 )^2 } .</math>

Thus stimulated emission at frequencies away from  <math>\nu_0</math> is reduced by this factor. In practice there may also be broadening of the line shape due to [[doppler broadening|inhomogeneous broadening]], most notably due to the [[Doppler effect]] resulting from the distribution of velocities in a gas at a certain temperature. This has a [[Gaussian]] shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through a [[convolution]] of the individual line shape functions involved. Therefore, optical amplification will add power to an incident optical field at frequency <math>\nu</math> at a rate given by
<math display="block">P =h\nu \, g(\nu) \, B_{21} \, \rho(\nu) \, \Delta N .</math>