Stimulated emission

Electrons and their interactions with electromagnetic fields are important in our understanding of chemistry and physics. In the classical view, the energy of an electron orbiting an atomic nucleus is larger for orbits further from the nucleus of an atom. However, quantum mechanical effects force electrons to take on discrete positions in orbitals. Thus, electrons are found in specific energy levels of an atom, two of which are shown below:-

When an electron absorbs energy either from light (photons) or heat (phonons), it receives that incident quantum of energy. But transitions are only allowed between discrete energy levels such as the two shown above. This leads to emission lines and absorption lines.

When an electron is excited from a lower to a higher energy level, it is unlikely for it to stay that way forever. An electron in an excited state may decay to a lower energy state which is not occupied, according to a particular time constant characterizing that transition. When such an electron decays without external influence, emitting a photon, that is called "spontaneous emission". The phase and direction associated with the photon that is emitted is random. A material with many atoms in such an excited state may thus result in radiation which has a narrow spectrum (centered around one wavelength of light), but the individual photons would have no common phase relationship and would also emanate in random directions. This is the mechanism of fluorescence and thermal emission.

An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom without being absorbed. As the electron in the atom makes a transition between two stationary states (neither of which shows a dipole field), it enters a transition state which does have a dipole field, and which acts like a small electric dipole, and this dipole oscillates at a characteristic frequency. In response to the external electric field at this frequency, the probability of the electron entering this transition state is greatly increased. Thus, the rate of transitions between two stationary states is increased beyond that of spontaneous emission. A transition from the higher to a lower energy state produces an additional photon with the same phase and direction as the incident photon; this is the process of stimulated emission.

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 E₁ and E₂ 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 ν₀ and energy hν₀, given by:

${\displaystyle E_{2}-E_{1}=h\,\nu _{0}}$

where h is Planck's constant.

Alternatively, if the excited-state atom is perturbed by an electric field of frequency ν₀, 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 N₂, the rate at which stimulated emission occurs is given by

${\displaystyle {\frac {\partial N_{2}}{\partial t}}=-{\frac {\partial N_{1}}{\partial t}}=-B_{21}\,\rho (\nu )\,N_{2}}$

where the proportionality constant B₂₁ is known as the 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 N₂, 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 given by an essentially identical equation,

${\displaystyle {\frac {\partial N_{2}}{\partial t}}=-{\frac {\partial N_{1}}{\partial t}}=B_{12}\,\rho (\nu )\,N_{1}}$.

The rate of absorption is thus proportional to the number of atoms in the lower state, N₁. Einstein showed that the coefficient for this transition must be identical to that for stimulated emission:

${\displaystyle B_{12}=B_{21}}$.

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 E₂ to E₁ due to this combined process can be found by adding their respective rates, given above:

${\displaystyle {\frac {\partial N_{1}^{\text{net}}}{\partial t}}=-{\frac {\partial N_{2}^{\text{net}}}{\partial t}}=B_{21}\,\rho (\nu )\,(N_{2}-N_{1})=B_{21}\,\rho (\nu )\,\Delta N}$.

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: ${\displaystyle \Delta N>0}$. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition ${\displaystyle N_{2}>N_{1}}$ 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 coherent. When a population inversion (${\displaystyle \Delta N>0}$) 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 ${\displaystyle \nu _{0}}$ corresponding to the energy of the transition. At frequencies offset from ${\displaystyle \nu _{0}}$ the strength of stimulated (or spontaneous) emission will be decreased according to the so-called line shape. Considering only homogeneous broadening affecting an atomic or molecular resonance, the spectral line shape function is described as a Lorentzian distribution

${\displaystyle g'(\nu )={1 \over \pi }{(\Gamma /2) \over (\nu -\nu _{0})^{2}+(\Gamma /2)^{2}}}$

where ${\displaystyle \Gamma \,}$ is the full width at half maximum or FWHM bandwidth.

The peak value of the Lorentzian line shape occurs at the line center, ${\displaystyle \nu =\nu _{0}}$. A line shape function can be normalized so that its value at ${\displaystyle \nu _{0}}$ is unity; in the case of a Lorentzian we obtain

${\displaystyle g(\nu )={g'(\nu ) \over g'(\nu _{0})}={(\Gamma /2)^{2} \over (\nu -\nu _{0})^{2}+(\Gamma /2)^{2}}}$.

Thus stimulated emission at frequencies away from ${\displaystyle \nu _{0}}$ is reduced by this factor. In practice there may also be broadening of the line shape due to 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 ${\displaystyle \nu }$ at a rate given by

${\displaystyle P=h\nu \,g(\nu )\,B_{21}\,\rho (\nu )\,\Delta N}$.

The stimulated emission cross section is

${\displaystyle \sigma _{21}(\nu )=A_{21}{\frac {\lambda ^{2}}{8\pi n^{2}}}g(\nu )}$

where

A₂₁ is the Einstein A coefficient,

λ is the wavelength in vacuum,

n is the refractive index of the medium (dimensionless), and

g(ν) is the spectral line shape function.

Stimulated emission can provide a physical mechanism for optical amplification. If an external source of energy stimulates more than 50% of the atoms in the ground state to transition into the excited state, then what is called a population inversion is created. When light of the appropriate frequency passes through the inverted medium, the photons are either absorbed by the atoms that remain in the ground state or the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction. Since more atoms are in the excited state than in the ground state then an amplification of the input intensity results.

The population inversion, in units of atoms per cubic meter, is

${\displaystyle \Delta N_{21}=N_{2}-{g_{2} \over g_{1}}N_{1}}$

where g₁ and g₂ are the degeneracies of energy levels 1 and 2, respectively.

• Saleh, Bahaa E. A. & Teich, Malvin Carl (1991). Fundamentals of Photonics. New York: John Wiley & Sons. ISBN 0-471-83965-5.
• Alan Corney (1977). Atomic and Laser Spectroscopy. Oxford: Oxford Uni. Press. ISBN 0-19-921145-0. ISBN 978-0-19-921145-6.

.3 Laser Fundamentals, William T. Silfvast