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Mathematics for Physics

148 Stimulated Emission

A radiative transition in which an incoming photon induces an excited atom to decay that is used to release a second photon of the same frequency, phase, and direction.

The stimulated frequency condition. An applied field of frequency \(f\) can force an atom from \(E_{i}\) to \(E_{f}\) with \(E_{i}>E_{f}\), creating a photon. This principle is used to amplify light in a laser.

The stimulated-emission condition is

\[ hf = E_{i}-E_{f} \]

where

  • \(h\) is Planck’s constant.
  • \(f\) is the frequency of the photon.
  • \(E_{i}\) and \(E_{f}\) are the upper and lower energies.

The Einstein \(B\) rate. The stimulated-emission rate is proportional to the radiation energy density and to the Einstein \(B\) coefficient. This principle is used to write the downward rate induced by the field.

The stimulated-emission rate is

\[ W_{\mathrm{stim}} = B_{fi}\rho(f) \]

where

  • \(W_{\mathrm{stim}}\) is the stimulated-emission rate.
  • \(B_{fi}\) is the Einstein \(B\) coefficient for emission.
  • \(\rho(f)\) is the radiation energy density.

Coherence of the emitted photon. The emitted photon matches the incoming photon in frequency, phase, and direction. This principle is used to produce coherent amplification.

148.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — stimulated emission.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — Einstein \(B\) coefficient.