# Füchtbauer–Ladenburg Equation

Definition: an equation used for calculating emission cross-sections of laser gain media

German: Füchtbauer-Ladenburg-Gleichung

Category: physical foundations

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/xy1

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The Füchtbauer–Ladenburg equation (often called *Fuchtbauer–Ladenburg equation*) is part of a procedure for determining emission cross-sections of laser gain medium. The procedure is based on the analysis of fluorescence related to an electronic transition of a medium. The wavelength-dependent fluorescence intensity is essentially proportional to the emission cross-section times the third power of the optical frequency. In situations with a small emission bandwidth, the latter factor may be regarded as constant, so that the fluorescence intensity is regarded as simply proportional to the emission cross-sections. It is further assumed that the spectral shape of the recorded fluorescent light is not modified e.g. by wavelength-selective absorption and amplification processes in the medium.

While the spectral shape of fluorescence light is relatively easily measured, it is much more challenging to measure absolute values because various factors such as the doping concentration, degree of electronic excitation, collection efficiency and the responsivity of the used photodetector would have to be known. Therefore, the absolute scaling of the obtained cross-section spectrum is often obtained in some other way. According to the Füchtbauer–Ladenburg method, one exploits the fact that the quantum efficiency of a laser transition is often near unity. This means that the upper-state lifetime is close to the radiative lifetime, which itself is determined by the emission cross-sections for transitions to any lower-lying energy levels. This is quantitatively described by the equation

$$\frac{1}{{{\tau _{{\rm{rad}}}}}} = \frac{{8\pi \;{n^2}}}{{{c^2}}}\;\int {{\nu ^2}{\sigma _{{\rm{em}}}}(\nu )\;{\rm{d}}\nu } = 8\pi \;{n^2}c\;\int {\frac{{{\sigma _{{\rm{em}}}}(\lambda )}}{{{\lambda ^4}}}\;{\rm{d}}\lambda } $$for the inverse radiative lifetime, where <$\nu$> is the optical frequency, <$n$> is the refractive index, <$c$> is the vacuum velocity of light, and <$\sigma_\rm{em}(\nu )$> denotes the frequency-dependent emission cross-sections. The equation can be considered as an extension of the relation between the Einstein <$A$> and <$B$> coefficients. With the above-mentioned approximation for a narrow emission bandwidth, this leads to

$$\frac{1}{{{\tau _{{\rm{rad}}}}}} \approx \frac{{8\pi c\;{n^2}}}{{{{\bar \lambda }^4}}}\;\int {{\sigma _{{\rm{em}}}}(\lambda )\;{\rm{d}}\lambda } $$where the dominator contains the mean wavelength of the considered transition. Using the fact that the fluorescence intensity <$I(\lambda )$> (e.g. in units of W/nm) is proportional to the emission cross-section (within a narrow frequency interval), this leads to the Füchtbauer–Ladenburg equation

$${\sigma _{{\rm{em}}}}(\lambda ) = \frac{{{{\bar \lambda }^4}}}{{8\pi c\;{n^2}{\tau _{{\rm{rad}}}}}}\frac{{I(\lambda )}}{{\int {I(\lambda )\;{\rm{d}}\lambda } }}$$This equation could be made more accurate for cases with broader bandwidth by including a factor <$(\lambda / \lambda_\textrm{mean})^5$>, taking into account the wavelength-dependent factors as already mentioned above. The power in that correction factor has to be reduced to 4 if the intensity is understood as a photon flux.

Note that the fluorescence intensity function <$I(\lambda )$> may be substantially modified when analyzing only guided light in an optical fiber; it is better to measure the spectra in the side light in order to avoid any possible distorting influences of absorption, amplification and also waveguide effects.

If the intensity is normalized to be 1 at the peak of the optical spectrum, the integral in the denominator can be interpreted as the effective emission bandwidth (which can be verified easily for a spectrum with rectangular shape).

One should keep in mind the approximations used for the Füchtbauer–Ladenburg equation. For example, the formula works only for relatively narrow spectra. Also, it is important to consider all lines of a fluorescence spectrum. If these are spread over a substantial spectral range, the approximation of a narrow bandwidth is not fulfilled (even if the single lines are narrow). It is not difficult, though, to generalize the equation for that situation.

### Bibliography

[1] | W. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids”, Phys. Rev. 128 (5), 2154 (1962); https://doi.org/10.1103/PhysRev.128.2154 |

[2] | W. F. Krupke, “Induced-emission cross-sections in neodymium laser glasses”, IEEE J. Quantum Electron. 10, 450 (1974); https://doi.org/10.1109/JQE.1974.1068162 |

[3] | B. F. Aull and H. Jenssen, “Vibronic interactions in Nd:YAG resulting in nonreciprocity of absorption and stimulated emission cross-sections”, IEEE J. Quantum Electron. 18 (5), 925 (1982); https://doi.org/10.1109/JQE.1982.1071611 |

See also: upper-state lifetime, radiative lifetime, transition cross-sections, reciprocity method

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