# Quantum Defect

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: energy difference between pump and laser photons

Category: physical foundations

Units: %

Formula symbol: <$q$>

In most lasers, the laser wavelength is longer than the pump wavelength (exception: upconversion lasers). This means that the energy of the laser photons is smaller than that of the pump photons – there is a so-called *Stokes shift*. As a consequence, the power efficiency of the laser could not be 100% even if every pump photon could be converted into a laser photon.

The quantum defect is defined as the difference in photon energies:

$$q = h{\nu _{{\rm{pump}}}} - h{\nu _{{\rm{laser}}}} = h{\nu _{{\rm{pump}}}} \cdot \left( {1 - \frac{{h{\nu _{{\rm{laser}}}}}}{{h{\nu _{{\rm{pump}}}}}}} \right) = h{\nu _{{\rm{pump}}}} \cdot \left( {1 - \frac{{{\lambda _{{\rm{pump}}}}}}{{{\lambda _{{\rm{laser}}}}}}} \right)$$It is also often specified as a percentage of the pump photon energy, effectively using only the parentheses in the equation above. In any case, it sets a lower limit to the loss in the conversion from pump power to laser power, i.e. an upper limit to the power efficiency.

In the same way, the quantum defect can be calculated for an optical amplifier.

As an example, consider the quantum defect of a Nd:YAG laser, which is pumped at 808 nm and lasing at 1064 nm; the quantum defect is 0.368 eV or 24.1%.

Some laser crystals (e.g. those doped with ytterbium) have a particularly small quantum defect of only a few percent of the pump photon energy, leading to potentially very high power efficiency. However, a small quantum defect also leads to quasi-three-level behavior of the gain medium, which makes certain aspects of laser design more sophisticated, and may even make it more difficult to achieve a high wall-plug efficiency.

There are special cases, for example upconversion lasers, where the definition of the quantum defect needs to be adapted, e.g. because multiple pump photons are involved.

The quantum defect is *not* related to the quantum efficiency. The latter refers to the average number of output photons per pump photon, rather than to the photon energies. There are cases with high quantum efficiency but large quantum defect, or vice versa.

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