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# Saturation Energy

Definition: a measure of the incident optical pulse energy required for achieving significant saturation of an absorber or a gain medium

German: Sättigungsenergie

Units: J

Formula symbol: <$E_\textrm{sat}$>

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The saturation energy of a laser gain medium is the pulse energy of an incident short signal pulse which leads to a reduction in the gain to <$1/e$> (≈ 37%) of its initial value. Similarly, the saturation energy of a saturable absorber is defined based on the loss reduction.

Precisely speaking, the mentioned gain or loss reduction refers to the gain or loss directly after the pulse – not to the average gain or loss for the pulse itself. The pulse duration is assumed to be so short (typically shorter than the upper-state lifetime) that both spontaneous emission and the addition of energy from the pump source (for continuous pumping of a laser gain medium) are negligible.

Usually, the gain or loss is assumed to be small, i.e. input and output pulse energies are similar. (In the case of large gain, the quantity can be related to the input or output pulse energy, leading to different values.)

When the pulse energy equals the saturation energy, the gain is reduced to ≈ 37% of the initial value.

The saturation fluence is the saturation energy per unit area. (See also the article on the term fluence.)

For a low-gain laser amplifier, saturation fluence and energy can be calculated according to

$${F_{{\rm{sat}}}} = \frac{{h\nu }}{{{\sigma _{{\rm{em}}}} + {\sigma _{{\rm{abs}}}}}},\quad {E_{{\rm{sat}}}} = A\;{F_{{\rm{sat}}}} = \frac{{A\;h\nu }}{{{\sigma _{{\rm{em}}}} + {\sigma _{{\rm{abs}}}}}}$$

where <$h \nu$> is the photon energy at the signal wavelength, <$\sigma_\rm{em}$> and <$\sigma_\rm{abs}$> are the emission and absorption cross-sections at the emission wavelength, and <$A$> is the mode area. The quantity <$\sigma_\rm{abs}$> is zero for four-level gain media (exhibiting no reabsorption on the laser transition) but should not be forgotten for quasi-three-level laser gain media.

## Calculator for the Saturation Energy

 Wavelength: Transition cross-section: Beam radius: Saturation energy: calc

Enter input values with units, where appropriate. After you have modified some inputs, click the “calc” button to recalculate the output.

When <$N$> passes of a pulse through an amplifier medium are arranged, an effective saturation energy can be defined, which is reduced by a factor <$N$>.

These quantities can be defined in an analogous way for saturable absorbers. For example, a pulse with a fluence equal to the saturation fluence reduces the saturable loss of a SESAM to <$1/e$> of its initial value.

Note that the saturation fluence of a saturable absorber does not depend on the thickness of the absorber layer, unless the thickness is so large that the fluence is reduced substantially within the device.

## Importance of the Saturation Energy

The saturation energy plays an important role in various areas of laser physics and laser design. Some examples are:

• It determines the pulse energy required for extracting most of the stored energy from a gain medium of an amplifier. A problem with some media having low emission cross-sections is that the saturation fluence is higher than the damage fluence, so that complete energy extraction is not possible, except with multiple passes through the gain medium.
• The saturation energy determines the relation between gain and stored energy: the logarithmic gain coefficient of a low-gain four-level laser amplifier equals the stored energy divided by the saturation energy.
• The pulse energy obtained from a Q-switched laser can not be much higher than the saturation energy, except if the gain is very high. Particularly for passive Q switching, the saturation energy has a strong impact on the pulse energy and pulse repetition rate.
• The ratio of pulse energy and saturation energy of a saturable absorber, as used e.g. in a mode-locked laser, is called the saturation parameter. It determines how strong the saturation by a single pulse is, and is one of the most important design parameters of a passively mode-locked laser. Its value depends on both the saturation fluence of the absorber and the mode area on the absorber in the laser resonator.

## Questions and Comments from Users

2022-10-11

What is the exact relationship between the seed energy and extractable energy?

Further, the extractable energy also depends on the pulse duration, pulse shape and repetition rate, but above the formula does not consider any of these, but still it can be used for all pulse durations/shapes/reprates?

As the energy extraction is directly related to the gain reduction, Fig. 1 answers the first question.

The gain reduction is independent of pulse duration and pulse shape; it only matters how much energy (how many photons) a pulse has. Of course, it is assumed that the pulse duration is short enough that no substantial amount of energy is lost by spontaneous emission. The pulse repetition rate also does not matter; the amplifier or absorption behavior should not depend on what amplification or absorption processes have been done in the past – except under more complicated circumstances, e.g. when heating of the device matters.

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