# Slope Efficiency

Definition: differential power efficiency of a laser

Alternative term: differential efficiency

German: differenzielle Effizienz

Formula symbol: η_{sl}

Units: %, dimensionless number

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Author: Dr. RĂ¼diger Paschotta

An important property of an optically pumped laser is its slope efficiency (or *differential efficiency*), defined as the slope of the curve obtained by plotting the laser output versus the pump power (Figure 1).
In many cases, this curve is close to linear, so that the specification of the slope efficiency as a single number makes sense.
The output power of the laser with threshold pump power *P*_{th} for a given pump power *P*_{p} (above that threshold) can then be simply calculated with the following equation:

Note that the slope efficiency may be defined with respect to *incident pump power* or *absorbed pump power*.
For comparisons of power efficiency, it is usually fair to compare slope efficiencies with respect to incident powers, so that the pump absorption efficiency is taken into account.
However, there are cases where values based on absorbed pump power are useful, e.g. for judging the intrinsic efficiency of the gain medium.

The slope efficiency can also be defined for other laser-like devices such as Raman lasers and optical parametric oscillators.

For electrically pumped lasers, one may in principle define the slope efficiency with respect to electrical pump power or pump current, in the latter case obtaining units of W/A. However, the term is usually applied to optically pumped lasers.

## Factors Influencing the Slope Efficiency

In simple situations (e.g., for some diode-pumped YAG lasers), the slope efficiency is essentially determined by the product of the following factors:

- pump absorption efficiency
- the ratio of laser to pump photon energy (→
*quantum defect*) - the quantum efficiency of the gain medium
- the output coupling efficiency of the laser resonator

For lamp-pumped lasers, it can be difficult to calculate the slope efficiency due to the difficulties of determining the fraction of pump power which is absorbed in the laser crystal, the position-dependent extraction efficiency and the complicated spectral dependence.

The optimization of the laser output power for a given pump power usually involves a trade-off between high slope efficiency and low threshold pump power. The optimum is usually a situation where the pump power is a few times the threshold pump power, and the slope efficiency is reduced below the value attainable with a stronger degree of output coupling.

## Nonlinear Dependence of Output Power on Pump Power

A linear relation between output and input power (as in the equation shown above) is often found in lasers despite certain nonlinearities in the system. For example, one may have some amount of saturation of pump absorption. However, the “clamping” of the round-trip gain to 0 dB often suppresses the influence of such nonlinearities, typically by also clamping the upper-state population (averaged along the propagation direction in the gain medium). As a result, the pump absorption can no longer be more strongly saturated above laser threshold.

However, nonlinear curves can occur under certain circumstances, e.g. as a consequence of quasi-three-level characteristics of the gain medium in combination with a transverse redistribution of excitation density, or in situations with substantial thermal effects. For example, there can be a thermal roll-over, if the gain medium becomes hot at high pump powers, and this decreases the power conversion efficiency. A laser may even stop working for too high pump powers, for example when it leaves the stability zone of the laser resonator due to excessive thermal lensing.

In case of such nonlinear curves, the slope efficiency is often determined from some approximately linear part.
Alternatively, one can define the *differential slope efficiency* as the derivative of the output power with respect to the pump power.

While a simple linear dependence often results for simple laser models, a more or less nonlinear dependence may be found when using more sophisticated models. For example, a simple model of a quasi-three-level laser, which is based on flat-top beams in the resonator, may show a linear dependence, and some nonlinearity is obtained only when fully treating the transverse dimensions.

In some cases, e.g. for optical parametric oscillators,the differential slope efficiency with respect to incident pump power can even well exceed 100% under certain circumstances.

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See also: lasers, laser threshold, threshold pump power, output coupling efficiency, quantum efficiency, The Photonics Spotlight 2006-08-23

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