# Gain

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a measure of the strength of optical amplification

More specific terms: small-signal gain, laser gain, Raman gain

Categories: laser devices and laser physics, optical amplifiers

Units: %, [[decibel|dB]] or dimensionless number

Formula symbol: <$g$>, <$G$>

In photonics, the term *gain* is usually used to quantify the amplification of optical amplifiers or of a laser gain medium. Different meanings occur in the literature:

- The gain can simply be an amplification factor, i.e., the ratio of output power and input power.
- Particularly for small gains, the gain is often specified as a percentage. For example, 3% correspond to a power amplification factor of 1.03.
- Particularly large gains are often specified in decibels (dB), i.e., as 10 times the logarithm (to base 10) of the amplification factor. For example, a fiber amplifier may have a small-signal gain of 40 dB, corresponding to an amplification factor of 10
^{4}= 10 000. - One also often specifies a gain per unit length, or more precisely the natural logarithm of the amplification factor per unit length, or alternatively the decibels per unit length.

Apart from its magnitude, important properties of gain are its spectral bandwidth and its saturation characteristics.

The gain achieved e.g. in a fiber amplifier or the gain medium of a laser depends on the population densities in different electronic levels, which themselves depend on the optical intensities. Rate equation modeling may be used for calculating the gain and investigating its dependence on various influences. A basic equation for the local gain coefficient in an excited laser gain medium is

$$g = N_{\rm exc} \: \sigma_{\rm em}$$where <$g$> is in units of 1/m, <$N_{\rm exc}$> is the density of laser ions in the upper state (which generally depends on pump and signal intensities and may be time-dependent), and <$\sigma_{\rm em}$> is the emission cross-section at the relevant signal wavelength. If there are reabsorption and/or other propagation losses, these must be subtracted. In an optical fiber, where the excitation density applies to the fiber core only, an additional overlap factor may be included to take into account that now all signal light propagates in the fiber core. For the gain over some propagation length, that gain coefficient can be integrated, resulting in a dimensionless exponential gain factor. Applying the natural exponential function to that, one obtains the power amplification factor.

Other equations need to be used for other mechanisms of providing amplification, for example for parametric amplification.

## Fiber Amplifiers | rare earth ions, gain and pump absorption, ASE, pumping, double-clad fibers, pulse amplification, noise

Part 2: Gain and Pump Absorption | Signal Reabsorption

We explain how the local gain and pump absorption depend on the degree of excitation of the laser-active ions.

## More to Learn

- Tutorial on Fiber Amplifiers | rare earth ions, gain and pump absorption, ASE, pumping, double-clad fibers, pulse amplification, noise, Part 2: Gain and Pump Absorption | Signal Reabsorption

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