# Noise Figure

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a measure of the excess noise added in an amplifier

Categories: optical amplifiers, fluctuations and noise

Units: [[decibel|dB]]

Formula symbol: <$F$>

The *noise factor* <$F$> of an (electronic or optical) amplifier is a measure of how much excess noise the amplifier adds to the signal. More precisely, it is a factor which indicates how much higher the intensity noise power spectral density of the amplified output is compared with the input noise power spectral density times the amplification factor <$G$>, assuming that the input noise is at the standard noise level:

The mentioned condition is essential: for input noise which is not at the standard noise level, the formula does not hold.

What standard noise means depends on the type of amplifier:

- For optical amplifiers, it is the shot noise, an aspect of quantum noise. Shot noise is the limiting factor for this type of amplifier.
- For electronic amplifiers, it is the thermal noise level, normally at room temperature. This is because thermal noise is the limiting factor for electronics, where the quantum energy is far below the thermal energy <$k_{\rm B} T$> due to the much lower frequencies.

The *noise figure* (in decibels) is defined as 10 times the logarithm (to base 10) of <$F$>:

A (hypothetical) noiseless amplifier would have a noise factor of 1, corresponding to a noise figure of 0 dB. Any additional noise may be called *excess noise*.

For an amplifier consisting of two amplifier stages, where the noise factor of each one is known, one can calculate the noise factor of the combined amplifier:

$$F_{\rm tot} = F_1 + \frac{F_2 - 1}{G_1}$$where <$G_1$> is the gain (power amplification factor) of the first stage. This shows that the noise factor of the second stage is not very relevant if the gain of the first one is large. This is essentially because the apparently input noise of the second stage competes with the already amplified signal and noise from the first stage.

The equation can be generalized to systems with more stages and is known as Friis's formula, referring to the engineer Harald T. Friis, who worked on noise of electronic amplifiers. The formula can also be used for double-pass amplifiers, as occur mostly in photonics.

Generally, the significance of excess noise of an amplifier is lower in cases with higher input noise level. For example, if two amplifier stages are used in series, the second stage is fed with noise well above the shot noise limit (assuming that the first stage has significant gain). Therefore, the excess noise of the second stage is not very relevant (unless it is very strong), and it does not contribute significantly to the noise figure of the amplifier chain.

## Noise Figure of Optical Amplifiers

For any kind of phase-insensitive optical amplifier (for example, a fiber amplifier or semiconductor optical amplifier), the noise figure is significantly increased by the influence of quantum noise. According to an important fundamental result of quantum optics [1], in the limit of high gain the noise factor must be at least 2, corresponding to a noise figure of at least 3 dB. Remarkably, that limit applies to any physical mechanism providing phase-insensitive amplification – be it stimulated emission, a nonlinear interaction or anything else.

The fundamental limit can under ideal circumstances (in particular, zero parasitic propagation losses) be reached for a four-level laser amplifier, where amplified spontaneous emission is the essential noise mechanism. The limit can also be reached with a non-degenerate optical parametric amplifier, where the vacuum noise input of the idler wave is essential, or a Raman amplifier. On the other hand, quasi-three-level amplifiers necessarily have higher noise figures, associated with a higher spontaneous emission factor. (In simple terms, the reabsorption on the laser transmission requires more upper-state population for a given amount of gain, which also causes stronger spontaneous emission.) Further, the noise figure can always be increased by excess losses, particularly when occurring at or near the amplifier input. For example, the noise performance of a fiber amplifier may be spoiled with a faulty fiber splice or non-perfect fiber connector at the input port.

Noise figures below 3 dB (in the high-gain limit) are only possible for phase-sensitive amplifiers, which can be based on degenerate parametric amplification, for example. In that case, the amplifier noise in the other quadrature component (optical phase) is correspondingly higher.

The noise figure is often relevant for amplifiers used in optical fiber communications. Erbium-doped fiber amplifiers, fiber Raman amplifiers and semiconductor optical amplifiers have non-ideal noise figures, which depend on design details like the direction of pump light injection and parasitic propagation losses. For example, in quasi-three-level amplifiers, the noise figure is larger for backward pumping, i.e., when the pump light propagates in the direction opposite to that of the signal light. The reason for that is the resulting lower excitation of the laser-active ions at the signal input end.

## More to Learn

Encyclopedia articles:

Blog articles:

- The Photonics Spotlight 2007-01-27: “Noise Figure of Amplifiers”

### Bibliography

[1] | C. M. Caves, “Quantum limits on noise in linear amplifiers”, Phys. Rev. D 26 (8), 1817 (1982); https://doi.org/10.1103/PhysRevD.26.1817 |

[2] | R. Olshansky, “Noise figure for erbium-doped optical fibre amplifiers”, Electron. Lett. 24, 1363 (1988); https://doi.org/10.1049/el:19880933 |

[3] | E. Desurvire, “Analysis of noise figure spectral distribution in erbium-doped fiber amplifiers pumped near 980 nm and 1480 nm”, Appl. Opt. 29 (21), 3118 (1990); https://doi.org/10.1364/AO.29.003118 |

[4] | E. Desurvire, “Spectral noise figure of Er^{3+}-doped fiber amplifiers”, IEEE Photon. Technol. Lett. 2 (3), 208 (1990); https://doi.org/10.1109/68.50891 |

[5] | H. A. Haus, “The noise figure of optical amplifiers”, IEEE Photon. Technol. Lett. 10 (11), 1602 (1998); https://doi.org/10.1109/68.726763 |

[6] | E. Desurvire, “Comments on 'The noise figure of optical amplifiers'”, IEEE Photon. Technol. Lett. 11 (5), 620 (1999); https://doi.org/10.1109/68.759418 |

[7] | R. Noe, “Consistent optical and electrical noise figure”, IEEE J. Lightwave Technol. 41 (1), 137 (2023); https://doi.org/10.1109/JLT.2022.3212936 |

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