# Phase Noise

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: noise of the optical phase of a beam or of an electric signal

More general term: laser noise

The light output of a single-frequency laser is not perfectly monochromatic but rather exhibits some *phase noise*, i.e., fluctuations of the optical phase. This leads to a finite linewidth of the laser output. The same applies to the frequency components of the output of a mode-locked laser, i.e. to the lines of the emitted frequency comb.

The fundamental origin of phase noise is quantum noise, in particular spontaneous emission of the gain medium into the resonator modes, but also quantum noise associated with optical losses. In addition, there can be technical noise influences, e.g. due to vibrations of the resonator mirrors or to temperature fluctuations in the gain medium. In many cases, there is also a coupling of intensity noise to phase noise, e.g. via nonlinearities (see, e.g., the article on the linewidth enhancement factor).

Phase noise of a single-frequency laser usually occurs in the form of a quasi-continuous frequency drift, not as sudden substantial phase jumps, since a large number of laser-active atoms or ions is involved.

## Quantification of Phase Noise

Phase noise can be quantified by the power spectral density <$S_{\varphi }(f)$> of the phase deviations, having units of rad^{2}/Hz (or simply Hz^{−1}, as radians are dimensionless). This power spectral density often diverges for zero frequency, so that an r.m.s. value with integration down to zero frequency can not be specified. For simple random-walk processes, the specification of a coherence time or coherence length or of a linewidth value can be appropriate. See the article on noise specifications for more details.

Phase noise is directly related to frequency noise, as the instantaneous frequency is essentially the temporal derivative of the phase. For example, white (frequency-independent) frequency noise corresponds to phase noise with <$S_{\varphi }(f) \approx 1 / f^2$>.

For a laser operating on multiple resonator modes, an evolving and fluctuating optical phase may be attributed to each individual resonator mode (related to a spectral line), but not to the laser output as a whole. Therefore, phase noise would also have to be quantified for individual modes.

## Measurement of Optical Phase Noise

Phase noise measurements are often based on a recorded beat note between two lasers on a fast photodiode. (This requires that the difference of the optical frequencies is not too large.) Alternatively, it is possible to record a beat note of the laser output with a different portion of the same laser output, which is subject to a long delay, e.g. by propagation through a long span of optical fiber (→ *self-heterodyne linewidth measurement*).

A flexible method is to digitize the beat note signal with a fast electronic sampling card. Further processing can then be done numerically on a computer, for example for calculating the time-dependent phase excursion and its power spectral density.

For more details on possible measurements, see the article on the term linewidth.

## Optical Phase Noise Versus Timing Jitter

Sometimes there is a confusion between optical phase noise (as discussed above) and timing jitter of a mode-locked laser because timing jitter can also be seen as a kind of phase noise: a timing change by one pulse period (i.e. the inverse pulse repetition rate) can be interpreted as a phase change of 2<$\pi$>. Such a phase can be called *timing phase*. The corresponding phase noise may be denoted as “timing phase noise” in order to avoid confusion. Of course, mode-locked lasers exhibit optical phase noise in all lines of the emitted frequency combs, and that is not directly related to its timing phase noise. For example, if the phase fluctuations of all resonator modes would be identical, there would be no timing phase noise at all.

## Comparison of Normalized Phase Noise

When comparing the level of phase noise of different oscillators, it is often appropriate to normalize it to the corresponding oscillator frequencies. In frequency metrology, it is common to use the quantity

$$x(t) = \frac{{\Delta \varphi (t)}}{{2\pi {\nu _0}}}$$which is the magnitude of phase fluctuations divided by the mean angular frequency. The quantity of <$x(t)$> does not change e.g. when digital oscillating signal is sent through a frequency divider, which reduces the phase fluctuations in proportion to the mean frequency. A comparison of optimized low-noise lasers, which are stabilized to some optical frequency standard, with high-quality microwave oscillators in terms of phase noise, shows that lasers normally have a higher level of phase noise <$\delta \varphi (t)$> but a lower level of normalized phase noise <$x(t)$>, which means that they are superior e.g. when used as clocks. Even without any stabilization, a laser can have a very low phase noise level at high noise frequencies, making it very suitable as a *fly-wheel oscillator*.

## More to Learn

Encyclopedia articles:

## Suppliers

The **RP Photonics Buyer's Guide** contains five suppliers for phase noise measurement equipment. Among them:

### HighFinesse

HighFinesse linewidth analyzers are the ultimate high-end instruments for measuring, analyzing and controlling frequency, linewidth and intensity noise of laser light sources. The product series covers a measurement range from 450 nm up to 1625 nm and enables the analysis of both very narrow laser lines down to 1 kHz as well as broader spectra up to 100 MHz. Thanks to the included software, the instrument delivers real-time analysis, with no need for further calculation. The analyzers feature an extremely high resolution and accuracy and are ideal for optimizing the stability of laser setups.

### Bibliography

[1] | A. L. Schawlow and C. H. Townes, “Infrared and optical masers”, Phys. Rev. 112 (6), 1940 (1958); https://doi.org/10.1103/PhysRev.112.1940 (contains the famous Schawlow–Townes equation) |

[2] | R. Paschotta et al., “Optical phase noise and carrier–envelope offset noise of mode-locked lasers”, Appl. Phys. B 82 (2), 265 (2006); https://doi.org/10.1007/s00340-005-2041-9 |

[3] | D. R. Walker et al., “Frequency dependence of the fixed point in a fluctuating frequency comb”, Appl. Phys. B 89, 535 (2007); https://doi.org/10.1007/s00340-007-2882-5 |

[4] | J. K. Wahlstrand et al., “The quantum-limited comb lineshape of a mode-locked laser: Fundamental limits on frequency uncertainty”, Opt. Express 16 (23), 18624 (2008); https://doi.org/10.1364/OE.16.018624 |

[5] | T. Habruseva et al., “Optical linewidth of a passively mode-locked semiconductor laser”, Opt. Lett. 34 (21), 3307 (2009); https://doi.org/10.1364/OL.34.003307 |

[6] | R. Paschotta, "Noise in Laser Technology – Part 1: Intensity and Phase Noise" |

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