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# Frequency Noise

Definition: noise of the instantaneous frequency of an oscillating signal

German: Frequenzrauschen

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The term frequency noise refers to random fluctuations of the instantaneous frequency of an oscillating signal. The instantaneous frequency is defined as

$$\nu (t) = \frac{1}{{2\pi }}\frac{{{\rm{d}}\varphi }}{{{\rm{d}}t}}$$

i.e. essentially as the temporal derivative of the oscillation phase <$\varphi$>. Any random deviation from a purely linear phase evolution is seen as frequency noise.

The power spectral density of frequency noise (with units of Hz2/Hz) is directly related to that of the phase noise:

$${S_\nu }(f) = {f^2}\;{S_\varphi }(f)$$

where <$f$> is the noise frequency. For example, white frequency noise (<$S_{\nu }(f)$> = const) corresponds to phase noise with a power spectral density proportional to <$f^{-2}$>. In that case, the linewidth is <$\pi$> times the one-sided power spectral density of the frequency noise (or 2<$\pi$> times the two-sided power spectral density). Such a situation occurs e.g. in a single-frequency laser which is only subject to quantum noise and exhibits the Schawlow–Townes linewidth.

Some data sheets exhibit plots with units of Hz/sqrt(Hz) on the vertical axis; this is just the square root of the power spectral density of frequency noise.

Phase noise or frequency noise are just different ways of describing the same phenomenon. However, numerical processing of frequency noise rather than phase noise can have technical advantages in certain situations.

## Questions and Comments from Users

2020-09-20

I think there is an error in this sentence: “The power spectral density of frequency noise (with units of Hz2/Hz)...”

The unit should probably be rad2/Hz?

No, the instantaneous frequency has units of Hertz, not radians. However, you often find units of rad2/Hz in the literature; these refer to <$S_{\varphi}(f)$>, the noise PSD of the optical phase, which is directly related to frequency noise as explained in the article.

2021-06-21

Based on the definition of the instantaneous frequency in the first equation. To be more accurate, should the first sentence be something like this: The term frequency noise refers to random fluctuations of the frequency (or the instantaneous frequency) of an oscillating signal? Because the instantaneous frequency inherently includes the fluctuations of the frequency per definition. Share this with your friends and colleagues, e.g. via social media:   These sharing buttons are implemented in a privacy-friendly way!