Phase noise units: dBc/Hz --> dB(rad/√Hz)

Question

Trying to better understand phase noise units and characterize a device to compare with specs.

I'm used to measuring in dBc/Hz, but the specs given are in dB(rad/√Hz)/m, which I guess is fairly obviously related to the other common phase noise units dB(rad²/Hz) <-- which I almost never see with the length unit in the divisor.

Anyways, I'm super lost. Can anyone help me convert from what I measure on the ESA (dBc/Hz) to either of the other two units, dB(rad/√Hz)/m or dB(rad²/Hz).

If it matters, it's a fiber laser's phase noise that I'm measuring.

Answer 1

The power spectral density (PSD) of a random phase \$\varphi(f)\$ is denoted with \$S_{\varphi}(f)\$ in units of \$\mathrm{rad}^2/\mathrm{Hz}\$ or as \$10~\log_{10}(S_{\varphi}(f))\$ in units of \$\mathrm{dB~rad}^2/\mathrm{Hz}\$.

The quantity \$L(f)\$ is defined as $$L(f) = \frac{1}{2}~S_{\varphi}(f)$$ and is the most widely used measure for phase noise. \$L(f)\$ is generally given in units of \$\mathrm{dBc}/\mathrm{Hz}\$, i.e. the phase noise is then given as \$10~\log_{10}(L(f))\$ \$\mathrm{dBc}/\mathrm{Hz}\$.

\$S_{\varphi}(f)\$ and \$L(f)\$ are generally fully equivalent and differ only in the unit of angle.

According to the given equation above and if both quantities are given in logarithmic form they can be simply converted into each other by: $$L(f) = S_{\varphi}(f) - 3~\mathrm{dB}$$ $$S_{\varphi}(f) = L(f) + 3~\mathrm{dB}$$

For further details I recommend page 22 in the book: E. Rubiola, Phase noise and frequency stability in oscillators, Cambridge University Press, 2009