Understanding the link between phase noise and line width, and phase noise units (in the context of optical physics / lasers) Trying to understand the phase noise units, and what they mean / how to convert between them.
For instance, [dBc/Hz --> dB(rad/sqrt(Hz))]?
But also, linewidth measurements are often given for a variety of different integration bandwidths, but the choices seem very arbitrary, and I really don't get why or where they come from, and to be honest, linewidth seems like a terrible measurement and I don't understand the purpose it.
 A: The spectrum of anything is a "power" spectrum because these are the Fourier Transforms of mean square averages and correlations, etc. When calculating the phase noise spectrum of an oscillator it is usually assumed that the amplitude oscillation is negligible, so that the oscillator signal is modeled as $\underline{X} (t) = A \rm{cos}(\omega_0 t +  \underline{\Phi})$ with a constant $A$ amplitude and the underline "$\underline{}$" designates a stochastic variable.
 For *small* phase error $\underline{\Phi}$, 
$\rm{cos}(\underline{\Phi}) \approx 1$ and $\rm{sin}(\underline{\Phi}) \approx \underline{\Phi} $, so one can write 
$$\underline{X} (t) = A \rm{cos}(\omega_0 t +  \underline{\Phi})
= A\rm{cos}(\omega_0 t) cos(\underline{\Phi}) - A\rm{sin}(\omega_0 t) sin(\underline{\Phi}))\\
\approx A\rm{cos}(\omega_0 t) - A\underline{\Phi}\rm{sin}(\omega_0 t)$$
This has two terms both being proportional to $A$. The 1st term is the "noiseless carrier", is pure sinusoid of power $\frac{1}{2} A^2$, the 2nd term, whose "sin" carrier is in quadrature with the "cos" carrier, has a spectrum that is proportional to the spectrum of the phase noise $\underline{\Phi}$.
Let the spectrum of a stochastic process $\underline{Z}$ be $S_Z(u)$ where $u$ is the spectral frequency, then the spectrum of $\rm{cos}(\omega_0 t)$ is the discrete  term $\delta(u-\omega_0)+\delta(\omega_0+u)$ and the continuous term will be $S_{\Phi}(u-\omega_0)+S_{\Phi}(\omega_0+u)$, thus
$$S_X(u) \approx  A^2(\delta(u-\omega_0)+\delta(\omega_0+u)) + A^2(S_{\Phi}(u-\omega_0)+S_{\Phi}(\omega_0+u))$$
With the small phase noise approximation $S_X(u)$ can be measured by a narrow  bandpass (IF) filter and power detector. Center the filter at $\omega_1$ and let its transfer function and bandwidth be $H(\omega)$ and $B$ such that $B<< \omega_0$. Then after much averaging ("video" filter) one can show that the power $p(\omega_1)$ measured at the filter output is approximately
$$p(\omega_1) \approx |H(\omega_1)|^2 S_X(\omega_1) B$$
Notice that since both discrete and continuous parts are proportional to $A^2$ thus the "relative" noisiness of the signal, ie., phase noise density power relative to the carrier power is independent of the latter.
After scanning the filter center frequency $\omega_1$ in the range of interest you can map the spectrum $S_X(u)$ so if the latter is normalized to the total signal power $A^2$ you get the power spectrum density of the phase noise $\underline{\Phi}$. The former has units $\rm{watts/Hz}$ since it is power measured per unit bandwidth of the filter, while the latter being normalized to the power has units $\rm{Hz}^{-1}$ or since it represents "phase" its unit is $\rm{rad}^2/Hz$ representing the mean square phase fluctuation measured after the narrowband filter. If you rather deal with standard deviation (root mean square deviation) rather than with variance (mean square deviation) then yur units will be $\rm{rad/\sqrt{Hz}}$ instead of $\rm{rad^2/Hz}$.
(The "dBc" thing is just an indication that the result of the power measurement is given in logarithmically (decibel) relative to the total carrier power, the "c".)
