## Appendix A  
Assume that a carrier containing phase noise generated through an optical heterodyne process can be approximated in the spectral domain by a Lorentzian shaped profile. The expression for a Lorentzian distribution is given as:  
\[L(f)=\frac{1}{\gamma}\left[\frac{\gamma^{2}}{(f-f_{0})+\gamma^{2}}\right]\ \, \tag{1}\]  
where \(f_{0}\) is the frequency offset and \(\gamma\) specifies half width at half maximum (and \(2\gamma\) is the 3-dB linewidth of the carrier source with phase noise). Setting \(f_{0}=0\), \(\gamma=1\) and normalizing \(f\) with respect to \(\gamma\) for a standard distribution yields:  
\[L(f)=\frac{1}{1+f^{2}}\ \, \tag{2}\]Integrating this expression reveals the power of the normalized Lorentzian distribution:  
\[\int_{-\infty}^{\infty}\frac{1}{1+f^{2}}\ df=\arctan\left(\infty\right)-\arctan \left(-\infty\right)=\frac{\pi}{2}-\left(-\frac{\pi}{2}\right)=\pi\enspace. \tag{10}\]  
Consider a band-limited distribution with frequency bounds \(-B\leq f\leq B\). The single-sided power of the band-limited distribution can be expressed as:  
\[\int_{0}^{B}\frac{1}{1+f^{2}}\ df=\arctan\left(B\right)-\arctan\left(0\right) =\arctan\left(B\right)\enspace, \tag{11}\]  
For the purposes of enabling noise cancellation across the full spectral range of a secondary signal, using band-pass filtering of a carrier containing phase noise (as discussed in Section 5.2), an SNR metric can be defined, where the signal component is the power of the distribution retained by band-limiting, and the noise component is the power of the distribution lost by band-limiting. This can be expressed as follows:  
\[P_{sig}=2\cdot\arctan(B)\enspace\text{and}\enspace P_{noise}=\pi-2\cdot \arctan(B)\enspace, \tag{12}\]  
and hence,  
\[\text{SNR}=\frac{P_{sig}}{P_{noise}}=\frac{2\cdot\arctan(B)}{\pi-2\cdot \arctan(B)}\enspace, \tag{13}\]  
Since \(f\) is normalized to \(\gamma\), and \(B\in f\), the above expression can be generalized to:  
\[\text{SNR}=\frac{2\cdot\arctan(\frac{B}{7})}{\pi-2\cdot\arctan(\frac{B}{7})}\enspace, \tag{14}\]  
It is clear from the above expression that the SNR metric varies in proportion to the ratio of \(B\), the single sided band-limit, to \(\gamma\), half the linewidth. Setting \(B=\gamma\)--which is equivalent to setting the pass-band of an ideal band-limiting filter (\(2B\)) equal to the 3-dB linewidth of the Lorentzian (\(2\gamma\))--gives  
\[\text{SNR}=\frac{2\cdot\arctan(1)}{\pi-2\cdot\arctan(1)}=1\enspace, \tag{15}\]  
and so for \(B\
ot\gg\gamma\) and \(B\
ot\ll\gamma\) an approximation of the SNR metric is given by:  
\[\text{SNR}\approx\frac{B}{\gamma}\enspace. \tag{16}\]  
For \(B\gg\gamma\), which describes most practical scenarios, the following approximation can be made:  
\[\arctan\left(\frac{B}{\gamma}\right)\approx\frac{\pi}{2}-\frac{1}{B/\gamma}\enspace, \tag{17}\]  
which allows the Equation (14) to be approximated to:  
\[\text{SNR}\approx\left(\frac{B}{\gamma}\cdot\frac{\pi}{2}\right)-1\enspace. \tag{18}\]  
Compared to this approximation, the expression given in (16) can be viewed as a conservative estimation of the achievable system SNR, for given linewidth and ideal filter pass-band values.