§ DOWN-CONVERTED COMBS

Illustration of the down-conversion process expressed in Eq. (<ref>), in which the optical comb signal is converted into the detected electronic signal. In the upper panel, the comb lines of the CUT and the LO are schematically displayed. As these are optical frequencies, they are offset hundreds of  from zero. Through the mixing and detection process, the differences of the optical frequencies are detected electrically. Note that negative difference frequencies are projected onto their corresponding positive frequencies through aliasing (blue arrows), as the signal we detect is real-valued. This is indicated by the purple overlay that illustrates the detected bandwidth. It is crucial to identify beat notes that have been aliased as their negative frequency needs to be part of the state space model for the EKF.

Two main options are available to characterize frequency combs in the setup described in Fig. <ref>. In this section, we will focus on the case where the CUT is a comb and the LO is a single-frequency laser. We will refer to this scenario as the "down-converted comb" setup in this chapter. The general expression for the measured signal from Eq. (<ref>) reduces into

\begin{align}
\begin{split}
y(t_k) &= 2\eta \sum_{l=0}^\infty \sqrt{P^{CUT}_l P^{LO}}  \\
&\quad \times \text{sin}[(\omega^{CUT}_{ceo} + l\omega^{CUT}_{rep} - \omega^{LO})t_k\\
&\qquad  + \phi^{CUT}_l(t_k) - \phi^{LO}(t_k)] + \xi(t_k)\\
&= \sum_{l=0}^\infty A_l \text{sin}[\omega_{l} t_k + \phi^{total}_l(t_k)] + \xi(t_k)
\end{split}
\label{eq:downconverted_comb_signal}
\end{align}

where all quantities are the same as in Eq. (<ref>). As the LO is a single-frequency laser, one of the sums of the original expression is dropped. For compact notation we define $A_l=2\eta \sqrt{P^{CUT}_l P^{RS}}$, $\omega_l=\omega^{CUT}_{ceo} + l\omega^{CUT}_{rep} - \omega^{LO}$ and $\phi^{total}_l(t_k) = \phi^{CUT}_l(t_k) - \phi^{LO}(t_k)$. This general expression describes a multi-heterodyne measurement of any down-converted comb.

The beat note generation process described in Eq. (<ref>) is illustrated in Fig. <ref>.
The resulting signal is once again a comb with the same spacing as the original line spacing. To simultaneously detect as many lines as possible, we want to maximize the number of down-converted lines within the detection bandwidth. This is achieved by selecting a LO with a suitable frequency $\omega^{LO}$ which is close to the frequency of the highest power comb lines of the CUT. All lines for which $|\omega_l| < 2\pi f_{BW}$ can then be detected.

The maximum number of detectable lines within the detection bandwidth $f_{BW}$ is $N_{detected} \leq 2\lfloor 2\pi f_{BW} / \omega_{rep} \rfloor + 1$.
While we can always detect at least one comb line, this relation clearly illustrates that this setup is only suitable for combs that have a repetition frequency significantly below the available bandwidth. Only then can we detect multiple lines at the same time and benefit from the reduced experimental complexity of the setup to infer information about the comb phase noise from only a single measurement.

This condition disqualifies this setup for types of combs that have a repetition frequency $\omega_{rep}$ in the tens of  and above, as even state-of-the-art photodiodes are limited to around 100  of bandwidth <cit.>. Even with these extreme bandwidths, only a few comb lines would be detectable. Therefore, combs with large spacing can effectively only be characterized in dual-comb setups, which are discussed in <ref>.

For combs with $\omega_{rep}$ up to 10, utilizing this single-frequency laser setup is more advantageous compared to employing a dual-comb configuration. With a single-frequency laser acting as the local oscillator (LO), it is still possible to detect numerous lines while the experimental setup is simplified significantly.
Furthermore, assessing the contribution of the LO towards the total phase noise per beat note $\phi^{total}_l(t_k)$ is more straightforward. As derived in Eq. (<ref>) the LO phase noise $\phi^{LO}(t_k)$ is an additive contribution independent of the comb line index $l$. It, therefore, only adds a common phase noise offset that does not add any index-dependent dynamics. In case an independent characterization of the LO is available, the phase noise offset can, in principle, be removed afterward.
This distinction is generally not as easy in a dual-comb setup, as the LO itself is a comb and its phase noise is therefore dependent on the line index.

To show how the EKF-based method can be used to extract the phase noise terms $\phi^{total}_l(t_k)$ in this scenario, we will first demonstrate its application on a simulated signal. An electro-optic (EO) comb is chosen as the CUT, as it is well-studied in literature and has a known phase noise model.
After showcasing the method on simulated data, the method's application is also demonstrated on experimental data.

(a) Setup for the generation of an EO comb and subsequent detection of a multi-heterodyne beat note signal.

CW: continuous wave laser, EOM: electro-optic modulator, RF: radio-frequency generator.

(b) PSD of a simulated multi-heterodyne signal of an EO comb. The colored numbers correspond to the comb line index $l$, where negative indices indicate aliased lines.

§.§ Simulating an EO comb

The setup that is used to simulate a down-converted EO comb signal is displayed in <ref> (a). The EO comb is generated by applying a sinusoidal phase modulation to a single-frequency seed laser (CW). Both the seed laser as well as the RF source that drives the electro-optica modulator are considered to exhibit phase noise.
The electric field of this EO comb has the form <cit.>

\begin{align}
E_{CUT}(t) &= \sqrt{P^{CW}} e^{i(\omega^{CW} t + \phi^{CW}(t))} e^{i K V(t)} \\
V(t) &= V_0 \sin{\left[(\omega^{RF} t + \phi^{RF}(t)) \right]}. \label{eq:RF_voltage}
\end{align}

Here, $V(t)$ is the sinusoidal voltage that is generated by the RF source and applied to the EOM, which has a peak voltage $V_0$ and the frequency $\omega^{RF}$. The EOM acts as a pure phase modulator.
$K$ is a material-dependent proportionality constant of the EOM which determines the phase change per applied voltage.
The LO is simply a single-frequency laser, and its electric field $E_{LO}(t)$ is given by Eq. (<ref>).

As detailed in <ref>, the electric fields $E_{CUT}(t)$ and $E_{LO}(t)$ are propagated through the setup using Eqs. (<ref>) and (<ref>). After balanced coupling, detection in the BDP and digitization by the ADC, the final signal then has the form

\begin{align}
\begin{split}
y(t_k) &= 2\eta \sqrt{P^{CW} P^{LO}} \text{sin}\left[(\omega^{CW} - \omega^{LO}) t_k + K V(t_k) \right. \\
&\qquad \left.+ \phi^{CW}(t_k) - \phi^{LO}(t_k)\right] + \xi(t_k) \label{eq:BPD_downconverted_comb}
\end{split}
\end{align}

We neglect amplitude noise, and we assume that the phase noise $\phi_{LO}(t_k)$ of the LO is insignificant compared to the phase noise of the seed laser $\phi_{CW}(t_k)$ for simplicity.

For the phase noise of the seed laser $\phi_{CW}(t_k)$ we use the same phase noise model as in the laser beat note study in <ref>.
We use the model described by Eq. (<ref>), in which the phase noise is composed of a random walk with added low-frequency flicker noise. For simplicity, we further use the same simulation parameters for the model; explicitly, we choose $\Xi_{flicker}=\SI{10}{\mega\hertz\squared}$ and $\Delta\nu=\SI{100}{\hertz}$.

The third phase noise term that appears in Eq. (<ref>) is hidden in $V(t_k)$, the output of the RF source. As Eq. (<ref>) states, $\phi^{RF}(t_k)$ is the phase noise in the RF signal. RF source phase noise generally has different and more complicated characteristics compared to laser phase noise. Therefore, we will use a different phenomenological model to generate $\phi^{RF}(t_k)$.

Ultra-low phase noise RF sources often have multiple plateaus in their phase noise power spectral density caused by multiple independent locking mechanisms <cit.>. A physically accurate simulation would heavily depend on implementation details and is out of the scope of this work.
To still roughly emulate this behavior and to intentionally deviate from the laser phase noise model, $\phi_{RF}(t_k)$ is simulated using

\begin{align}
S_f(\phi_{RF})(\omega)= \Xi_{Plateau,RF}\sin{(\log_{10}(\omega))^8} + \Xi_{Lorentz,RF}. \label{eq:RF_pn_model}
\end{align}

which has two free scale parameters $\Xi_{Lorentz,RF}$ and $\Xi_{Plateau,RF}$. The former models an underlying random walk of the phase noise. The latter models the scale of a sinusoidal deviation that is added to the random walk. The sinusoidal shape of the perturbation introduces
For this numerical study, we choose the free parameters as $\Xi_{Lorentz,RF} = \SI{100}{\hertz}$ and $\Xi_{Plateau,RF} = 10^4$ with the corresponding unit, which creates a phase noise PSD model that has multiple plateaus.

Note that this model and its parameters are phenomenologically chosen to emulate the phase noise of a specific device, which we have used as a reference [see Agilent E8257C datasheet].
The secondary purpose of this model is its explicit deviation from a simple phase noise random walk.
As discussed in <ref>, the EKF state-transition function we use to approximate the phase noise behavior does not consider any deviations from a random walk. An intentional mismatch between the true and approximated model can, therefore, give an indication of whether the random walk state-space model can still capture complicated phase noise dynamics.

To simulate time series realizations $\phi^{CW}(t_k)$ and $\phi^{RF}(t_k)$ based on the PSD models in Eqs. (<ref>) and (<ref>), we again use the algorithm given in <cit.>.
Eq. (<ref>) can now be used to directly simulate a time domain signal of a down-converted EO comb.
The parameters chosen for this numerical study are
$2\eta \sqrt{P^{CW} P^{LO}}=\SI{100}{\milli\volt}$, $K V_0=\num{1,6}$, $\omega_{CW}=2\pi\cdot\SI{1.1}{\giga\hertz}$ and $\omega_{RF}=2\pi\cdot\SI{500}{\mega\hertz}$. Here, we have combined some individual parameters into effective combined quantities, as only total values, as opposed to the individual contributions, matter.
The spectrum of the resulting signal $y(t_k)$ is shown in <ref> (b). The resulting comb is centered around the line with index $l_{center}=2$ and features $N_{alias} = 3$ and $N_{reg.} = 8$ aliased and non-aliased comb lines, respectively.

§.§ Latent phase noise state-space model

To use the EKF-based phase noise estimation method, a state-space model needs to be defined.
The measurement equation of the model, which describes the detected signal, is based on the general analytical expression in Eq. (<ref>).

The simplest approach to tracking the comb's phase noise is to track the phase noise terms of each beat note individually.
In analogy to the laser beat note case, the hidden variables of the state space model are the phase noise terms $\phi_l(t_k)$ of the individual beat notes. They are approximated by a multi-dimensional random walk.
The state-space model takes the form

\begin{align}
\begin{split}
\mathbf{f}_{indiv.}(\mathbf{x}_k, \mathbf{\theta}) &= \mathbf{x}_k + \mathbf{q}_k \\
\mathbf{q}_k &\sim \mathcal{N}(0, \mathbf{Q}_N)
\end{split}  \label{eq:model_comb_transition}\\
\begin{split}
\mathbf{h}_{indiv.}(\mathbf{x}_k, \mathbf{\theta}) &= \sum_{l=-N_{alias}}^{N_{reg.}-1} A_{l} \text{sin}\left[ \left( \omega_{0} + l\omega_{rep} \right) t_k + \mathbf{x}_k \right] + r_k \\
r_k &\sim \mathcal{N}(0, \sigma_R)
\end{split}  \label{eq:model_comb_measurement}
\end{align}

where we identify the $N = N_{alias} + N_{reg.}$ dimensional vector of hidden variables $\mathbf{x}_k$ as the beat note phase noise terms $\mathbf{\phi}(t_k)$. $N_{alias}$ and $N_{reg.}$ are the number of aliased and non-aliased comb lines with significant amplitude in the recorded spectrum, respectively.
$\omega_{0}$ is chosen as the lowest of the beat note frequencies that is not aliased (as indicated in Fig. <ref>).
The static parameters are $\mathbf{\theta}_{indiv.} = \{ \mathbf{A}, \omega_{0}, \omega_{rep}, \mathbf{x}_0, \mathbf{P}_0, \mathbf{Q}_N, \sigma_R\}$. This model has $|\theta_{individual}| = \nicefrac{1}{2} \left( N^2 + 7N \right) + 3$ free parameters that need to be optimized.

Using this state-space model for EKF-based phase noise tracking has been shown to produce accurate estimations that outperform the conventional method in an EO comb <cit.>.
The approach does, however, have multiple drawbacks that have to be addressed.

In this individual line tracking approach, the correlations between the phase noise terms $\phi^{total}_l(t_k)$ are found through the optimization of $\mathbf{Q}_N$. This matrix represents the differential phase noise covariance, from which the differential correlations can be calculated.
No prior information about the phase noise in the system is used in this model.

The state vector $\mathbf{x}_k$ has the same dimension as the number of detected comb lines $N$. This causes the associated covariance matrices $\mathbf{P}_k$ as well as the process noise covariance matrix $\mathbf{Q}_N$ of the state vector to be a matrix of size $\left( N \times N\right)$. The EKF filtering equations (<ref>) include matrix operations that have an undesirable computational complexity scaling of $\mathcal{O}(N^3)$. Applying the EKF on a large number of comb lines $N$ can make this approach computationally infeasible.
Additionally, the covariance matrix $\mathbf{Q}_N$ needs to be optimized, which requires the optimization of $\mathcal{O}(N^2)$ free parameters.

To solve the complexity scaling problem, we can exploit the fact that the individual comb line phase noise terms $\phi^{total}_l$ is a combination of few underlying phase noise sources.
The expression for the signal in Eq. (<ref>) shows that only three phase noise sources contribute. The $N$ individual beat note phase noise terms $\phi^{total}_l(t_k)$ are a combination of the three sources, which means that they are correlated.

For the EO comb, these correlations are well studied <cit.>. The phase noise of each beat note is

\begin{align}
\phi^{total}_l(t_k) = \phi^{CW}(t_k) - \phi^{LO}(t_k) + ( l - l_{center})\phi^{RF}(t_k). \label{eq:EO_comb_phase_noise}
\end{align}

In matrix notation, it becomes

\begin{align}
% \mathbf{\phi}^{total} = \mathbf{C}_{true} \left[ \phi^{CW} - \phi^{LO}(t_k), \phi^{RF}\right]^T \\
\mathbf{\phi}^{total}(t_k) &= \mathbf{C}_{true} \begin{bmatrix} \phi^{CW}(t_k) - \phi^{LO}(t_k)\\ \phi^{RF}(t_k)\end{bmatrix} \label{eq:EO_comb_latent_space} \\
\mathbf{C}_{true} &= \begin{bmatrix}
... & 1 & 1 & 1 & 1 & 1 &...\\
... & -2& -1 & 0 & 1 & 2 &...
\end{bmatrix}^T. \label{eq:C_true}
\end{align}

The second column of $\mathbf{C}_{true}$ is zero at the index of the central comb line $l_{center}$.
We have introduced a mixing matrix $\mathbf{C}$ which describes the linear transformation that connects the true phase noise sources with the phase noise per beat note. Independent of the number of beat notes $N$, each $\phi^{total}_l(t_k)$ is described by only two independent contributions: A common mode noise term $\phi^{CW}(t_k) - \phi^{LO}(t_k)$ that is added to all beat notes independent of $l$, and a differential mode noise term $\phi^{RF}(t_k)$ which has a linear dependence on $l$.

The structure of Eq. (<ref>) can be used to design a more efficient state-space model.
Instead of tracking the phase noise of each beat note, we can track the phase noise sources directly. Using Eq. (<ref>), $\mathbf{\phi}^{total}(t_k)$ can be constructed if $\mathbf{C}_{true}$ is known.
In more formal terms, we introduce a linear latent space into the state-space model, which is a lower-dimensional representation of the comb line phase noises $\mathbf{\phi}^{total}(t_k)$. In general, this can be expressed as

\begin{align}
\mathbf{\phi}(t_k) = \mathbf{C} \mathbf{\psi}(t_k) \label{eq:latent_space_eq}
\end{align}

where $\mathbf{\psi}(t_k)$ is a $L < N$ dimensional vector of latent phase noise sources.

Including this latent space into the measurement function of the state-space model given in Eq. (<ref>) yields an improved model:

\begin{align}
\begin{split}
\mathbf{f}_{lat.}(\mathbf{x}_k, \mathbf{\theta}) &= \mathbf{x}_k + \mathbf{q}_k \\
\mathbf{q}_k &\sim \mathcal{N}(0, \mathbf{Q}_L)
\end{split}  \label{eq:latent_model_comb_transition}\\
\begin{split}
\mathbf{h}_{lat.}(\mathbf{x}_k, \mathbf{\theta}) &= \sum_{l=-N_{alias}}^{N_{reg.}-1} A_{l} \text{sin}\left[ \left( \omega_{0} + l\omega_{rep} \right) t_k + \mathbf{C}\mathbf{x}_k + \phi^0_l \right] + r_k \\
r_k &\sim \mathcal{N}(0, \sigma_R)
\end{split}  \label{eq:latent_model_comb_measurement}
\end{align}

The hidden state $\mathbf{x}_k$ is the $L$ dimensional vector of latent phase noise sources $\mathbf{\psi}(t_k)$.
$\phi^0_l$ is a global initial phase of the l-th line. While not strictly required as the initial values $\mathbf{x}_0$ also can contribute to an initial phase, the inclusion of $\phi^0_l$ improves optimization convergence.

The static parameters of this model are $\mathbf{\theta}_{lat.} = \{ \mathbf{A}, \mathbf{\phi}^0, \omega_0, \omega_{rep}, \mathbf{x}_0, \mathbf{P}_0, \mathbf{C}, \mathbf{Q}_L, \sigma_R\}$.
The dimension $L$ of the latent space is a hyperparameter of the model. This means that it can not be optimized directly, but instead needs to be chosen based on prior information.
In the EO comb case, Eq. (<ref>) shows that a two dimensional latent space with $L=\num{2}$ can explain the phase noise of all comb lines. Further, the form of the mixing matrix $\mathbf{C}$ is given by Eq. (<ref>). The only free parameter in this expression is the index of the central comb line, which shifts the zero-crossing point of the second column of the matrix. As $\mathbf{C}$ is part of $\mathbf{\theta}_{lat.}$, the index offset can be optimized alongside the other parameters.
To perform the optimization, we define the optimization transformation function $\Lambda$ as introduced in <ref>:

\begin{align}
\Lambda(\mathbf{\theta}) = \left\{
\begin{array}{llrl}
\Lambda_{A_l}&: &A_l &\rightarrow 10^{A_l}\\
\Lambda_{\phi_l^0}&: &\phi_l^0 &\rightarrow \phi_l^0 \\
\Lambda_{x_0}&: &x_0 &\rightarrow x_0\\
\Lambda_{\delta\omega_0}&: &\delta\omega_0 &\rightarrow \delta\omega_{max}\cdot \mathrm{tanh}(\delta\omega_0) \\
\Lambda_{\delta\omega_{rep}}&: &\delta\omega_{rep} &\rightarrow \delta\omega_{max}\cdot \mathrm{tanh}(\delta\omega_{rep}) \\
\Lambda_{\mathbf{P}_0}&: &\mathbf{P}_{0, mm} &\rightarrow 10^{\mathbf{P}_{0, mm}} \\
\Lambda_{\mathbf{Q}_{L, mm}}&: &\mathbf{Q}_{L, mm} &\rightarrow 10^{\mathbf{Q}_{L, mm}} \\
\Lambda_{\sigma_{R}}&: &\sigma &\rightarrow 10^\sigma \\
\Lambda_{\mathbf{C}}&: &l_{center} &\rightarrow \begin{bmatrix}
1 & 0-l_{center} \\
1 & 1-l_{center} \\
1 & 2-l_{center} \\
\vdots & \vdots
% -l_{center} & 1-l_{center}& 2-l_{center} & ...
\end{bmatrix}\\
\end{array}\right. \label{eq:optimization_function_sim_eo_comb}
\end{align}

The function is very similar compared to the laser beat note case in Eq. (<ref>). A notable difference is that $\mathbf{P}_0$ and $\mathbf{Q}_L$ are matrices, as the hidden state dimension is larger than one. With the corresponding transformation, we force both of them into a diagonal form and optimize their diagonal elements in logarithmic space.
While it is possible to allow non-zero off-diagonal elements, this complicates the optimization process, as $\mathbf{Q}_L$ has to be constrained to the space of positive semi-definite matrices as it is a covariance matrix. Only allowing diagonal matrices avoids this problem and additionally reduces the number of parameters to optimize.
This is equivalent to constraining the latent phase noise sources to be uncorrelated. For the EO comb this is a reasonable assumption as the true phase noise sources are also uncorrelated.
To optimize the comb frequencies, we again choose $\delta\omega^{max}=\SI{5}{\kilo\hertz}$.

Application of the EKF-based phase noise estimation method on a simulated EO comb.
(a) Column vectors (indicated by the superscripts $^{(0)}$ and $^{(1)}$) of the true and optimized mixing matrix $\mathbf{C}_{c.}$ which describe the latent phase noise space of the EO comb. During the optimization, $\mathbf{C}_{c.}$ is constrained to only exhibit common mode and differential mode phase noise.
(b) Frequency noise PSDs $S_f(f)$ of the latent phase noise sources, based on the mixing matrix shown in (a).
(c) Column vectors of the true and optimized mixing matrix $\mathbf{C}_{uc.}$ where the entries of the matrix are unconstrained during optimization.
(d) Frequency noise PSDs $S_f(f)$ of the latent phase noise sources corresponding to the unconstrained mixing matrix shown in (c).
All CIs shown are smoothed for visibility.

The function $\Lambda_{\mathbf{C}}$ that transforms the mixing matrix is based on Eq. (<ref>). Only a single free parameter, $l_{center}$, is required to define the matrix. The latent phase noise space is constrained to only have a common mode and a differential mode phase noise contribution. We will call this the constrained latent space model, indicated by the subscript $\mathbf{C}_{c.}$.

The constrained model has $|\mathbf{\theta}_{c.}| = 2N + 4L + 2$ free parameters and $L$ tracked variables.
For large $N$ the linear scaling of the number of parameters causes it to have significantly fewer free parameters compared to the individual line approach.
Notably, the number of parameters no longer scales quadratically with the number of comb lines. Further, the EKF computational complexity reduces to $\mathcal{O}(L^3)$, which is no longer dominated by scaling with the number of comb lines.
This means that it becomes feasible to use this model to apply the EKF-based phase noise estimation method on combs with a large number of comb lines.

*Unconstrained latent spaces
While the latent phase noise space of an EO comb can be explicitly described by Eq. (<ref>), a similar expression might not be available for all types of combs.
In particular, evidence for latent phase noise sources beyond the common mode and differential mode has been reported for mode-locked lasers <cit.>.

In these scenarios, we can relax the requirement for prior information about the phase noise while still benefiting from the latent state-space model. Instead of using the transformation function $\Lambda_{\mathbf{C}}$ as defined in Eq. (<ref>), it can be replaced by the identity function

\begin{align}
\Lambda_{\mathbf{C}}: \mathbf{C}_{lm} \rightarrow \mathbf{C}_{lm}. \label{eq:optimization_function_uc}
\end{align}

If this function is used instead, each entry of $\mathbf{C}$ becomes a free parameter of the state-space model. The latent space decomposition becomes fully data-driven as the optimizer learns the entries of $\mathbf{C}$. No assumptions about the comb line phase noise are made, except for the assumption that only a few latent sources contribute to each comb line.
We will call this variant the unconstrained model, indicated by the subscript $\mathbf{C}_{uc.}$.

In this form, the model has $|\theta_{uc.}| = NL + 2N + 3L + 3$ parameters, which still has a favorable scaling compared to the individual line tracking approach as $N$ becomes large. In addition, the complexity of applying the EKF is still $\mathcal{O}(L^3)$.

§.§ Application on simulated EO combs

To evaluate the latent space phase noise model, we apply it to the simulated EO comb discussed in <ref>.
We will compare two variants of the latent space model against the ground truth phase noise.

The constrained latent state-space model assumes that the comb line phase noise is explained by a common and a differential phase noise mode. This model is derived from the known behavior of EO comb phase noise as expressed in Eq. (<ref>).
The state-space model is defined in (<ref>) and (<ref>). Together with the optimization transition function which is defined in Eq. (<ref>), the parameters $\mathbf{\theta}_{c}$ are optimized, as detailed in <ref>, and used to estimate the latent phase noise sources $\widehat{\boldsymbol{\psi}}^{c}(t_k)$.

For comparison, we also test the unconstrained state-space model variant on the same simulated signal. This variant uses the same state-space model. The difference lies in the optimization transformation function, where we use the replacement detailed in Eq. (<ref>). While the latent space is still kept two-dimensional, it is not constrained to common and differential noise modes. Using the optimization framework, we find the parameters $\mathbf{\theta}_{uc.}$ and use them to estimate the latent phase noise $\widehat{\psi}^{uc.}_m(t_k)$.

Accuracy comparison of the different phase noise estimation methods on a simulated EO comb.
(a) Integrated phase noise PSD per comb line for each method. The estimates of the EKF using the constrained ($\mathbf{C}_{c.}$) and unconstrained ($\mathbf{C}_{uc.}$) latent space models, as well as the conventional method, are compared against the ground truth.
Additionally, the SNR per comb line is shown with the purple, triangular markers which are measured on the right y-axis.
(b) and (c) illustrate the phase noise PSDs of selected comb lines with indices $l=\num{-1}$ and $l=\num{3}$ respectively. The gray shaded area corresponds to the integrated PSD displayed in (a) for each line.

<ref> displays the results of the EKF-based estimation.
In <ref> (a) the columns of the optimized mixing matrix $\mathbf{C}_{c.}$ are compared to the true matrix. The single free parameter of $\mathbf{C}_{c.}$ in Eq. (<ref>) which identifies the index of the central comb lines was optimized. Its final value $l^{MAP}_{center} \approx \num{2}$ is very close to the true value. Therefore the columns of $\mathbf{C}_{c.}$ align perfectly with the true values. In (b), the frequency noise PSDs of the latent phase noise sources are shown. As $\mathbf{C}_{c.}$ matches $\mathbf{C}_{true}$, the latent noise sources map directly to the physical phase noises $\phi^{true}_{CW}$ and $\phi^{true}_{RF}$. At high offset frequencies $f>\SI{100}{\mega\hertz}$, the EKF estimates deviate from the true PSDs. The CI around the PSD, however, captures this estimation inaccuracy.

In <ref> (c) the optimized mixing matrix $\mathbf{C}_{uc.}$ of the unconstrained model is shown. The column vectors do not display a well-defined separation into common and differential noise modes. The latent phase noise sources will, therefore, not correspond to physical noise sources. This can be seen in (d), where their PSDs are shown. The shape of the PSDs are similar to the true ones, but they deviate considerably. In addition, both latent phase noise PSDs display strong spikes at high frequencies which are not present in the true phase noise sources.

While it can be useful to isolate the physical noise sources in a comb, this is only possible if a prior model of the comb phase noise exists, as in Eq. (<ref>). In all other cases, especially in the unconstrained mixing matrix case, the optimizer will find any $\mathbf{C}$ that can approximately decompose the comb line phase noise into latent noise sources.

Here, the latent space is not interpreted as a physical quantity. However, the observable of interest are the phase noise per comb line $\phi^{total}_l(t_k)$. The latent space simply serves as a compressed representation. Using Eq. (<ref>), $\phi^{total}_l(t_k)$ can easily be computed from the latent phase noise.

§.§.§ Estimation accuracy

In <ref> the estimation accuracy of both the constrained and unconstrained EKF-based approach and the conventional method are compared against the ground truth.
The phase noise per comb line can also be estimated using the conventional method discussed in <ref>. The accuracy is estimated based on the estimated variance of the phase noise.

In (a) the integrated phase noise PSDs of each individual comb line for the different methods are shown. This measure is similar to the empirical phase noise time domain variance as it has the same unit, except that it considers a wide part of the spectrum instead. The PSD of each comb line $S_{\phi}(\phi_i^{total})$ is integrated from $f_{min}=\SI{100}{\kilo\hertz}$ to $f_{max}=\SI{100}{\mega\hertz}$. The value of $f_{max}$ is chosen since 100 is the maximum offset frequency we can evaluate using the conventional method in this case.

Estimated differential phase noise correlation matrices of a simulated EO comb. The matrices are estimated empirically from the time domain phase noise traces.

The constrained EKF variant produces estimates that perfectly align with the true value. This is unsurprising as the latent space estimate shown in <ref> (a,b) is accurate as well.
However, the estimate also is accurate for the outermost lines, which have less than -20 SNR.
Here we define the electrical SNR per beat note $l$ as

\begin{align}
\text{SNR}_l = \frac{A_l^2 / 4}{\sigma_R^2 \Delta T_s f_{BW}} \label{eq:SNR}
\end{align}

where $A_l$ is its amplitude, $\sigma_R^2$ is the variance of the white measurement noise, $\Delta T_s$ is the sampling period and $f_{BW}$ is the detection bandwidth.
While this seems counterintuitive, this impressive measurement noise rejection can be explained by the restrictive state-space model.
Since $\mathbf{C}_{c.}$ is determined by only one parameter, there are not enough degrees of freedom in the model to allow for phase noise overestimation. Intuitively, the single free parameter is determined during the optimization based on the high SNR lines. The phase noise of the noisy low SNR lines is then extrapolated based on the constrained phase noise model, such that the low SNR of the lines does not impact the estimation process.

The unconstrained variant with $\mathbf{C}_{uc.}$ produces accurate estimates for the central comb lines which have high SNR. For the outer lines with $\text{SNR}<\SI{15}{\decibel}$, the estimated variance is larger than the true value, which indicates that the phase noise magnitude is overestimated.
The conventional method estimates are similar where the high SNR comb line phase noise is estimated accurately. For the outer lines, however, the method completely fails. The very low SNR indicates that the measurement noise dominates over the beat notes, a regime in which the conventional method can not produce a good estimate.

In <ref> (b,c) two exemplary phase noise PSD of the comb lines with indices $l=-1$ and $l=3$ are shown. The former beat note has low SNR, which is, however, still enough such that the conventional method does not fail completely. Beyond $f=\SI{2}{\mega\hertz}$ its PSD estimate is dominated by measurement noise. Both EKF variants follow the true PSD, which is below the noise floor.
The beat note with index $l=3$ is one of the highest SNR notes. The conventional estimate is considerably closer to the true PSD, as the noise floor is lower than that of line $l=-1$.

§.§.§ Correlation matrices

Finally, we investigate the correlations of the comb lines phase noise in the time domain.
First, the covariance matrices are estimated using the sample covariance matrix defined as

\begin{align}
\widehat{\mathbf{Q}}_N &= \frac{1}{1-T} \sum_{k=1}^T (\Delta \mathbf{\phi}(t_k) - \overline{\Delta \mathbf{\phi}})(\Delta \mathbf{\phi}(t_k) - \overline{\Delta \mathbf{\phi}})^T \label{eq:cov_matrix}\\
\overline{\Delta \mathbf{\phi}} &= \frac{1}{T} \sum_{k=1}^T \Delta \mathbf{\phi}(t_k) \\
\Delta \mathbf{\phi}(t_k) &= \mathbf{\phi}(t_k) - \mathbf{\phi}(t_{k-1})
\end{align}

where $\overline{\Delta \mathbf{\phi}}$ is the sample mean. $\Delta \mathbf{\phi}(t_k)$ is the N-dimensional vector of differential comb line phase noise at time $t_k$.

The correlation matrices $\mathbf{R}$ are then calculated component-wise from the covariance matrices using

\begin{align}
\mathbf{R}_{nm} = \frac{\mathbf{Q}_{nm}} {\sqrt{\mathbf{Q}_{nn}\mathbf{Q}_{mm}}}
\label{eq:corr_matrix}
\end{align}

which returns the Pearson correlation coefficients.

In <ref>, the correlation matrix estimates for the different phase estimation methods are shown. Both correlation matrices from the EKF-based variants are similar to the true matrix. In the unconstrained case, however, the correlations for the line with index $l=7$ deviate from the ground truth. This line has a very low SNR, which causes problems in the optimization process. The optimized parameters that correspond to this line are harder to find, which causes the inaccuracy.

The conventional phase estimations' correlations match the true correlations for the central lines. As discussed before, the method fails for low SNR lines, resulting in completely uncorrelated estimates for the outer comb lines.
In summary, the accuracy of the correlation matrix estimated seems to follow the phase noise variance estimate that is shown in <ref>. This is not surprising as inaccurate phase noise estimates will lead to inaccurate estimates of their correlations.

(a) PSD of an experimental mutli-heterodyne EO comb signal. The number next to each line is its index, where negative numbers indicate aliased lines.
To showcase the EKF's ability to decompose the signal, the contribution from each line is isolated and displayed in a slightly different color. Individual comb lines can be isolated without the need for a bandpass filter. The signal, as detected in the experiment, is the sum of all lines and the residual noise.

(b) Frequency noise PSDs of the latent phase noise sources, as estimated by the EKF. The PSDs are shown with a 99% CI. Also shown are frequency noise PSDs of the seed laser and RF source used in the EO comb generation, extracted from independent reference measurements.

§.§ Application to an experimental EO comb

In this section, we demonstrate the robustness of the method and its applicability to real-world experimental data. The EKF-based method will be applied on experimental traces of a down-converted EO comb with similar characteristics to the simulated comb discussed in <ref>.

The experimental setup for the measurement considered in this section is identical to the setup displayed in <ref> (a). A NKT Photonics Koheras BASIK E15 fiber laser is mixed with an EO comb. The EO comb is generated using a separate NKT Photonics Koheras BASIK E15 laser and an EO modulator, driven by an Agilent E8247C signal generator. The mixed signal is detected by a balanced photodetector and digitized using a sampling oscilloscope with $f_{BW}=\SI{13}{\giga\hertz}$ and $f_{sampling}=\SI{40}{\giga\hertz}$. After digitization, the recorded signal is downsampled digitally by a factor 3, as no beat notes appear in the high-frequency part of the signal spectrum. This step reduces the the bandwidth of the measurement which increases the SNR.
The spectrum of the resulting measured signal is displayed in Fig. <ref> (a). We identify 13 non-aliased down-converted comb lines and 13 aliased lines, which are centered around the line with index $l_{center}=6$.
Notably, all lines except lines four to eight have low powers and, therefore, low SNRs. Additionally, each aliased line happens to be only about 36 away from a neighboring non-aliased line. Since most of the regular lines have higher SNR, they dominate over the aliased lines which makes them hard to characterize.
This is a problem when using the conventional method since it relies on bandpass filtering around each line. Because of the narrow spacing, the maximum bandwidth of each filter is limited to these 36. This limits the maximum measurable phase noise PSD offset frequency to half the bandwidth. For this measurement, this means that the conventional method can be used to characterize each comb line only up to 18. While in practice higher bandwidths could be used, the resulting phase noise estimate would be biased since it would include an aliased line.

Accuracy comparison between the EKF-based and conventional phase noise estimation method on an experimental EO comb.
(a) Integrated phase noise PSD per comb line for each method. For the EKF, the constrained model was used which only allows common and differential mode phase noise.

Additionally, the SNR per comb line is shown with the purple, triangular markers which are measured on the right y-axis. The SNR is calculated as in Eq. (<ref>).
(b) Differential phase noise correlation matrix of the EKF-based estimate.
(c) Differential phase noise correlation matrix of the conventional method's estimate.

As the EKF operates in the time domain, this problem is entirely circumvented as no bandpass filtering is required.
As long as the state space model correctly describes the detected signal, it does not matter how close the lines are to each other in frequency. We can, therefore, separate the contribution of each line and, in principle, isolate each line from the signal.
However, here we are interested in their phase noise properties.

To apply the EKF to the detected signal, we use the same steps as in <ref>. The same state-space model as defined in Eqs. (<ref>) and (<ref>) are used, which describe the latent space phase noise model.
As shown in the aforementioned section, the constrained model variant produces a more accurate phase noise estimate, especially for very low SNR comb lines. In this section we will, therefore, only use this variant. The corresponding optimization transformation function $\Lambda$ is defined in Eq. (<ref>).
After optimizing the model parameters $\mathbf{\theta}$, the EKF is applied to estimate the phase. As we used the constrained model for the mixing matrix $\mathbf{C}_{c.}$, the latent phase noise sources can be identified as the common and differential mode noise sources.
For the EO comb, the common mode corresponds to the phase noise of the comb seed laser, and the differential mode is the phase noise originating from the RF source.
As these are easily accessible, we can record independent phase noise reference measurements.
The seed laser phase noise is estimated by turning off the RF source and applying the conventional method. For the RF phase noise measurement, the source is connected directly to the ADC. The phase noise of the resulting signal spectrum is again estimated using the conventional method.

The frequency noise PSDs of the latent phase noise sources are shown in
<ref> (b), alongside the reference measurements.
The PSDs of the EKF estimates are accurate over a wide frequency range, as they match the reference measurements very well.
The common mode latent phase noise $\psi_0$ matches the seed laser phase noise over the full frequency range.
Up to 1 offset frequency, the differential mode phase noise $\psi_1$ matches the RF source phase noise.
Beyond 1, the reference measurement is limited by measurement noise, which can be seen by the $\propto f^2$ dependency of the frequency noise PSD.
At the same frequency, the PSD of $\psi_1$ flattens out and its CI grows larger.
Still, the interval predicts the PSD to be significantly below the noise floor of the reference measurement. A flat frequency noise PSD corresponds to the random walk behavior that is part of the EKF state space model in Eq. (<ref>). It is, therefore, not surprising that the EKF PSD estimate takes on this behavior: Once the phase noise is limited by measurement noise, extrapolating from the state-space model is the only remaining information the EKF can use.
The CI does not reflect this model assumption. Its calculation, as presented in the appendix, does not include uncertainty in the parameters of the model itself. It represents uncertainty originating only from the measurement noise while assuming completely certainty about the state-space model.
While a random walk is a valid assumption for laser phase noise, this is not necessarily the case for an RF source, as discussed in <ref>.
The latent phase noise PSDs below the measurement noise floor should only be trusted if the phase noise behavior is well-modeled in the state-space model.
Since we can not assume that the random walk is a good model for the RF source phase noise, we can not trust the PSD of $\psi_1$ beyond 1.

Next, we investigate the estimation accuracy with respect to the phase noise of the individual comb lines. Just as for the simulated EO comb, the phase noise per comb line can be calculated using Eq. (<ref>).
They are compared to estimation using the conventional method.

<ref> (a) shows the integrated phase noise variance per comb line for the EKF and conventional method.
The phase noise variance per comb line is calculated by integrating the phase noise PSDs of each comb line between $f_{min}=\SI{4}{\kilo\hertz}$ and $f_{max}=\SI{15}{\mega\hertz}$.
For the central comb lines with SNR > 15, both methods agree on the phase noise variance.
As we saw for the simulated EO comb in <ref>, both methods produce accurate estimates in high SNR regimes. Therefore, we can assume that both methods produce an accurate estimate for lines four to eight.
Beyond these lines the conventional method fails. Since we have used the constrained EKF model, its estimate follows the theoretical phase noise model of the EO comb in Eq. (<ref>). Just as in the simulation, the variance follows the expected parabolic form.
Since this is an experimental measurement, we do not have access to ground truth values. The simulation results from the previous section, however, indicate an accurate prediction.

In <ref> (b,c), the estimated correlation matrices of both methods are shown, which are calculated according to Eqs. (<ref>) and (<ref>).
As expected, the EKF predicts strongly correlated comb line phase noise. The conventional method can only recover the correlations for the few central lines that exhibit enough SNR.

To summarize this section, we have shown how to use the EKF to characterize the phase noise of down-converted frequency combs using the example of an EO comb.
Both in simulation and on experimental data we demonstrate that the EKF outperforms the conventional DSP method.
Especially for very low SNR comb lines, the EKF can still recover the phase noise while the conventional method fails. While the EO comb was used as a benchmark system, the latent space EKF method is much more general. Two variants of the latent space were introduced, which, in principle, allows for its application on any comb.