Nanometric dual-comb ranging using photon-level microcavity solitons

Here, we use mutually coherent CP solitons generated in a foundry-manufactured Si₃N₄ microresonator Liu_PR2023foundry ; Bao_arXiv2024rhythmic for DCR and reach the 1-nm-precision regime. We derive and verify equations establishing the relationship between DCR precision and spectral signal-to-noise ratio in the radiofrequency (RF) domain (denoted as rSNR). It not only fills the longstanding theoretical understanding gap for DCR, but also impacts precise spectroscopy using dual-comb phase measurement Newbury_Optica2016 . The coherent soliton pair also enables precise DCR with a received power as low as 7 pW or 5.5$\times$10^-4 photon per pulse for our 100 GHz soliton stream. The power used for photodetection is more than 60 dB lower than reported microcomb-based ranging systems Kippenberg_Science2018Range ; Vahala_Science2018Range ; Kippenberg_Nature2020massively ; Zhang_PR2020long ; Wong_PRL2021nanometric ; Kippenberg_NP2023chaotic ; Wang_NP2023breaking , and the photon number is an order of magnitude lower than the time-programmable frequency combs DCR system Newbury_Nature2022time . Our system’s reliance on the relative phase stability of CP solitons makes it immune to intensity fluctuations, setting it apart from other high-precision ranging techniques including the electro-optical sampling (EOS) Kim_NP2020ultrafast , time-of-flight (ToF) Vahala_Science2018Range , chaotic ranging Wang_NP2023breaking ; Kippenberg_NP2023chaotic and dispersive interferometry Zhang_PR2020long ; Wong_PRL2021nanometric techniques. In addition, we perform DCR at a rate of 1.83 MHz without optical amplification and measure fast vibrations up to 900 kHz with a sensitivity of 0.4 nm/$\sqrt{\rm Hz}$. The fast dual-comb vibrometry (DCV) is possible due to the high power per comb line in microcombs, although low total power has been perceived as a significant drawback for soliton microcombs. Our work highlights the advantages of microcombs for precise ranging, and can be used in surface profiling of samples with low reflection (e.g., integrated circuits and thin films for PICs) Kim_NP2020ultrafast ; joo2013femtosecond , strain sensing Kim_NP2020ultrafast ; gagliardi2010probing , and human voice recovery Boudreau_OE2014range .

Our setup to generate CP solitons with VFL is shown in Fig. 1a, see Methods. The generated soliton microcombs (Fig. 1b) were heterodyne beat on balanced photodetectors (BPDs) to generate the DCR signals. An example of the interferogram with $\delta f_{r}$=1.62 MHz is shown in Fig. 1c. We then Fourier transform the RF pulses to have the power spectrum and $\phi_{\rm sig}(\omega)$ shown in Fig. 1d. A power rSNR (signal over average noise floor) exceeding 80 dB can be obtained within a measurement time $t$=0.5 s. To show the phase stability between the CP solitons, we plot the Allan deviation of $\phi_{\rm sig}(9\omega_{r})-\phi_{\rm sig}(4\omega_{r})$ in Fig. 1e, which scales as $t^{-1/2}$ and reaches 4.5 mrad at 0.1 s. The relative timing stability between CP solitons can be estimated as $(\phi_{\rm sig}(9\omega_{r})-\phi_{\rm sig}(4\omega_{r}))/5\omega_{r}$, reaching 0.1 fs at 0.1 s.

Then, we analyze $\phi_{\rm sig}-\phi_{\rm ref}$ using RF pulses measured within 50 $\mu$s and observe an excellent linearity for the used $N$=30 lines (excluding the pump, see Fig. 1f). Fitting the relative phase yields the distance in multiple 50 $\mu$s slots (Fig. 1g). The DCR precision is evaluated by Allan deviation and shows a $t^{-1/2}$ scaling with a normalized precision of 0.6 nm$\cdot\sqrt{\rm s}$ (Fig. 1h). We further changed the used comb line number from $N$=30 to other numbers to evaluate the precision. Normalized precision as high as 0.5 nm$\cdot\sqrt{\rm s}$ (reaching 1.1-nm-precision at 0.2 s) is possible for our system. The inset of Fig. 1h confirms that our measurement always yields the same distance when using different comb line number $N$.

where rSNR_eff is the effective rSNR for the signal arm, $B$=$N$$f_{r}$ is the used comb bandwidth, and $\alpha$ is the ratio between the rSNR_eff in the reference and the signal arms. The phase of a measured RF tone has contribution from both signal and noise (inset of Fig. 2a). Thus, the phase deviation of a measured RF line ($\sigma_{\phi}$) is determined by the rSNR determines as,

$\sigma_{\phi}(4,t)$ plotted in Fig. 2a confirms this relationship. $\sigma_{\phi}$ divided by 2$\pi B$ yields the timing (thus, distance) deviation. The $1/\sqrt{N}$ term is added as fitting multiple points leads to higher stability montgomery2021introduction (Supplementary Sec. 1). To verify it, we analyzed the DCR precision for a fixed $B$=3.2 THz but selecting lines with different spacings (thus, different $N$). A $1/\sqrt{N}$ scaling is observed for the normalized precision (Fig. 2b). Finally, $\alpha$ is included as the DCR signal is derived from the phase difference between two arms (Eq. 1).

Our measured DCR precision is in excellent agreement with the theory (Fig. 2c). The precision no longer improves evidently for $N$$>$40. This is because rSNR($m$) decreases for large $|m|$, resulting in a reduced rSNR_eff. We numerically calculated rSNR_eff based on rSNR($m$) of the used microcomb lines (Methods and Supplementary Fig. S1). After taking the decrease of rSNR_eff into account, a scaling of $N^{-3/2}$ is observed (see Eq. 2 and inset of Fig. 2b).

Therefore, the high DCR precision in our system can be attributed to the high rSNR and broad usable bandwidth. rSNR_eff can be further written as,

where $S_{\rm n}$ is the spectral power density of the RF noise floor, and $f_{\rm BW}$=1/$t$ is the resolution bandwidth; $P_{\rm sig(LO)}$ and $E_{\rm sig(LO)}$ are the received signal (local) optical comb power and energy, respectively; $K(B)$ is a conversion coefficient that includes response from the BPD and variation of rSNR($m$) within the used bandwidth. By combining Eq. 2 and Eq. 4, it can be seen that $\sigma$ is proportional to $\sqrt{N}$ for a given bandwidth $B$ and comb powers. Note that the comb power is ultimately limited by the saturation of the BPDs. Thus, small comb line numbers for microcombs are beneficial for enhancing DCR precision.

The DCR Allan deviation still scales as $t^{-1/2}$, but with a lower normalized precision of 300 nm$\cdot\sqrt{\rm s}$ (top curve in Fig. 3c). It means a micron-precision can be obtained in 0.1 s using only 5.5$\times$10^-4 photon per pulse. The Allan deviation for a series of received powers is plotted in Fig. 3c, all exhibiting the $t^{-1/2}$ scaling. The normalized DCR precision decreases with the received power in a square-root trend, consistent with Eqs. 2, 4 (Fig. 3d).

Since interferometric DCR relies upon optical phase measurements, our system is immune against intensity fluctuations. To showcase this feature, we inserted an intensity modulator (IM) and drove it by a broadband noise in the signal arm (Fig. 1a). The measured RF pulses have strong fluctuations in the amplitude (Fig. 3e). Despite these fluctuations, the measured average distance remains the same for five 5 ms slots separated by 0.1 s ($t$=50 $\mu$s for a single data point, see Fig. 3f). The distribution of the RF pulse amplitudes subject to different intensity noise levels is shown in the inset of Fig. 3g (the blue one is the case without IM noise). The added intensity noise can cause the RF pulses to have a 30% intensity fluctuation (see the labelled standard deviation $\sigma_{\rm int}$). DCR Allan deviation retains the $t^{-1/2}$ scaling in the presence of the intensity noise (Fig. 3g, average received power is 15 $\mu$W). Nanometre-scale precision is still possible with $\sigma_{\rm int}$=30%.

The Allan deviation does increase with $\sigma_{\rm int}$, as the intensity noise deteriorates rSNR. rSNR_eff for the used 30 lines decreases with a slope of $-$0.26 in a log-log plot versus $\sigma_{\rm int}$, while the change of the DCR Allan deviation has a slope of 0.13 (Fig. 3h). Such a $-$0.5 relationship between them further strengthens our theory in Eq. 2. Note that the rSNR deterioration does not impact the DCR measured distance (Fig. 3f); it only means a longer measurement time is needed to reach a desired precision. The intensity noise immunity distinguishes interferometric DCR from other comb-based ranging techniques, that rely upon power detection Kim_NP2020ultrafast ; Vahala_Science2018Range ; Zhang_PR2020long ; Wong_PRL2021nanometric ; Wang_NP2023breaking ; Kippenberg_NP2023chaotic . The immunity can be quite useful when the measured target surface has varying reflection coefficients and may be used to distinguish amplitude and phase fluctuations induced by air turbulence Newbury_PRL2015broadband .

The noise floor in Fig. 4b provides a measure of the DCV sensitivity. We summarize this sensitivity versus received powers in Fig. 4c. For a nearly white noise floor, its noise spectral amplitude density has a linear relationship with the Allan deviation in DCR Barnes1971characterization . Since the DCR precision has an inverse square-root relationship with the received power (Fig. 3d and Eqs. 2, 4), the measured DCV sensitivity also follows an inverse square-root scaling with the received power, and reaches a sensitivity of 0.4 nm/$\sqrt{\rm Hz}$ (Fig. 4c).

To measure vibration at $\delta f_{r}$/2, the received power should guarantee a sufficient rSNR($m$, 1/$\delta f_{r}$) for single-interferogram DCR. It is technically challenging to quantify rSNR($m$, 1/$\delta f_{r}$) directly, as $f_{\rm BW}$=$\delta f_{r}$ for a single-interferogram measurement. In practice, we derive it from rSNR at $t$=100 $\mu$s, using the relationship that rSNR scales linearly with $t$ for VFL solitons Bao_arXiv2024rhythmic . The minimum rSNR($m$, 1/$\delta f_{r}$) among the used 30 lines under different received powers is plotted in Fig. 4d. Our data process practice suggests that the minimum rSNR($m$, 1/$\delta f_{r}$) should exceed 3 dB to enable a reliable measurement. The received power needed to reach this threshold is about 100 nW (about 10 photon per pulse). For lower powers, we need to average longer time to reach the required rSNR, and the needed time $t$ increases linearly with decreasing $P_{\rm sig}$. Therefore, the highest vibration frequency decreases linearly with the received power for $P_{\rm sig}<100$ nW (Fig. 4e). For the current system, tens of kHz vibration can be measured with nW-level received power (over 40 dB loss in the signal arm) without any microcomb amplification. Based on Eq. 4, the highest measurable DCV frequency decreases linearly with $N$ for a given received microcomb power (Fig. 4e).

However, as a characteristics of rhythmically interacting CP solitons Bao_arXiv2024rhythmic , a systematic temporal motion needs to be considered and calibrated when measuring over the non-ambiguity range (Supplementary Sec. 2). This soliton motion does not impact the DCR measurement strongly when the measurement range is below micron-scale (Fig. 4b and Supplementary Fig. S2).

In conclusion, we have demonstrated rapid, intensity noise immune, nanometric absolute ranging using photon-level received soliton pulses. The simplified DCR system tolerates 67 dB power loss to retain the precise ranging capability without using any optical amplification. Mutual coherence between CP solitons is essential to achieve this performance. The simplified DCR system can be used monitor the operation condition of nanodevices. Our theory and measurements elucidate why microcombs fit for dual-comb applications. The theory also provides an invaluable tool to optimize DCR and dual-comb phase spectroscopy Newbury_PRL2015broadband ; Newbury_Optica2016 for all types of comb systems. Recent progress of microcombs bodes well for the full photonic integration of our DCR system Bowers_Science2021laser , which could enable nanometric LiDAR on compact, mass-produced nanophotonic chips.

According to the linear regression theory montgomery2021introduction , the slope deviation is $\sigma_{k}=2\sqrt{3}/({\rm rSNR_{eff}^{1/2}}N^{3/2})$ in the homoscedasticity case (i.e., all the RF tones have the same phase deviation), when $N$ is large (see Supplementary Sec. 2). Thus, Eq. 2 can be derived by swapping dimensionless $m$ to $m\omega_{r}$.