### Near-ultraviolet frequency-comb generation, photon-counting dual-comb interferometer and experimental spectra

Table of Contents

A continuous-wave extended-cavity widely tuneable diode laser emits at a centre frequency of 193 THz (1,550 nm) with an average power of 40 mW. The continuous-wave-laser beam is split into two beams. In one beam, the frequency of the continuous-wave-laser is shifted using an acousto-optic modulator^{12,20,43,44}, to avoid aliasing in the dual-comb interferograms. Here the frequency shift is *f*_{AOM} = 40.0 MHz. The centre frequency of the near-ultraviolet dual-comb spectrum is therefore mapped at 160 MHz. In addition, we choose *f*_{AOM} as a multiple of δ*f*_{rep}, so that the condition δ*f*_{ceo} = 0(modulo δ*f*_{rep}) is met and the individual interferograms are strictly periodic waveforms with a period of 1/δ*f*_{rep.} Each beam is modulated by an electro-optic amplitude modulator followed by an electro-optic phase modulator. The phase modulator is driven at a voltage of roughly 4.4 *V*_{π} where *V*_{π} is the voltage required to induce a phase change of π. The amplitude modulator gates the linear part of the up- or down-chirp induced by the phase modulator, leading to a flat-top spectral intensity distribution. To drive repetition frequencies at around 500 MHz, roughly 27 comb lines are generated within a 3 dB bandwidth. Each near-infrared beam is amplified up to 400 mW by an erbium-doped fibre amplifier and it is frequency doubled to 384 THz (780 nm) in a 40 mm-long periodically poled lithium-niobate crystal with a conversion efficiency of 4 × 10^{−3} W^{−1} cm^{−1}. The residual 192 THz light is filtered out with a dichroic mirror and the 384 THz beam is focused onto a 10 mm-long BIBO crystal. The conversion efficiency in BIBO is 1 × 10^{−4} W^{−1} cm^{−1}. Its low value is due to the low input power in quasi continuous-wave conditions. Two near-ultraviolet combs of 100 lines each, with repetition frequencies *f*_{rep} and *f*_{rep} + δ*f*_{rep} respectively, are generated at around 772 THz. In these experiments, we tune the centre frequency of the combs between 770 and 774 THz but the specifications of the involved instrument allow for tunability between 750 and 784 THz. Typically, we chose *f*_{rep} between 200 and 500 MHz and δ*f*_{rep} between 1.6 MHz and 500 kHz. The span is 50 GHz at a line spacing of 500 MHz (Figs. 2 and 3), and 26 GHz at a line spacing of 200 MHz (Extended Data Fig. 5). The maximum detected count rate of each comb is 2.5 × 10^{7} counts s^{−1}, corresponding to an average power for each comb beam of up to 5 × 10^{−11} W.

The intensity fluctuations of each of the ultraviolet comb generators show a standard deviation of 1 × 10^{−3} over 350 s. For accurate frequency measurements, the frequency of the continuous-wave laser is measured against a commercial Er-doped-fibre frequency-comb synthesizer referenced to a global-positioning-system disciplined hydrogen maser. The continuous-wave laser can be phase-locked to a comb line, as in the results in Fig. 3 and Extended Data Fig. 3. The in-loop instabilities amount to less than 1 Hz over half an hour, which means that the optical width of the comb line (of the order of 100 kHz) is transferred to the continuous-wave laser. Alternatively, the continuous-wave laser can be free-running with instabilities monitored by the commercial comb synthesizer and its drifts are up to 2 MHz over 200 s. We did not observe any differences in the signal-to-noise ratio of the dual-comb spectra or in the quality of the spectral line shapes between these two configurations.

The beam of one comb passes through a heated caesium-vapour cell. The two near-ultraviolet beams are then superimposed on a beam splitter to form an interferometer. At one of the outputs of the beam splitter, an optical short-wavelength-pass filter further selects the ultraviolet light and the combined beam is detected by a photon-counting detector. The detector is a photomultiplier of a single-electron response width of 600 ps and a quantum efficiency of 25% at optical frequencies around 772 THz. Photon rates of several 10^{7} photons s^{−1} are detected, corresponding to optical powers incident on the detector of 10^{−11} W. The counts are counted as a function of time by a multiscaler. The multiscaler is started with a trigger signal generated by a frequency division of the 10 MHz clock that synchronizes all electronics in the experiment. Typically, for *f*_{rep} = 500 and δ*f*_{rep} = 1.6 MHz, we set the trigger to 200 kHz. At each trigger signal, the multiscaler adds up the counts to those of previous scans over a total duration of 5 μs, with a time resolution of 160 ps. By accumulating photon-count statistics, an interferogram is reconstituted. The interferograms are zero-filled with a factor of 16 (Fig. 2a,b) or eight (Figs. 2c and 3a,c). A complex Fourier transform of the interferogram reveals the amplitude spectrum and the phase spectrum.

In Fig. 2a, the count rate is 2.7 × 10^{7} counts s^{−1}. The strong line at about 770.71 THz (at a radio frequency of 80 MHz) is an artefact corresponding to the frequency-doubled frequency shift of the acousto-optic modulator. Part of the beam that is not deflected by the acousto-optic modulator is coupled into the output fibre. It is then frequency doubled in the periodically poled lithium-niobate crystal and detected by the photon counter. This additive spurious signal can be avoided, for example, by using an acousto-optic modulator with a better diffraction efficiency or by filtering out the 384 THz residual light with a filter of a higher optical density.

In the experiments in Fig. 3a, a caesium-vapour cell of a length of 75 mm is used. For the weak 6S_{1/2}–8P_{1/2} transition at around 770 THz (Fig. 3a,b), the Cs vapour pressure is 2 × 10^{−1} Pa (cell temperature 392 K), whereas in Extended Data Fig. 4a,b, both combs experience absorption by Cs and the observed absorption depth is stronger, therefore the cell temperature could be lowered to 378 K (vapour pressure 7 × 10^{−2} Pa). For the 6S_{1/2}–8P_{3/2} transition at around 773 THz (Fig. 3c,d and Extended Data Fig. 4c,d), the pressure is 2 × 10^{−2} Pa (cell heated to 359 K). If the signal-to-noise ratio in Fig. 3b and Extended Data Fig. 4 is normalized to an accumulation time of 1 s, one obtains a signal-to-noise ratio at 1 s on the order of 20 s^{−1/2}.

For measuring the frequency of the positions of the four 6S_{1/2}(*F* = 3,4)–8P_{1/2} and 6S_{1/2}(*F* = 3,4)–8P_{3/2} transitions in ^{133}Cs (Fig. 3), Doppler profiles are adjusted to the experimental transmittance spectra using a nonlinear least-square fit program. Extended Data Fig. 3 shows the observed spectra of Fig. 3, the results of the fit and the difference ‘observed-fitted’. The difference, at the noise level, does not show any specific signatures. The line centres returned by the fits provide the position of the lines. The positions in ten spectra are averaged for each transition and the results are presented in Extended Data Table 2. We estimate the relative frequency uncertainty of 6 × 10^{−9} for the two most intense lines; This uncertainty is dominated by the statistical uncertainty: the width of the profiles is on the order of 1 GHz and their signal-to-noise ratio is less than 100 and this limits how precisely the line positions can be determined. Our 6S_{1/2}(*F* = 3,4)–8P_{1/2} positions are in good agreement with that computed from the more precise measurements in ref. ^{34}, in which Doppler-free saturation spectroscopy was implemented. Future work will include reducing the line widths and enhancing the signal-to-noise ratio by background-free Doppler-free spectroscopy^{26}.

For *f*_{rep} = 200 MHz, we set δ*f*_{rep} = 0.5 MHz and the frequency of the trigger to 100 kHz. The spectrum (Extended Data Fig. 5) spans over 26 GHz and includes more than 130 comb lines. Whereas the spectral line shapes are more densely sampled at 200 MHz line spacing, the signal-to-noise ratio is 260 at an accumulation time of 328 s.

### Visible-range photon-counting dual-comb interferometer and experimental spectra

Two erbium-doped-fibre mode-locked lasers of a repetition frequency of 100 MHz emit at 192 THz (Extended Data Fig. 6). The repetition frequency *f*_{rep} = 100 MHz and the carrier-envelope offset frequency *f*_{ceo} of the first comb laser (called the master comb generator) are stabilized against the radio-frequency signal of a hydrogen maser, using self-referencing with a *f*-2*f* interferometer. The second comb (called slave comb) has a repetition frequency *f*_{rep} + δ*f*_{rep} with δ*f*_{rep} = −12.5 kHz and a carrier-envelope offset frequency *f*_{ceo} + δ*f*_{ceo}. It is stabilized against the first comb through feed-forward control of the relative carrier-envelope offset frequency. This scheme allows long coherence times for the interferometer and direct averaging of the time-domain interferograms over more than 1 hour. The feed-forward control scheme, which uses an external acousto-optic modulator, has been described in detail in ref. ^{36}. In addition, as one comb is fully referenced to a radio-frequency clock, absolute calibration of the frequency scale is directly achieved.

In our setup, each laser beam is frequency doubled to 384 THz in a 40-mm-long periodically poled lithium-niobate crystal. The span is limited to 120 GHz by the long crystals to reduce the volume of data, and to adapt to the capabilities of our multiscaler. At the output of the periodically poled lithium-niobate crystals, dichroic mirrors filter out the 192 THz radiation. The 384 THz beam of the master comb generator passes through a 3-cm-long cell with rubidium in natural abundance. The cell is heated to 315.5 K (vapour pressure of Rb 3.3 × 10^{−4} Pa). The beam is then combined on a beam splitter with the beam of the second comb generator. One output of the beam splitter is attenuated to an average optical power of 3 × 10^{−12} W. It is detected by a fibre-coupled single-photon-counting module based on an avalanche photodiode. The counting module detects an average rate of 8.4 × 10^{6} counts per second. The detection efficiency of the module is about 70%. The second output of the beam splitter is detected by a fast silicon photodiode and the central interference fringe provides a trigger signal. The counts of the single-photon-counting module are acquired by a multiscaler triggered by the the fast silicon photodiode. The sampling rate of the multiscaler is 156.25 × 10^{6} samples s^{−1}. On average, one count is detected every twelfth laser pulse. As many as 1.4 × 10^{6} triggered sequences, each of 3.28 ms, are summed up to provide, from the photon-counting statistics, a time-domain interference signal (Extended Data Fig. 7) accumulated over a total time of 4,592 s. The interferogram comprises 41 individual interferograms that recur at a period of 1/δ*f*_{rep} = 8 × 10^{−5} s for a total of 512,500 samples, at present limited by the multiscaler capabilities.

The raw interferometric signal shows a significant non-interferential part (Extended Data Fig. 7a); owing to dark counts of the detector, stray light, parasitic light leaking through the fibre before the counting module, spectra of the two combs not perfectly overlapping spectrally, the fringe visibility *V* is 36%. The amplitude of the 41 zero optical-delay bursts remains constant, illustrating that the sequences are efficiently averaged over the time of the experiment. The bandwidth of the electronics does not entirely filter out the pulses at the detector, explaining the residual pulse pattern that is maximum around zero optical delay and minimum at the largest optical delay, in the middle of the interferometric sequence. Simple numerical filtering returns the usual interferogram shape (Extended Data Fig. 7b), where the modulation due to the absorbing rubidium is clearly visible even in the region of the largest optical retardations of 5 ns (Extended Data Fig. 7c). The complex Fourier transform of the interferogram (Extended Data Fig. 7b) provides the complex response (amplitude and phase) of the sample. The unapodized amplitude spectrum, interpolated through fourfold zero-filling of the interferogram, shows well-resolved comb lines with the imprint of the Doppler-broadened 5S_{1/2}–5P_{3/2} transitions in ^{85}Rb and ^{87}Rb (Fig. 5). The spectral envelope of the spectrum has a Gaussian shape determined by the phase-matching conditions in the long periodically poled lithium-niobate crystal. The full-width at half-maximum of the spectral envelope is 38.4 GHz. This corresponds to 384 comb lines spaced at 100 MHz. The individual comb lines show the instrumental line shape of the interferometer, a cardinal sine, which is induced by the finite measurement time. Owing to the good signal-to-noise ratio, more than 1,200 comb lines are measurable over the entire spectral span. Sampling the spectra at the comb line positions reveals the transmittance and dispersion spectra of the Rb transitions (Extended Data Fig. 8).

### Derivation of the quantum-noise-limited signal-to-noise ratio in photon-counting mode

We adapt the formalism developed in refs. ^{45,46} to our experimental situation. We consider a dual-comb interferometer, in which only one output of the interferometer is detected by a photon counter. The time-domain interferogram is composed of a sequence of *L* individual interferograms. An individual interferogram spans over laboratory times between −1/(2δ*f*_{rep}) and +1/(2δ*f*_{rep}), corresponding to optical delays ranging between −1/(2*f*_{rep}) and +1/(2*f*_{rep}). As explained above, an individual interferogram is acquired over an accumulation time *T*_{indiv}, resulting from the addition, for each time bin, of photon counts over many triggered scans to statistically reconstruct the individual interferogram. Assuming that sufficient statistics have been accumulated, by a proper selection of the sampling and comb parameters, all the individual interferograms are expected to be identical, but for the noise.

At zero optical delay (*t* *=* 0), the quantum-noise-limited signal-to-noise ratio in one individual interferogram is given by

$${\left(\frac{S}{N}\right)}_{t=0}=\frac{{n}_{{\rm{interf}}}}{\sqrt{n+{n}_{{\rm{interf}}}}}$$

where *n* is the number of detector counts corresponding to non-interferometrically modulated signal, accumulated for the time bin at zero optical delay over an integration time of 1/*f*_{rep} and *n*_{interf} is the number of counts that contribute to the interferometric signal for the same time bin (that is, the total number of counts minus the number of counts for non-interferometric contributions).

In an ideal interferometer, one would expect \({n}_{{\rm{interf}}}=n\), leading to \({\left(\frac{S}{N}\right)}_{t=0}=\sqrt{\frac{n}{2}}\). Experimentally, however, many factors contribute to deviations in an additive fashion or in a multiplicative one. Additive contributions include residual stray light or the dark counts of the photon counter. Multiplicative contributions can be due to optical misalignment; the beam splitter may not have the optimal reflection and transmission coefficient; the two interfering combs may not be identical: they may show different power, spectral intensity distribution, polarization and so on.

The fringe visibility *V* can conveniently be introduced: \(V=\frac{{n}_{{\rm{interf}},\max }-{n}_{{\rm{interf}},\min }}{{n}_{{\rm{interf}},\max }+{n}_{{\rm{interf}},\min }}\,\), with \({n}_{{\rm{interf}},\max }\) and \({n}_{{\rm{interf}},\min }\) being the maxima and minima of the interference counts, respectively. In an ideal interferometer, *V* = 1. In our experiments, the additive noise is negligible.

In such a case, \({n}_{{\rm{interf}},\max }=n+{n}_{{\rm{interf}}}\) and \({n}_{{\rm{interf}},\min }=n-{n}_{{\rm{interf}}}\), thus \({n}_{{\rm{interf}}}=V\,n\).

Consequently, the signal-to-noise ratio at zero optical delay can be written:

$${\left(\frac{S}{N}\right)}_{t=0}=\frac{V}{\sqrt{1+V}}\sqrt{n},$$

The signal-to-noise ratio \({\left(\frac{S}{N}\right)}_{\nu }\) at the frequency *ν* in the spectrum is related to the signal-to-noise ratio \({\left(\frac{S}{N}\right)}_{t=0}\) at zero optical delay in the time-domain interferogram by the expression:

$${\left(\frac{S}{N}\right)}_{\nu }=\,\sqrt{\frac{2}{K}}\,\frac{B(\nu )}{\bar{{B}_{{\rm{e}}}}}\,{\left(\frac{S}{N}\right)}_{t=0}$$

where *B*(*ν*) is the spectral distribution at the frequency \(\nu ,\,{\bar{B}}_{{\rm{e}}}\) is the mean value of the spectral function \({B}_{{\rm{e}}}(\nu )=\frac{1}{2}(B(\nu )+B(-\nu ))\), which accounts for the unphysical negative frequencies. The number of time bins *K* in the interferogram is twice the number of spectral elements in the actual spectral distribution that, according to our sampling conditions here, spans from a frequency 0 to a frequency *f*_{rep}/2 on the radio-frequency scale.

Under the simplifying hypothesis that the spectrum is made of *M* comb lines, all of equal intensity, one can express the ratio *B*(*ν*) to \(\bar{{B}_{{\rm{e}}}}\) at a frequency *ν* corresponding to a comb line position:

$$\frac{B\left(\nu \right)}{\bar{{B}_{{\rm{e}}}}}=\frac{1/M}{1/K}=\frac{K}{M}$$

Consequently,

$${\left(\frac{S}{N}\right)}_{\nu }={\frac{\sqrt{2K}}{M}\left(\frac{S}{N}\right)}_{t=0}=\sqrt{2}\frac{V}{\sqrt{1+V}}\frac{\sqrt{K}}{M}\sqrt{n}$$

On summing up *L* individual interferograms, the quantum-limited signal-to-noise ratio at the comb line positions becomes:

$${\left(\frac{S}{N}\right)}_{\nu }=\sqrt{2}\frac{V}{\sqrt{1+V}}\frac{\sqrt{K}}{M}\sqrt{n\,L}$$

(1)

Equation (1) can also be written using the detected photon rate *N*_{phot} (in photons s^{−1}) in the interferogram:

$$n=\frac{{N}_{{\rm{phot}}}\,{T}_{{\rm{indiv}}}}{K}$$

where *K* is the number of time bins in the individual interferogram and *T*_{indiv} is the accumulation time for one individual interferogram that has optical delays −1/(2*f*_{rep}) to +1/(2*f*_{rep}) with time bins of 1/*f*_{rep}.

Equation (1) becomes

$${\left(\frac{S}{N}\right)}_{\nu }=\sqrt{2}\frac{V}{\sqrt{1+V}}\frac{1}{M}\sqrt{{N}_{{\rm{phot}}}\,{T}_{{\rm{indiv}}}\,L}$$

(2)

Moreover, the measurement of *N*_{phot} in the interferogram enables to infer the average power *P* incident on the photon counter.

$${N}_{{\rm{p}}{\rm{h}}{\rm{o}}{\rm{t}}}=\frac{P\,{\rm{Q}}{\rm{E}}\,}{h\nu }$$

where *P* is the average optical power incident on the counter and QE is the counter quantum efficiency.

One can also write the quantum-limited signal-to-noise ratio at the optical frequency *ν* of a comb line as an equation involving the average power rather than the photon counts:

$${\left(\frac{S}{N}\right)}_{\nu }=\frac{\sqrt{2}}{M}\frac{V}{\sqrt{1+V}}\sqrt{\frac{P\,{\rm{Q}}{\rm{E}}\,}{h\nu \,}\,{T}_{{\rm{i}}{\rm{n}}{\rm{d}}{\rm{i}}{\rm{v}}}\,L}=\frac{\sqrt{2}}{M}\frac{V}{\sqrt{1+V}}\sqrt{\frac{P\,{\rm{Q}}{\rm{E}}\,}{h\nu \,}\,{T}_{{\rm{t}}{\rm{o}}{\rm{t}}}\,}$$

where \({T}_{{\rm{tot}}}={T}_{{\rm{indiv}}}\,L\) is the total accumulation time in the entire recording.