## Abstract

We present an original instrument designed to accomplish high-speed spectroscopy of individual optical lines based on a frequency comb generated by pseudo-random phase modulation of a continuous-wave (CW) laser. This approach delivers efficient usage of the laser power as well as independent control over the spectral point spacing, bandwidth and central wavelength of the comb. The comb is mixed with a local oscillator generated from the same CW laser frequency-shifted by an acousto-optic modulator, enabling a self-heterodyne detection scheme. The current configuration offers a calibrated spectrum every 1.12 µs. We demonstrate the capabilities of the spectrometer by producing averaged, as well as time-resolved, spectra of the D1 transition of cesium with a 9.8-MHz point spacing, a 50-kHz resolution and a span of more than 3 GHz. The spectra obtained after 1 ms of averaging are fitted with complex Voigt profiles that return parameters in good agreement with expected values.

© 2015 Optical Society of America

## 1. Introduction

Mode-locked frequency combs have proven to be invaluable tools for the measurement of optical frequencies [1]. When used for spectroscopy, these tools are however often too rigid and their adaptation to specific applications represents a challenge. Indeed, there are significant difficulties associated with modifying the repetition rate, the central wavelength and the spectral power distribution of a comb [2–4]. Moreover, while wideband comb spectroscopy techniques are considered fast compared to conventional spectrometers, they do not make optimal use of combs considering that a full spectrum is theoretically available at each pulse repetition. This is due to the necessity to extract the wideband and dense optical information in a relatively narrow electrical bandwidth. Direct comb spectroscopy using dispersive elements [5–7] and multiheterodyne spectroscopy [8–10] both suffer from speed limitations due respectively to the readout time of the camera and the construction rate of the interferogram, which is dictated by the required spectral compression ratio [9].

When the optical bandwidth of interest is narrow enough that no compression is required from the optical to the electrical domain, it should be possible to reach the shortest theoretical measurement time. This could be used to provide fast coherent averaging for applications such as the absolute measurement of single optical lines for the determination of fundamental constants [11–13] as well as the development of frequency standards [14, 15]. Faster instruments would also allow time-resolved spectroscopy to access shorter time scales, enabling the study of rapid phenomena such as chemical reactions [16] and population transfer dynamics [17]. In all these cases, efficient use of the comb power as well as convenient tunability of the spectral point spacing, central wavelength and bandwidth are paramount to optimize the signal-to-noise ratio for short measurement times.

A new approach called frequency comb interference spectroscopy (FCIS) has recently been proposed to accomplish fast spectroscopy of single optical lines with a few GHz of bandwidth [18, 19]. This approach essentially relies on fast electronics to detect the heterodyne beat between a mode-locked frequency comb and a single-frequency laser. In contrast to multiheterodyne spectroscopy, the spectrum is not compressed once downmixed to the electrical domain. This approach can thus only recover a small proportion of the original comb that is twice as large as the electrical bandwidth available at the detection stage. Since the majority of comb lines are unused and only contribute to shot noise if no optical filtering is in place, the mode-locked comb is not the optimal probe for this kind of experiment.

Comb-like sources with adjustable bandwidth can be readily created using a continuous-wave (CW) laser and an electro-optic modulator (EOM) driven by a periodic waveform [20]. This design offers the possibility to redistribute the optical power from the CW laser over a frequency range matching the detection bandwidth, allowing the FCIS technique to be more efficient in terms of power. EOMs have been used previously to create combs accomplishing multiheterodyne spectroscopy with a limited number of modes [21, 22], or with denser combs generated from pulsed modulation [23], the latter design making an inefficient utilization of the available CW power by switching the laser on only a small fraction of the time.

Modulation using pseudo-random bit sequences (PRBS) shows promise for the efficient generation of adjustable optical combs. This kind of excitation has already been used for the characterization of systems in many branches of science such as acoustics [24], control engineering [25], nuclear magnetic resonance spectroscopy [26] and radar detection [27]. It has also been used in optics to incoherently measure temporal responses of molecules [28] and for laser ranging [29]. However, PRBSs have only found limited applications in optical spectroscopy. Dinse et al. [30] explored the idea with narrow combs to extend the concept of frequency-modulation spectroscopy [31]. However, this system lacked a dedicated local oscillator to perform a heterodyne measurement and thus only the degenerate beats obtained from all possible combinations of optical modes could be detected. Moreover, the data was cross-correlated with the initial PRBS, an operation which does not yield a proper amplitude calibration. These conditions are not acceptable for applications requiring high-precision quantitative spectroscopy [12].

This paper demonstrates a new kind of fast, absolute and phase-sensitive spectrometer based on self-heterodyne detection of an optical comb generated from a pseudo-randomly phase-modulated CW laser. This modulation scheme offers flexible and fully decoupled control over the mode spacing and bandwidth of the comb, which addresses the optical power waste issues of the FCIS approach. Furthermore, it reduces the dynamic range and nonlinearity constraints associated with pulsed signals. The simple design suggested here allows power-efficient generation of wavelength-tunable combs matched to the detection bandwidth and offers high-speed measurements with an innovative in-line calibration method. If needed, a comparison of the CW laser with a referenced mode-locked frequency comb allows the retrieval of an absolute frequency axis. This technique, which can be described as Self-Hetef eventually performing absolute thermometry [11] and measuring population transfer dynamics [17].

## 2. Methodology

#### 2.1. Experimental setup

The method introduced here is based on the use of a single continuous-wave laser which is phase modulated using an EOM driven by an electrical PRBS. Figure 1 shows a schematic of the experimental setup. The light from the CW laser, emitting at frequency *ν _{c}*, is split in two paths via a fiber coupler. One tenth of the power is sent to an EOM to create a comb-like signal used as an optical probe. The remaining power is sent to an acousto-optic modulator (AOM) to generate a local oscillator (LO) whose frequency (

*ν*−

_{c}*f*) does not coincide with any of the comb modes. This design is similar to the one suggested in [29] for a different application. A 10/90 ratio is chosen as it makes the most efficient use of the available CW power, while avoiding saturation in the sample and appreciating that the quantity of interest is the geometric mean of the comb power and LO power. Both signals are mixed in a 50/50 coupler where one output is used as a reference and the other probes a sample, which is a cesium-133 vapour cell in our case. The resulting signals are finally detected by two fast photodiodes and are acquired simultaneously with a high-speed oscilloscope. Note that, unlike a balanced detection setup, the LO does pass through the sample but is detuned from the atomic resonance to minimize their interaction. This approach simplifies the setup, although it is not fundamentally necessary. The rough wavelength adjustment of the laser can be done using a wavemeter. It can then be precisely tuned by monitoring the position of the absorption feature on the Fourier transform of the electrical signal acquired by the oscilloscope.

_{AOM}A Virtex 6 field-programmable gate array (FPGA) mounted on a Xilinx ML605 board outputs a PRBS through its 5-Gb/s GTX digital transmitter, which is fed to the EOM. Using a maximum length sequence generates the smoothest power envelope, which has a theoretical sinc^{2} shape due to ideally rectangular bits [33]. Such a maximum length sequence can be obtained from a linear feedback shift register having *n* taps, yielding a 2* ^{n}* − 1 bit-long sequence. Figure 2(a) illustrates the hypothetical case of an electrical PRBS having a bit rate

*f*and a sequence rate

_{B}*f*. When fed to the EOM with an amplitude corresponding to the half-wave voltage (

_{R}*V*), the optical carrier is phase modulated between 0 and

_{π}*π*. The result, shown on Fig. 2(b), is a sine wave having discrete phase jumps. This can equivalently be seen as a discrete amplitude modulation of the sine wave between −1 and 1. The power spectrum of the electrical PRBS is a sinc

^{2}-shaped comb whereas the power spectrum of the optical signal measured after the EOM is identical, but centered on

*ν*. The latter is shown on Fig. 2(c). The power of the carrier relative to the envelope is dependant on the sequence length. The bandwidth and the mode spacing of the resulting comb are readily and independently adjustable in software to best suit the user’s application. The bit rate (

_{c}*f*) sets the bandwidth of the comb whose first zero falls exactly at

_{B}*f*from the optical carrier (

_{B}*ν*). The sinc

_{c}^{2}-shaped comb envelope then has a full width at half maximum of approximately 0.886 ×

*f*. The mode spacing is controlled via the sequence rate (

_{B}*f*) and sets the spectral point spacing of the spectrometer, analogous to the repetition rate with a conventional mode-locked frequency comb. For a given

_{R}*f*, the mode spacing is then defined by

_{B}*f*=

_{R}*f*/(2

_{B}*− 1).*

^{n}The FPGA also receives and distributes a clock signal to all other electrical components. It can make use of an atomic clock or other reference signal. For this experiment, we chose to use the 250-MHz photodetected repetition rate of a stabilized mode-locked frequency comb (Menlo Systems FC1500). The carrier-envelope offset frequency and repetition rate have relative stabilities of < 10^{−11} and < 10^{−13} for a 1-s observation time, respectively, and are both known with a relative accuracy of < 10^{−11} [7]. The FPGA divides the clock and distributes a standard 10-MHz synchronization signal to the oscilloscope and the function generator driving the AOM. The PRBS is also referenced to the mode-locked comb repetition rate. As explained further in subsection 2.2, making sure that all components in the system have the same time base substantially eases the signal processing.

With the comb mode spacing fixed by the external reference clock, the value of *ν _{c}* is the only remaining parameter required in order to accomplish absolute spectroscopy. In our case, this is done by comparing the CW laser to the 895-nm output of the same mode-locked comb used to generate the clock. Using a separate frequency comb as a frequency ruler to perform absolute spectroscopy is a common approach that is used by other techniques. Knowing the carrier-envelope offset frequency and the repetition rate of the comb, as well as using the measured atomic resonance as a marker to extract the mode index closest to the CW laser, allows an absolute determination of

*ν*. If the rough frequency of the spectral feature of interest is not known a priori, one can scan the CW laser until the feature is found, measure the wavelength of the laser using a wavemeter having more resolution than half the mode-locked comb spacing and, finally, use that information to determine the closest mode index. Different bands of the optical spectrum can be probed simply by detuning the CW laser. As long as the wavelength is still in the operating band of the components, the EOM and AOM take care of generating a comb-like signal and a LO regardless of the central wavelength and without any other adjustment required.

_{c}#### 2.2. Signal processing

Figure 3 illustrates the heterodyning step occurring with the SHISCM approach for the power spectrum. The phase spectrum is not shown here but it is also generated simultaneously. On panel (a), an optical comb probing an atomic resonance is shown with a frequency shifted LO. Different colors are used to differentiate the low-frequency (blue) and high-frequency (orange) parts of the comb as well as the LO (green). Once it is photodetected, the comb is folded about the LO and is replicated at baseband, as seen on panel (b). As we use phase modulation to generate the optical comb, we should not see any signal at harmonics of *f _{R}* after it is photodetected. However, absorption and phase shifts due to the sample lead to generation of amplitude modulation, yielding an electrical comb having modes at harmonics of

*f*. It can be ignored since it occurs at different frequencies to those from the beat notes of interest and, for simplicity, is not shown on Fig. 3.

_{R}Although it is not fundamentally necessary, having a fully synchronized system enables easy and robust extraction of the useful information from the data. Parameters can be carefully chosen so that the beat notes measured by the oscilloscope are all resolved and fall exactly on the bins of the discrete Fourier transform. To achieve this, one must first acquire an integer multiple of *q* sequences. In a second step, to ensure that comb modes below and above the LO fall precisely on distinct frequency bins, it is required that the AOM shift be of an integer multiple *p* of spectral bins, such that the corresponding frequency be *f _{AOM}* =

*p/q*×

*f*. The beat notes then fall at frequencies (

_{R}*m*+

*p/q*) ×

*f*and (

_{R}*m*+ (1 −

*p/q*)) ×

*f*, with m an integer representing the mode index. The spectral sampling is simply accomplished by reading the complex amplitudes of bins corresponding to beat notes. Furthermore, since a single laser is used, its linewidth is eliminated at the heterodyning step if delays are properly matched. We verified that the electrical linewidth was transform limited to 1 kHz in our case.

_{R}We can deduce from the predicted beat frequencies and from Fig. 3 that *p/q* = 1/3 sets a lower bound to the number of sequences required to resolve all beat notes and isolate them from the modes at *m* × *f _{R}*. It is the first scenario in which they all fall on independent bins. Indeed,

*p/q*= 1/2 cannot be used as both sides of the comb would overlap in the electrical domain, while

*p/q*= 1 places the LO exactly on an existing optical mode. Theoretically, a single packet of

*q*= 3 sequences is thus sufficient to obtain a full double-sided spectrum of the sample. This is a considerable advantage over multiheterodyne spectroscopy where an interferogram can only be constructed with a large number of successive pulses for a given repetition rate and mode spacing. However, there is still an inevitable trade-off between spectral point spacing and minimum measurement time because long sequences are required for dense spectral sampling.

PRBS signals are often processed with a cross-correlation operation to obtain temporal signals which approximate an impulse response [30]. This can be useful for some applications like control engineering and radar, but is an inappropriate tool for spectroscopy. Cross-correlating the measured signal with the initial PRBS squares the spectral envelope instead of flattening it. Furthermore, the downmixing step used here adds another level of complexity by modifying the appearance of the temporal PRBS. Calibration data is therefore mandatory to obtain spectra having accurate transmittance and phase values.

We undertake an innovative two-step calibration procedure. The first step is to divide the complex mode amplitudes from the sample channel by those from the reference channel (see Fig. 1). This cancels common-mode optical distortions encountered before the last fiber coupler and removes the spectral shape of the probe comb. We thus obtain a spectrum similar to the one illustrated on Fig. 3(c). It is worth mentioning here that comb modes generated from a PRBS have pseudo-random phases, which are also cancelled at this step. This calibration step also corrects linear electrical distortions common to both channels and only the ratio of their transfer functions remains.

Large dissimilarities are expected between the two electrical channels when working with SHISCM and thus a second calibration is required to compensate for the remaining ratio of transfer functions mentioned above. Indeed, no spectral compression occurs at the downmixing step and thus the required electrical bandwidth is approximately half that of the optical comb. Electrical signals with GHz bandwidth push currently available electronics to their limits and strong spectral distortions are likely to appear. Figure 4(a) illustrates an example of such a distortion as it appears after the first calibration step. By wisely choosing the AOM parameter *p*, we can make the low and high-frequency parts of the initial comb fall closely to one another in the electrical domain. They then suffer almost identical electrical deviations and can be used in pairs to perform a second calibration. The optical spectrum of interest is ideally imprinted on a single side of the LO, as seen on Fig. 3(a), while the other side is farther from the resonance. Hence, we use half of the comb modes created by the EOM as references for the remaining half. Modes that serve as references are interpolated at the locations of modes carrying information for increased precision. The result is shown on Fig. 4 (b). This two-step calibration procedure cuts the useable band in half, but it conveniently allows single-shot measurements without the need for an off-line measurement where the sample is taken out of the setup.

Note that the case illustrated by Figs. 3 and 4 does not produce a perfect representation of the atomic line because its wings extend into the region of the comb used as a reference (blue region). In Fig. 4, the measured line and the true line disagree slightly near DC because the line is effectively divided by the complex conjugate of its own folded wing during the second calibration. However, this effect is well understood and can easily be taken into account when fitting a theoretical line profile to the data, allowing the retrieval of the exact line centre, width and depth. The effect can also be reduced with further detuning of the spectral feature. However, if a model for the line profile is not available or if slight variations in the profile are investigated, the proposed calibration method is not appropriate. In that case, a separate more conventional calibration can be performed with an empty cell and with the same electrical channel used for the sample measurement.

## 3. Demonstration

#### 3.1. Experimental conditions

We measure the D1 transition (6^{2}*S*_{1}_{/}_{2} *→* 6^{2}*P*_{1}_{/}_{2}) of cesium-133 located around 894.6 nm as a demonstration for the SHISCM technique. The transition is in fact composed of four hyperfine levels producing two groups of lines separated by *∼* 9.2 GHz, each containing two lines *∼* 1.2 GHz apart [32]. A two-layer magnetic shield is placed around the cell to prevent Zeeman splitting. The CW laser, a tunable titanium-sapphire laser (M Squared SolsTiS) with a 50-kHz linewidth, is tuned near the group of interest involving either the ground state *F _{G}* = 4 and the excited states

*F*= 3,4, or the ground state

_{E}*F*= 3 and the same excited states. The EOM (Photline NIR-MPX800-LN-10) outputs 400 µW of total comb power, which yields a total intensity of 5.2 mW/cm

_{G}^{2}in the sample. The experiment is performed at a temperature of 22 ± 1

*°*C.

The digital transmitter is used at its full rate, *f _{B}* = 5 Gb/s, to obtain an optical comb with full width at half maximum of 4.4 GHz. The sequence length is set to 2

^{9}− 1 = 511 bits, providing a spectral point spacing of

*f*= 5/511 GHz ≈ 9.8 MHz. The sinc

_{R}^{2}-shaped comb envelope thus contains approximatively 1000 modes between its two first zeros. The optical resolution of the spectrometer is determined by the 50-kHz linewidth of the comb modes, which is limited by the initial frequency noise of the CW laser. The ratio

*p/q*for the AOM frequency is fixed to 5/11 so that the downmixed modes fall at frequencies (

*m*+ 5/11) ×

*f*and (

_{R}*m*+ 6/11) ×

*f*. This ensures that the folded modes are close enough in the electrical domain, namely

_{R}*f*/11 ≈ 0.89 MHz, to perform a high-quality electrical calibration. Reducing that gap by a factor of 10 did not show any improvement on the final spectra. This choice forces the acquisition of an integer number of

_{R}*q*= 11 sequences to fully resolve the modes in the discrete Fourier transform. Therefore, the minimum measurement time with that configuration is of 11

*/f*≈ 1.12 µs. Because the operating band of our AOM is centered on 100 MHz, the AOM frequency is fixed to

_{R}*f*= (10 + 5/11) ×

_{AOM}*f*≈ 102.3 MHz.

_{R}The sample and reference signals are detected by two photodiodes (Discovery Semiconductors DSC-R401HG) with 20-GHz bandwidth and are acquired simultaneously by an 8-GHz oscilloscope (Keysight MSOS804A) at a rate of 20 GS/s. Using a moderately faster digital transmitter would allow to create a wider comb and fully exploit the bandwidth available at the detection stage. As the high-speed oscilloscope only has two input channels, another oscilloscope synchronized with the data acquisition is used to record the beat note between the CW laser and the mode-locked frequency comb to later retrieve an absolute frequency axis (see Fig. 1). A total averaging time of 1 ms is chosen so as to preserve a good balance between speed and random noise. The data is intentionally truncated to the nearest integer number of *q* = 11 sequences.

Figure 5 shows the optical comb obtained with the above parameters. It is folded about the LO and thus all frequencies are relative to *ν _{c}* −

*f*. The first large peak corresponds to the optical carrier, which remains high because the driving signal has a lower amplitude than the EOM’s

_{AOM}*V*voltage. The two peaks near 5 GHz are found at the expected locations for the two first zeros of the sinc

_{π}^{2}once folded. Because the modulation signal is not composed of two perfect discrete levels, the harmonics of the 5-Gb/s bit stream are not completely nulled by the zeros of the expected sinc

^{2}. An inset also shows a close-up of the spectrum and reveals two families of comb modes, as depicted in Fig. 3(b). Within a same family, the mode spacing is

*f*≈ 9.8 MHz, while the spacing between the two families is

_{R}*f*/11 ≈ 0.89 MHz. The modulation observed on the modes also appears on the spectrum of the measured electrical PRBS. Therefore, it is common to both the sample and reference arms and it is calibrated using the procedure presented above.

_{R}#### 3.2. Spectral analysis

Figure 6(a) shows calibrated transmittance and phase spectra for the *F _{G}* = 4 line pair of cesium obtained from 1 ms of processed data. The measurement window covers 3.2 GHz of the spectrum, which extends beyond the 3-dB bandwidth of the system limited by the digital transmitter. The bottom axis is relative and represents the electrical frequencies associated with the down-mixed comb modes. The absolute frequency axis is also retrieved and indicated at the top. A fit based on a complex Voigt profile is superimposed with the data points and residuals are shown under each curve. The cesium line pair involving the ground state

*F*= 3 is also measured and shown on Fig. 6(b).

_{G}The model used for the fit is derived from the Faddeeva function *w*(*z*), also known as the complex error function [34]. It can be decomposed as

*V*(

_{R}*z*) and

*V*(

_{I}*z*) are the real and imaginary Voigt profiles when evaluated at $z=((f-{f}_{0})+i\gamma )/(\sqrt{2}\sigma )$, with

*f*the electrical frequency,

*f*

_{0}the line centre,

*γ*the Lorentzian half width at half maximum and

*σ*the Gaussian (Doppler) half width at one standard deviation. Introducing

*w*(

*z*) into the Beer-Lambert law [35] yields a real and a complex exponential respectively describing the absorption and the dispersion in the sample. From there, we can easily model the measured transmittance

*T*(

*f*) and phase shift

*ϕ*(

*f*) caused by two atomic lines of strengths

*α*with the expressions

_{j}As mentioned earlier, the optical spectrum is divided by the flipped complex conjugate of itself when performing the electrical calibration developed for SHISCM. This is reproduced in the model and the corrected quantities are

*γ*= 2.28 MHz [32] and the remaining parameters are retrieved via a nonlinear least squares fit using Eq. 4 and 5 that simultaneously minimizes the transmittance and phase residuals. Those parameters can be used to plot the true lines without the undesired effect introduced by the calibration.

The residuals shown on Fig. 6 are of the order of 1 % of their respective full-scale range. A coarse structure can nonetheless be observed, which indicates the presence of unmodeled effects on the line shape. This could potentially be due to saturation [36], but a more elaborate model that considers the excitation of the atoms with a dense comb would first be required to truly understand the nature of those residuals. One could question if they could be produced by some parasitic optical effects such as undesired reflections creating etalons. We verified that it was not the case by measuring spectra without the cesium cell in the sample arm that did not show the slowly varying residual structure. We also verified that the cell itself was not responsible for parasitic reflections by obtaining flat spectra while using the cell and the CW laser far detuned from the atomic resonances. This reinforces the hypothesis that the residuals are mainly due to an imperfect line profile model. The fact that the slow variations are strongly correlated with the spectral features also leads to the same conclusion.

Beyond this modeling issue, the spectra generated with the SHISCM technique can be biased by undesirable phenomena that systematically add energy at frequencies of beat notes. Simulations and experimental validations have shown that photodetector nonlinearities as well as the presence of other frequencies created by the AOM, particularly *ν _{c}* +

*f*for the case depicted by Fig. 3, can perturb those specific frequencies. Signal integrity after digitization is also crucial here as nonlinear or imperfectly interleaved analog-to-digital converters can produce harmonics and modulated copies of the signal. These systematic effects combined with random digitization noise are believed to be responsible for the deviations remaining within the coarser residual structure of Fig. 6.

_{AOM}Table 1 gathers the line parameters retrieved after fitting the two previous data sets. The Gaussian (Doppler) widths *σ* and the line splittings Δ*ν*_{34}, between *F _{E}* = 3 and

*F*= 4, are given. The expected values for

_{E}*σ*, determined by calculation [35], and Δ

*ν*

_{34}, given in [13], are included in braces. Those numbers apply to all interrogated lines. The measured absolute line centres

*ν*

_{0}and their expected values are also included. The uncertainties for all measured values are the 95% confidence intervals computed from fit residuals assuming a normal distribution. However, residuals are dominated by systematic and modeling errors rather than by random noise. The observed deviations from expected values, which are of a few MHz, at most, are consistent with those residuals.

#### 3.3. Time-resolved spectroscopy

To demonstrate the rapidity of this instrument, we monitor the *F _{G}* = 4 line pair while a second pump laser scans the measurement window. The pump beam is generated with a laser diode operating around 895 nm and it has an intensity of

*∼*25 mW/cm

^{2}. It is pointed at the vapour cell so as to be nearly co-propagating with the probe comb beam with an angle of

*∼*2

*°*, crossing it in the middle of the cell. Figure 7 shows a frame taken from Visualization 1 in which the pump laser scans both lines in the direction of increasing frequencies. The laser is first coincident with the

*F*= 4

_{G}*→ F*= 3 line, strongly pumping atoms to the excited state. Both lines suffer from an absorption reduction as they share a common depleted ground state. The pump is afterwards coincident with the

_{E}*F*= 4

_{G}*→ F*= 4 line for a similar effect.

_{E}For
Visualization 1, a *∼* 1-ms measurement was sliced in 80 pieces of 12.37 µs, corresponding to 11 packets of *q* = 11 sequences for each slice (11×11*/f _{R}* = 12.37 µs). The frames were computed by taking the Fourier transform of each slice. From
Visualization 1, the scanning rate of the pump laser is estimated to

*∼*3 THz/s. The spectra are dominated by random noise and have a frequency-dependent signal-to-noise ratio (SNR) due to the shape of the probe comb. The SNR for the transmittance data, computed as one over the standard deviation of spectral points not showing absorption, is of

*∼*135 on the left side of the spectrum and of

*∼*39 on the right side. Visualization 2 shows the case where 3 packets of

*q*= 11 sequences are used to compute each frame, for a temporal resolution of 3 × 11

*/f*= 3.37 µs. Here, SNRs at low and high frequencies are of

_{R}*∼*75 and

*∼*17.

## 4. Conclusion

We have demonstrated a new type of fast spectrometer that uses a single laser and pseudo-random phase modulation for the generation of an optical comb. In conjunction with an innovative calibration method, this simple instrument is especially useful for the accurate measurement of individual optical lines with independently adjustable bandwidth and spectral point spacing. This bandwidth is obviously limited by the electrical components of the system. With the fastest components now commercially available, this technique could potentially reach a bandwidth of a few tens of GHz. As it is now, taking multiple measurements while tuning the central wavelength of the comb is the easiest way to obtain broader spectra.

We showed that this instrument could easily provide high-quality spectra with an absolute frequency axis after only 1 ms of averaging by measuring the D1 transition of cesium. This technique also shows potential for thermometry using Doppler-broadened optical lines. The accuracy of the instrument can be improved with a more elaborate model of the line shape considering subtle effects such as the ones associated with a simultaneous and dense interrogation of the atomic resonance. A reduction of other systematic contributions with an improved setup, by replacing the AOM by a single sideband EOM for instance, would also be helpful.

Moreover, we showed that short time scales could be accessed while preserving an adequate measurement quality, opening the way to time-resolved spectroscopy. Temporal resolutions of 3.37 µs and 12.37 µs were demonstrated, with different compromises between speed and SNR. The speed is ultimately limited by the sequence length, which fixes the spectral point spacing.

## Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Fonds de Recherche du Québec - Nature et Technologies, the NIST Precision Measurement Grants Program and the Australian Research Council ( FT0991631, DP1094500). A. L. acknowledges support by the South Australian Government through the Premiers Science and Research Fund. The authors also thank TRIO Test & Measurement for providing the Keysight oscilloscope, as well as Christopher Perrella and Thomas M. Stace for helpful discussions.

## References and links

**1. **T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature **416**, 233–237 (2002). [CrossRef] [PubMed]

**2. **Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. **4**, 3816 (2014). [CrossRef]

**3. **N. B. Hébert, S. Boudreau, J. Genest, and J.-D. Deschênes, “Coherent dual-comb interferometry with quasi-integer-ratio repetition rates,” Opt. Express **22**, 29152–29160 (2014). [CrossRef] [PubMed]

**4. **F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics **5**, 770–776 (2011). [CrossRef]

**5. **S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature **445**, 627–630 (2007). [CrossRef] [PubMed]

**6. **M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express **16**, 2387–2397 (2008). [CrossRef] [PubMed]

**7. **N. B. Hébert, S. K. Scholten, R. T. White, J. Genest, A. N. Luiten, and J. D. Anstie, “A quantitative mode-resolved frequency comb spectrometer,” Opt. Express **23**, 13991–14001 (2015). [CrossRef] [PubMed]

**8. **F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. **29**, 1542–1544 (2004). [CrossRef] [PubMed]

**9. **I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. **100**, 013902 (2008). [CrossRef] [PubMed]

**10. **J. Roy, J.-D. Deschênes, S. Potvin, and J. Genest, “Continuous real-time correction and averaging for frequency comb interferometry,” Opt. Express **20**, 21932–21939 (2012). [CrossRef] [PubMed]

**11. **G.-W. Truong, E. F. May, T. M. Stace, and A. N. Luiten, “Quantitative atomic spectroscopy for primary thermometry,” Phys. Rev. A **83**, 033805 (2011). [CrossRef]

**12. **A. Castrillo, G. Casa, A. Merlone, G. Galzerano, P. Laporta, and L. Gianfrani, “On the determination of the boltzmann constant by means of precision molecular spectroscopy in the near-infrared,” C. R. Phys. **10**, 894–906 (2009). [CrossRef]

**13. **V. Gerginov, K. Calkins, C. E. Tanner, J. J. McFerran, S. Diddams, A. Bartels, and L. Hollberg, “Optical frequency measurements of 6s^{2}s_{1}_{/}_{2} − 6p^{2}p_{1}_{/}_{2} (d_{1}) transitions in ^{133}Cs and their impact on the fine-structure constant,” Phys. Rev. A **73**, 032504 (2006). [CrossRef]

**14. **M. De Labachelerie, K. Nakagawa, and M. Ohtsu, “Ultranarrow 13 c 2 h 2 saturated-absorption lines at 1.5 µ m,” Opt. Lett. **19**, 840–842 (1994). [CrossRef] [PubMed]

**15. **D. J. Robichaud, J. T. Hodges, P. Masłowski, L. Y. Yeung, M. Okumura, C. E. Miller, and L. R. Brown, “High-accuracy transition frequencies for the o 2 a-band,” J. Mol. Spectrosc. **251**, 27–37 (2008). [CrossRef]

**16. **E. T. Nibbering, H. Fidder, and E. Pines, “Ultrafast chemistry: using time-resolved vibrational spectroscopy for interrogation of structural dynamics,” Annu. Rev. Phys. Chem. **56**, 337–367 (2005). [CrossRef] [PubMed]

**17. **A. Marian, M. C. Stowe, D. Felinto, and J. Ye, “Direct frequency comb measurements of absolute optical frequencies and population transfer dynamics,” Phys. Rev. Lett. **95**, 023001 (2005). [CrossRef] [PubMed]

**18. **J.-D. Deschênes and J. Genest, “Frequency-noise removal and on-line calibration for accurate frequency comb interference spectroscopy of acetylene,” Appl. Opt. **53**, 731–735 (2014). [CrossRef] [PubMed]

**19. **K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and single-wavelength laser,” Appl. Phys. Lett. **101**, 051105 (2012). [CrossRef]

**20. **T. Sakamoto, T. Kawanishi, and M. Izutsu, “Widely wavelength-tunable ultra-flat frequency comb generation using conventional dual-drive machzehnder modulator,” Electron. Lett. **43**, 1039–1040 (2007). [CrossRef]

**21. **D. A. Long, A. J. Fleisher, K. O. Douglass, S. E. Maxwell, K. Bielska, J. T. Hodges, and D. F. Plusquellic, “Multiheterodyne spectroscopy with optical frequency combs generated from a continuous-wave laser,” Opt. Lett. **39**, 2688–2690 (2014). [CrossRef] [PubMed]

**22. **V. Duran, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” http://arxiv.org/abs/1507.04288.

**23. **M. Yan, S. Pitois, T. Hovannysyan, A. Bendahmane, T. W. Hänsch, N. Picqué, and G. Millot, “Dual-comb spectroscopy with frequency-agile lasers,” in “*CLEO: Science and Innovations*,” (Optical Society of America, 2015), pp. JTh5C–6.

**24. **E. Mommertz and S. Müller, “Measuring impulse responses with digitally pre-emphasized pseudorandom noise derived from maximum-length sequences,” Appl. Acoust. **44**, 195–214 (1995). [CrossRef]

**25. **T. Roinila, M. Huovinen, M. Vilkko, and T. Helin, “Continuous monitoring of industrial processes through cross-correlation techniques,” in Proceedings of the “18th IFAC World Congress” (2011), pp. 12171–12176.

**26. **R. R. Ernst, “Magnetic resonance with stochastic excitation,” J. Mag. Reson. **3**, 10–27 (1970).

**27. **R. Zetik, J. Sachs, and R. S. Thomä, “Uwb short-range radar sensing-the architecture of a baseband, pseudo-noise uwb radar sensor,” IEEE Instrum. Meas. Mag. **10**, 39–45 (2007). [CrossRef]

**28. **H. Baba, K. Sakurai, and F. Shimizu, “Measurement system for temporal response of atomic and molecular systems using the correlation method with pseudorandomly modulated laser light,” Rev. Sci. Instrum. **54**, 454–457 (1983). [CrossRef]

**29. **G. De Vine, D. S. Rabeling, B. J. Slagmolen, T. T. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express **17**, 828–837 (2009). [CrossRef] [PubMed]

**30. **K. Dinse, M. Winters, and J. L. Hall, “Doppler-free optical multiplex spectroscopy with stochastic excitation,” J. Opt. Soc. Am. B **5**, 1825–1831 (1988). [CrossRef]

**31. **G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. **5**, 15–17 (1980). [CrossRef] [PubMed]

**32. **D. A. Steck, “Cesium d line data,” http://steck.us/alkalidata.

**33. **S. Haykin, *Communication Systems*(John Wiley & Sons, 2001).

**34. **S. Abrarov and B. M. Quine, “Efficient algorithmic implementation of the voigt/complex error function based on exponential series approximation,” Appl. Math. Comput. **218**, 1894–1902 (2011). [CrossRef]

**35. **W. Demtröder, *Laser spectroscopy: basic concepts and instrumentation*(Springer Science & Business Media, 2003). [CrossRef]

**36. **T. M. Stace, G.-W. Truong, J. Anstie, E. F. May, and A. N. Luiten, “Power-dependent line-shape corrections for quantitative spectroscopy,” Phys. Rev. A **86**, 012506 (2012). [CrossRef]