Frequency-noise removal and on-line calibration for accurate frequency comb interference spectroscopy of acetylene Jean-Daniel Deschênes and Jérôme Genest* Centre d’Optique, Photonique et de Laser, 2375 de la Terrasse, Université Laval, Québec, Quebec G1V 0A6, Canada *Corresponding author: [email protected] Received 30 October 2013; revised 28 December 2013; accepted 30 December 2013; posted 3 January 2014 (Doc. ID 200431); published 30 January 2014
We demonstrate that using appropriate signal-processing techniques allows us to greatly improve the signal-to-noise ratio and accuracy of frequency comb interference spectroscopy measurements. We show that the phase noise from the continuous wave laser used as local oscillator is common to all beat notes and can be removed, enabling longer coherent integration time. An on-line calibration of the spectrum normalizes the frequency response of the electronics. The signal power-to-noise ratio of the spectra thus obtained is a factor of 16,000 (42 dB) higher than in previously demonstrated results, and the quality of the spectra is much higher. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (300.6530) Spectroscopy, ultrafast; (300.6310) Spectroscopy, heterodyne; (300.6390) Spectroscopy, molecular. http://dx.doi.org/10.1364/AO.53.000731
1. Introduction
As the technology of frequency combs matures, more ways of taking advantage of their high spectral coherence and frequency accuracy slowly keep emerging. While the first applications are obviously in frequency metrology [1] and optical clockworks [2], another flourishing field of applications is in spectroscopy [3–8]. A new spectroscopic technique called frequency comb interference spectroscopy (FCIS) was recently described in [9,10] and uses a frequency comb as a probing source for a single molecular transition. In this technique (Fig. 1), a small number of N comb modes is selected by a bandpass filter and sent through the sample to be measured where the molecular absorption (amplitude) and dispersion (phase) characteristics of the sample are imprinted onto the comb modes. A CW laser provides a single, strong local oscillator mode and is superimposed 1559-128X/14/040731-05$15.00/0 © 2014 Optical Society of America
with the light filtered by the sample and sent to a photodetector. This field only serves to downconvert all the comb modes into the radio-frequency (RF) domain to enable recording the sample information with conventional electronics. Since the frequency of each comb mode can be known accurately (for example, when one uses a self-referenced comb), this yields potential for high spectral accuracy, as the exact frequencies at which the sample was probed are related only to the comb and not to the CW laser. This technique sits somewhere between multiheterodyne spectroscopy techniques, which instead use many modes from a second comb as the local oscillators to provide the optical-to-RF mapping, and linear optical sampling techniques, where the comb is used as an LO for a modulated CW laser [11,12]. It represents a previously unexplored and interesting configuration with intriguing possibilities, providing a valuable trade-off for certain measurement scenarios, sacrificing bandwidth for accuracy over a single molecular transition. Indeed, the spectral point spacing is set by the repetition rate of the comb, while the spectral resolution is instead limited by the comb’s 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
731
Fig. 1. Experimental setup. Solid lines are optical fibers; split into the fibers are fiber couplers; and dashed lines are electrical coaxial cables. BPF, optical bandpass filter; DL, optical delay line; PC, polarization controller.
linewidth, which is usually orders of magnitude better than the mode spacing. The span of the resulting spectra is limited to the bandwidth of the photodetection electronics. Two potential application areas where FCIS seems particularly well suited are the measurement of fast dynamical processes, where the dynamic changes of a single molecular transition are of interest, or the accurate determination of absolute line centers and line shapes [7]. Indeed, the shortest acquisition time for a complete FCIS spectrum is equal to the comb repetition rate, where hundreds of MHz or a couple GHz are possible, although this potential remains to be demonstrated. In this paper, we report on two improvements that we believe to be critical to ensure the accuracy of the FCIS technique. The first and most important technique is detailed in Section 2, which is the tracking of the phase noise of the CW laser used as the LO. Our second contribution to the FCIS technique is described in Section 3, which is the use of an on-line spectrum calibration technique. Section 4 shows experimental results of spectroscopy of acetylene C2 H2 obtained using the presented techniques. 2. Phase Correction Algorithm
In a typical comb spectroscopy experiment, the maximum coherent integration time is limited by the mutual coherence time of the two lasers [13], in the case of an FCIS measurement being the CW laser and the comb. For maximum flexibility, an FCIS setup should use a tunable CW laser, which usually shows more noise than a fixed-frequency laser. As such, as was pointed out in [9], most of the errors in an FCIS setup will come from the LO’s frequency noise unless some form of mitigation is used. For this work, we have adapted the phase-correction technique from [13,14] to work with the FCIS setup. This allows us to completely decouple the experiment integration time and the LO’s coherence time and enables measurements of much higher accuracy. Implementing the phase-tracking algorithm in an FCIS experiment is much easier than in a multiheterodyne measurement, however, as no additional acquisition channel and associated optics are necessary, and all the information is already available from the recorded interference signal. Indeed, assuming that most frequency noise comes from the LO, every beating mode produces a signal of the form 732
APPLIED OPTICS / Vol. 53, No. 4 / 1 February 2014
Af k exp j2πf k t − f cw t − ϕt c:c. on the photodetector, Af being the complex amplitude of the beating mode and containing the sample’s complex transmittance, f k being the frequency of mode number k from the comb, f cw the frequency of the CW laser, and ϕt the phase noise (integrated frequency noise) from the CW laser, and c.c. indicating the complex conjugate of the previous term. One such beating mode can be seen on the blue trace in Fig. 2. The signal at the output of the photodetector is comprised of a linear sum of many such beating modes, k, but separated by the repetition rate in frequency (usually hundreds of MHz) such that we can easily isolate them one at a time using a software bandpass filter. There are also spectral lines at the repetition rate and all its harmonics, but these can be safely ignored as long as they are well separated from the comb-cw beating modes. Extracting a single mode “m” with a Hilbert bandpass filter [15] and normalizing by the amplitude, we can isolate a reference signal, exp j2π f m t − f cw t − ϕt. Multiplying the original signal by the complex conjugate of the reference signal yields modes of the form: A f k exp j2πf k − f m t, free of the influence of the frequency noise of the CW laser, yielding the green trace on Fig. 2. This works as long as a bandpass filter can unambiguously extract a single beat component, which means that the beat frequency drifts must be much smaller than the Nyquist frequency (half the repetition rate). All that is left is a complex exponential with a linear phase ramp carrying the desired information in its complex amplitude, which can be conveniently estimated using a simple discrete Fourier transform (DFT) at the desired frequencies. We call this phase extraction and multiplication procedure “phase correction” of the signal. This technique, coupled with the parallel acquisition of all the beating modes using a fast oscilloscope, enables measurements with low random noise at short acquisition times. Note that the time axis of the DFT is given by the oscilloscope’s timebase, which is itself synchronized
Fig. 2. Spectrum of one beating mode before (blue) and after (green) phase correction. Resolution bandwidth is 5 kHz. The noise floor is additive noise from the oscilloscope’s front-end. The linewidth of the beat signal is dominated by the CW laser’s frequency noise at around 500 kHz. After phase correction, all the beat signal’s energy becomes concentrated in a single transform-limited narrow peak.
to the comb’s repetition rate. This makes the frequency axis directly linked to the spectral grid given by the comb and ensures transfer of the comb’s frequency accuracy to the spectral measurement. There is still, however, an arbitrary linear phase in the spectral domain, which is related to the path length difference between the calibration path and the sample path, which was not compensated for. If we try to avoid using such a phase-correction algorithm, we are then forced by the LO’s short coherence time to perform short FFTs to estimate the amplitude of each mode, and we will probably use a different frequency bin in each FFT to estimate the amplitude of the beat. It can be shown that this amounts to using phase tracking in piece-wise linear fashion. We thus argue that restricting the FFT length this way is an ad-hoc solution, and that continuous phase tracking solves the same problem in a much more elegant manner. 3. On-line Calibration
As a second contribution to the developing FCIS technique, we have implemented an on-line calibration technique in order to remove the impact of having a nonflat comb mode spectrum and electrical frequency response in the detection electronics (photodetector, any amplifiers, and the acquisition hardware). Indeed, the FCIS technique requires as much electrical bandwidth as the optical bandwidth over which the sample is to be probed, which is frequently over 1 GHz. At such high bandwidth, the shape of the electrical frequency response cannot be simply assumed to be flat if accurate measurements are to be produced. We have decided to use the concept presented in [16] and to measure at the same time the comb light that passed through the gas cell and the light that passed through a second, fiber only, calibration path. The calibration path was suitably delayed, so that the calibration and measurement pulses are offset by half a repetition period, in order to time-multiplex the calibration and measurement signal. The processing algorithm takes care of demultiplexing the measured trace into two traces by slicing in the time domain and computes the FCIS spectrum for both traces independently, after which the normalization can take place by simple spectral division of the two spectra. The advantage of this technique is that the calibration signal passes through the same electrical transfer function as the measurement signal. Note that this particular technique uses half the measurement trace for the calibration trace, and thus the maximum resolution of the measurement is halved. As long as the impulse response of the sample decays to zero in a time faster than half the repetition rate of the comb, no temporal aliasing will occur, and the measured spectrum will be unaffected by this process. In the spectral domain, this implies that the minimum width of the spectral line to be measured has to be twice as large as would be required to ensure proper sampling of the line shape. One could always use a second measurement
channel to record this calibration trace without suffering the factor of two reductions in resolution, but this brings the problem of mismatch between the two photodiodes and measurement channel frequency response. With the current implementation, both the calibration signal and the sample’s signal occur at the same exact frequency in the data-acquisition electronics and are acquired at close times (5 ns offset between the two), yielding good rejection of any common mode distortions of the amplitude and phase. 4. Experimental Results
The experimental setup is depicted in Fig. 1. The frequency comb used is a Menlo Systems C-Comb at 100 MHz repetition rate, with 20 mW of output power. The CW laser used is an Agilent 8164A tunable laser, with 5 mW output power, set at a fixed frequency next to the absorption line of interest, in our case the P(11) line at 194.748 THz [17]. The bandpass filter has 0.5 nm (60 GHz) of bandwidth and is centered on the CW laser’s frequency. The comb and LO light are combined in an optical hybrid coupler [11], although a regular fiber coupler could also have been used. Having both the in-phase (I) and in-quadrature (Q) outputs has the convenience of yielding the complete complex comb field and removing the ambiguity between the comb modes that are above and below the CW laser frequency. The signals are detected by two Bookham PT10G 10 GHz photodetectors and recorded on a WavePro 7Zi Lecroy oscilloscope with 3.5 GHz of bandwidth, 20 GS∕s sampling rate, and 256 × 106 samples of memory. The synchronization path ensures that the oscilloscope samples at an integer multiple of the repetition rate of the frequency comb, allowing simpler processing by assuming the oscilloscope’s clock is perfect. The sample consists of a Newport 2010WR acetylene wavelength reference cell. The cell length is 51 mm, and the pressure is 225 Torr, yielding mostly Doppler-broadened lines with 0.8 GHz linewidth. Recording the interference signal using a fast oscilloscope has two important characteristics: first, all the beating modes are acquired in parallel; second, it enables the measurement of the phase of every mode in addition to their amplitude. This second characteristic means that FCIS is part of the techniques that can measure the dispersion characteristic of a molecular transition rather than the absorption only [18]. Figure 2 shows that the phase correction completely cancels the phase and frequency noise on the beat note, leaving a transformlimited peak in the corrected signal spectrum; in this case improving the linewidth by a factor of 100, an improvement that is limited by our choice of resolution bandwidth for this spectrum. The important point, however, is that the choice of the measurement integration time is de-coupled from the laser’s coherence time. The task of estimating the peak’s amplitude is now much easier and less error-prone than doing so from the uncorrected spectrum. Note that 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
733
the peak is stronger than the maximum of the uncorrected peak because the phase correction has gathered all the signal energy in a single DFT bin, while leaving the noise level untouched, thus enhancing the SNR. Figure 3 shows the spectra obtained using the phase-corrected DFT. In (a), the raw power spectra are displayed for 11 consecutive measurements of 400 μs each, while (b) is the spectral phase of the signal. Each of these spectra represents 8 × 106 time domain points that were first time de-multiplexed into a measurement and a reference channel, phasecorrected using the procedure explained in Section 2, and then Fourier-transformed using a DFT and a Blackman window. The green trace corresponds to the signal that passed through the gas cell, while the blue trace is used for calibration. Normalization of the green trace with the blue trace removes most of the distortion that was imposed on the measurement by the electrical frequency response of the measurement channel, yielding traces (c) and (d). As mentioned previously, the phase response shows a linear phase between the calibration and signal spectra, which is due to the propagation delay difference between those two paths. More importantly, we can see that because of the high bandwidths involved, it is easy for the electrical frequency response of the measurement channel to impose severe distortions of the measured absorption profiles. Using detectors and oscilloscopes with higher bandwidth could potentially lower the amount of fixed distortion, but it is unreasonable to expect that the transfer function will be perfectly flat over all the measurement bandwidth as even small impedance mismatches can cause reflections and thus oscillation of the frequency response. Note that this error is usually negligible in a multiheterodyne measurement using two frequency combs as all the modes are all measured in a narrow electrical bandwidth of a couple MHz in the
Fig. 3. Beat spectra obtained using the phase-corrected DFT. Each subfigure contains 11 overlaid measurements of 400 μs each. (a) shows the power spectrum of the calibration path (blue) and the measurement path (green), while (b) shows the spectral phase of both signals. (c) and (d) are the result of normalizing the contents of (a) and (b), respectively. The frequency axis is given relative to the comb mode closest to the CW laser. 734
APPLIED OPTICS / Vol. 53, No. 4 / 1 February 2014
electrical measurement channel, as opposed to being spread out over GHz in an FCIS measurement. Following [19], the spectrum of acetylene at 225 Torr should be accurately modeled by a Voigt function, as this models both the Doppler and collisional broadening, which are the dominant physical mechanisms at play. Figure 4 shows the result of fitting a Voigt profile to the measured transmittance. The fit residuals are overlaid on both (a) and (b) in order to both show the scale of the errors relative to the measurement and the fine structure of the errors. The widths of the both Lorentzian and the Gaussian as well as the absorption depth and the constant baseline level are free parameters of the fit. The fine structure of the residuals reveals that systematic errors dominate this measurement, as the standard deviation of each spectral bin is approximately 10 times smaller than the residuals themselves. Note that this standard deviation is the sample standard deviation, estimated from our data set of 11 data points, and so this estimate is somewhat noisy, but the same conclusion can be inferred from the tight scatter of the measurements at each frequency versus the scatter of the measurement across the spectrum. We attribute those errors to temporal aliasing of the calibration and measurement traces, as the scale of these errors is too large to come from the sample’s spectrum. The fact that systematic errors dominate our measurements even for such short measurement times and careful calibration validates our hypothesis that the calibration process is critical for an FCIS system to yield high-accuracy spectra. An improved calibration process could potentially yield even higher quality data, although this is still to be demonstrated. Nevertheless, the signal-to-noise ratio of the obtained
Fig. 4. Measured transmittance profile and Voigt fit. In (a), the solid dots are the measured transmittance values of the 11 measurements; the red solid trace is the Voigt profile fit; and the x’s are the residual, offset for clarity. In (b), each point corresponds to the average residual of the 11 measurements, and the errors bars denote the four standard deviations range on each spectral bin. The frequency axis has an offset corresponding to the frequency of the comb mode closest to the CW laser, which was not measured in this work.
transmittance compares favorably with spectra obtained by previous multiheterodyne spectroscopy work [20], especially considering the short measurement time used here, although there are still worse systematic errors in the FCIS technique. Compared with previous work on FCIS [9,10], we can state with confidence that the random errors of our results are much lower, as their beat spectra show 13 dB of peak SNR, while we can see in Fig. 2 that our noise floor is 55 dB below our peak signal, a factor of 16,000 improvement. It is difficult to conclude on the amount of systematic errors present in the only other known work on FCIS, as the random noise is too high. Considering the fact that the authors do not mention any calibration of the frequency response, we can assume that such errors are likely present, but it is difficult to conclude on their level. 5. Conclusion
We have demonstrated two improvements that we believe to be critical for high-accuracy FCIS measurements. A phase-correction algorithm relaxes the constraints on the LO CW laser and enables unlimited coherent integration times, while a frequency-response normalization procedure flattens the response of the electronics. These two improvements were paired with parallel acquisition of all the beating modes using a high-speed oscilloscope to yield measurements with low random noise, and little distortion of the line shape. The full potential of the FCIS technique becomes available once we use a fully self-referenced comb with a 1f–2f setup. In this case, the comb is essentially perfect and probes the sample at accurate frequency points. The CW laser then simply down-mixes the electric field to measurable frequencies. The phase-noise cancellation technique that we presented allows using a simple CW laser with moderate noise specifications, rather than needing a state-of-the-art CW laser to match the self-referenced comb. The residual errors in this technique were shown to be systematic calibration issues, which highlights the importance of careful calibration of FCIS spectra. Thus the next step in the improvement of the technique would be to further improve the calibration step to reduce the systematic errors. The authors would like to thank the Natural Sciences and Engineering Council of Canada and the Fonds Québécois de la Nature et des Technologies for their financial support. References 1. J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172, 59–68 (1999).
2. S. A. Diddams, L. Hollberg, L. S. Ma, and L. Robertsson, “Femtosecond-laser-based optical clockwork with instability less than or equal to 6.3×10−16 in 1 s,” Opt. Lett. 27, 58–60 (2002). 3. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 13902 (2008). 4. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, and N. Picqu, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2009). 5. 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). 6. M. C. Stowe, M. J. Thorpe, A. Peer, J. Ye, J. E. Stalnaker, V. Gerginov, and S. A. Diddams, “Direct frequency comb spectroscopy,” Adv. At. Mol. Opt. Phys. 55, 1–60 (2008). 7. A. M. Zolot, F. R. Giorgetta, E. Baumann, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “Direct-comb molecular spectroscopy with accurate, resolved comb teeth over 43 THz,” Opt. Lett. 37, 638–640 (2012). 8. E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, and N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011). 9. K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and singlewavelength laser,” Appl. Phys. Lett. 101, 051105 (2012). 10. K. Urabe and O. Sakai, “Multiheterodyne interference spectroscopy using a probing optical frequency comb and a reference single-frequency laser,” Phys. Rev. A 88, 023856 (2013). 11. C. Dorrer, D. C. Kilper, H. R. Stuart, G. Raybon, and M. G. Raymer, “Linear optical sampling,” IEEE Photon. Technol. Lett. 15, 1746–1748 (2003). 12. P. A. Williams, T. Dennis, I. Coddington, W. C. Swann, and N. R. Newbury, “Vector signal characterization of high-speed optical components by use of linear optical sampling with milliradian resolution,” IEEE Photon. Technol. Lett. 20, 2007–2009 (2008). 13. J. D. Deschênes, P. Giaccarri, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express 18, 23358–23370 (2010). 14. P. Giaccari, J. D. Deschênes, P. Saucier, J. Genest, and P. Tremblay, “Active Fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method,” Opt. Express 16, 4347–4365 (2008). 15. J. McClellan, T. W. Parks, and L. Rabiner, “A computer program for designing optimum FIR linear phase digital filters,” IEEE Trans. Audio Electroacoust. 21, 506–526 (1973). 16. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34, 2153–2155 (2009). 17. A. Czajkowski, J. E. Bernard, A. A. Madej, and R. S. Windeler, “Absolute frequency measurement of acetylene transitions in the region of 1540 nm,” Appl. Phys. B, 79, 45–50 (2004). 18. M. Nikodem and G. Wysocki, “Molecular dispersion spectroscopy—new capabilities in laser chemical sensing,” Ann. NY Acad. Sci. 1260, 101–111 (2012). 19. W. C. Swann and S. L. Gilbert, “Pressure-induced shift and broadening of 1510–1540-nm acetylene wavelength calibration lines,” J. Opt. Soc. Am. B 17, 1263–1270 (2000). 20. N. R. Newbury, I. Coddington, and W. Swann, “Sensitivity of coherent dual-comb spectroscopy,” Opt. Express 18, 7929–7945 (2010).
1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
735
We demonstrate that using appropriate signal-processing techniques allows us to greatly improve the signal-to-noise ratio and accuracy of frequency comb interference spectroscopy measurements. We show that the phase noise from the continuous wave laser used as local oscillator is common to all beat notes and can be removed, enabling longer coherent integration time. An on-line calibration of the spectrum normalizes the frequency response of the electronics. The signal power-to-noise ratio of the spectra thus obtained is a factor of 16,000 (42 dB) higher than in previously demonstrated results, and the quality of the spectra is much higher. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (300.6530) Spectroscopy, ultrafast; (300.6310) Spectroscopy, heterodyne; (300.6390) Spectroscopy, molecular. http://dx.doi.org/10.1364/AO.53.000731
1. Introduction
As the technology of frequency combs matures, more ways of taking advantage of their high spectral coherence and frequency accuracy slowly keep emerging. While the first applications are obviously in frequency metrology [1] and optical clockworks [2], another flourishing field of applications is in spectroscopy [3–8]. A new spectroscopic technique called frequency comb interference spectroscopy (FCIS) was recently described in [9,10] and uses a frequency comb as a probing source for a single molecular transition. In this technique (Fig. 1), a small number of N comb modes is selected by a bandpass filter and sent through the sample to be measured where the molecular absorption (amplitude) and dispersion (phase) characteristics of the sample are imprinted onto the comb modes. A CW laser provides a single, strong local oscillator mode and is superimposed 1559-128X/14/040731-05$15.00/0 © 2014 Optical Society of America
with the light filtered by the sample and sent to a photodetector. This field only serves to downconvert all the comb modes into the radio-frequency (RF) domain to enable recording the sample information with conventional electronics. Since the frequency of each comb mode can be known accurately (for example, when one uses a self-referenced comb), this yields potential for high spectral accuracy, as the exact frequencies at which the sample was probed are related only to the comb and not to the CW laser. This technique sits somewhere between multiheterodyne spectroscopy techniques, which instead use many modes from a second comb as the local oscillators to provide the optical-to-RF mapping, and linear optical sampling techniques, where the comb is used as an LO for a modulated CW laser [11,12]. It represents a previously unexplored and interesting configuration with intriguing possibilities, providing a valuable trade-off for certain measurement scenarios, sacrificing bandwidth for accuracy over a single molecular transition. Indeed, the spectral point spacing is set by the repetition rate of the comb, while the spectral resolution is instead limited by the comb’s 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
731
Fig. 1. Experimental setup. Solid lines are optical fibers; split into the fibers are fiber couplers; and dashed lines are electrical coaxial cables. BPF, optical bandpass filter; DL, optical delay line; PC, polarization controller.
linewidth, which is usually orders of magnitude better than the mode spacing. The span of the resulting spectra is limited to the bandwidth of the photodetection electronics. Two potential application areas where FCIS seems particularly well suited are the measurement of fast dynamical processes, where the dynamic changes of a single molecular transition are of interest, or the accurate determination of absolute line centers and line shapes [7]. Indeed, the shortest acquisition time for a complete FCIS spectrum is equal to the comb repetition rate, where hundreds of MHz or a couple GHz are possible, although this potential remains to be demonstrated. In this paper, we report on two improvements that we believe to be critical to ensure the accuracy of the FCIS technique. The first and most important technique is detailed in Section 2, which is the tracking of the phase noise of the CW laser used as the LO. Our second contribution to the FCIS technique is described in Section 3, which is the use of an on-line spectrum calibration technique. Section 4 shows experimental results of spectroscopy of acetylene C2 H2 obtained using the presented techniques. 2. Phase Correction Algorithm
In a typical comb spectroscopy experiment, the maximum coherent integration time is limited by the mutual coherence time of the two lasers [13], in the case of an FCIS measurement being the CW laser and the comb. For maximum flexibility, an FCIS setup should use a tunable CW laser, which usually shows more noise than a fixed-frequency laser. As such, as was pointed out in [9], most of the errors in an FCIS setup will come from the LO’s frequency noise unless some form of mitigation is used. For this work, we have adapted the phase-correction technique from [13,14] to work with the FCIS setup. This allows us to completely decouple the experiment integration time and the LO’s coherence time and enables measurements of much higher accuracy. Implementing the phase-tracking algorithm in an FCIS experiment is much easier than in a multiheterodyne measurement, however, as no additional acquisition channel and associated optics are necessary, and all the information is already available from the recorded interference signal. Indeed, assuming that most frequency noise comes from the LO, every beating mode produces a signal of the form 732
APPLIED OPTICS / Vol. 53, No. 4 / 1 February 2014
Af k exp j2πf k t − f cw t − ϕt c:c. on the photodetector, Af being the complex amplitude of the beating mode and containing the sample’s complex transmittance, f k being the frequency of mode number k from the comb, f cw the frequency of the CW laser, and ϕt the phase noise (integrated frequency noise) from the CW laser, and c.c. indicating the complex conjugate of the previous term. One such beating mode can be seen on the blue trace in Fig. 2. The signal at the output of the photodetector is comprised of a linear sum of many such beating modes, k, but separated by the repetition rate in frequency (usually hundreds of MHz) such that we can easily isolate them one at a time using a software bandpass filter. There are also spectral lines at the repetition rate and all its harmonics, but these can be safely ignored as long as they are well separated from the comb-cw beating modes. Extracting a single mode “m” with a Hilbert bandpass filter [15] and normalizing by the amplitude, we can isolate a reference signal, exp j2π f m t − f cw t − ϕt. Multiplying the original signal by the complex conjugate of the reference signal yields modes of the form: A f k exp j2πf k − f m t, free of the influence of the frequency noise of the CW laser, yielding the green trace on Fig. 2. This works as long as a bandpass filter can unambiguously extract a single beat component, which means that the beat frequency drifts must be much smaller than the Nyquist frequency (half the repetition rate). All that is left is a complex exponential with a linear phase ramp carrying the desired information in its complex amplitude, which can be conveniently estimated using a simple discrete Fourier transform (DFT) at the desired frequencies. We call this phase extraction and multiplication procedure “phase correction” of the signal. This technique, coupled with the parallel acquisition of all the beating modes using a fast oscilloscope, enables measurements with low random noise at short acquisition times. Note that the time axis of the DFT is given by the oscilloscope’s timebase, which is itself synchronized
Fig. 2. Spectrum of one beating mode before (blue) and after (green) phase correction. Resolution bandwidth is 5 kHz. The noise floor is additive noise from the oscilloscope’s front-end. The linewidth of the beat signal is dominated by the CW laser’s frequency noise at around 500 kHz. After phase correction, all the beat signal’s energy becomes concentrated in a single transform-limited narrow peak.
to the comb’s repetition rate. This makes the frequency axis directly linked to the spectral grid given by the comb and ensures transfer of the comb’s frequency accuracy to the spectral measurement. There is still, however, an arbitrary linear phase in the spectral domain, which is related to the path length difference between the calibration path and the sample path, which was not compensated for. If we try to avoid using such a phase-correction algorithm, we are then forced by the LO’s short coherence time to perform short FFTs to estimate the amplitude of each mode, and we will probably use a different frequency bin in each FFT to estimate the amplitude of the beat. It can be shown that this amounts to using phase tracking in piece-wise linear fashion. We thus argue that restricting the FFT length this way is an ad-hoc solution, and that continuous phase tracking solves the same problem in a much more elegant manner. 3. On-line Calibration
As a second contribution to the developing FCIS technique, we have implemented an on-line calibration technique in order to remove the impact of having a nonflat comb mode spectrum and electrical frequency response in the detection electronics (photodetector, any amplifiers, and the acquisition hardware). Indeed, the FCIS technique requires as much electrical bandwidth as the optical bandwidth over which the sample is to be probed, which is frequently over 1 GHz. At such high bandwidth, the shape of the electrical frequency response cannot be simply assumed to be flat if accurate measurements are to be produced. We have decided to use the concept presented in [16] and to measure at the same time the comb light that passed through the gas cell and the light that passed through a second, fiber only, calibration path. The calibration path was suitably delayed, so that the calibration and measurement pulses are offset by half a repetition period, in order to time-multiplex the calibration and measurement signal. The processing algorithm takes care of demultiplexing the measured trace into two traces by slicing in the time domain and computes the FCIS spectrum for both traces independently, after which the normalization can take place by simple spectral division of the two spectra. The advantage of this technique is that the calibration signal passes through the same electrical transfer function as the measurement signal. Note that this particular technique uses half the measurement trace for the calibration trace, and thus the maximum resolution of the measurement is halved. As long as the impulse response of the sample decays to zero in a time faster than half the repetition rate of the comb, no temporal aliasing will occur, and the measured spectrum will be unaffected by this process. In the spectral domain, this implies that the minimum width of the spectral line to be measured has to be twice as large as would be required to ensure proper sampling of the line shape. One could always use a second measurement
channel to record this calibration trace without suffering the factor of two reductions in resolution, but this brings the problem of mismatch between the two photodiodes and measurement channel frequency response. With the current implementation, both the calibration signal and the sample’s signal occur at the same exact frequency in the data-acquisition electronics and are acquired at close times (5 ns offset between the two), yielding good rejection of any common mode distortions of the amplitude and phase. 4. Experimental Results
The experimental setup is depicted in Fig. 1. The frequency comb used is a Menlo Systems C-Comb at 100 MHz repetition rate, with 20 mW of output power. The CW laser used is an Agilent 8164A tunable laser, with 5 mW output power, set at a fixed frequency next to the absorption line of interest, in our case the P(11) line at 194.748 THz [17]. The bandpass filter has 0.5 nm (60 GHz) of bandwidth and is centered on the CW laser’s frequency. The comb and LO light are combined in an optical hybrid coupler [11], although a regular fiber coupler could also have been used. Having both the in-phase (I) and in-quadrature (Q) outputs has the convenience of yielding the complete complex comb field and removing the ambiguity between the comb modes that are above and below the CW laser frequency. The signals are detected by two Bookham PT10G 10 GHz photodetectors and recorded on a WavePro 7Zi Lecroy oscilloscope with 3.5 GHz of bandwidth, 20 GS∕s sampling rate, and 256 × 106 samples of memory. The synchronization path ensures that the oscilloscope samples at an integer multiple of the repetition rate of the frequency comb, allowing simpler processing by assuming the oscilloscope’s clock is perfect. The sample consists of a Newport 2010WR acetylene wavelength reference cell. The cell length is 51 mm, and the pressure is 225 Torr, yielding mostly Doppler-broadened lines with 0.8 GHz linewidth. Recording the interference signal using a fast oscilloscope has two important characteristics: first, all the beating modes are acquired in parallel; second, it enables the measurement of the phase of every mode in addition to their amplitude. This second characteristic means that FCIS is part of the techniques that can measure the dispersion characteristic of a molecular transition rather than the absorption only [18]. Figure 2 shows that the phase correction completely cancels the phase and frequency noise on the beat note, leaving a transformlimited peak in the corrected signal spectrum; in this case improving the linewidth by a factor of 100, an improvement that is limited by our choice of resolution bandwidth for this spectrum. The important point, however, is that the choice of the measurement integration time is de-coupled from the laser’s coherence time. The task of estimating the peak’s amplitude is now much easier and less error-prone than doing so from the uncorrected spectrum. Note that 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
733
the peak is stronger than the maximum of the uncorrected peak because the phase correction has gathered all the signal energy in a single DFT bin, while leaving the noise level untouched, thus enhancing the SNR. Figure 3 shows the spectra obtained using the phase-corrected DFT. In (a), the raw power spectra are displayed for 11 consecutive measurements of 400 μs each, while (b) is the spectral phase of the signal. Each of these spectra represents 8 × 106 time domain points that were first time de-multiplexed into a measurement and a reference channel, phasecorrected using the procedure explained in Section 2, and then Fourier-transformed using a DFT and a Blackman window. The green trace corresponds to the signal that passed through the gas cell, while the blue trace is used for calibration. Normalization of the green trace with the blue trace removes most of the distortion that was imposed on the measurement by the electrical frequency response of the measurement channel, yielding traces (c) and (d). As mentioned previously, the phase response shows a linear phase between the calibration and signal spectra, which is due to the propagation delay difference between those two paths. More importantly, we can see that because of the high bandwidths involved, it is easy for the electrical frequency response of the measurement channel to impose severe distortions of the measured absorption profiles. Using detectors and oscilloscopes with higher bandwidth could potentially lower the amount of fixed distortion, but it is unreasonable to expect that the transfer function will be perfectly flat over all the measurement bandwidth as even small impedance mismatches can cause reflections and thus oscillation of the frequency response. Note that this error is usually negligible in a multiheterodyne measurement using two frequency combs as all the modes are all measured in a narrow electrical bandwidth of a couple MHz in the
Fig. 3. Beat spectra obtained using the phase-corrected DFT. Each subfigure contains 11 overlaid measurements of 400 μs each. (a) shows the power spectrum of the calibration path (blue) and the measurement path (green), while (b) shows the spectral phase of both signals. (c) and (d) are the result of normalizing the contents of (a) and (b), respectively. The frequency axis is given relative to the comb mode closest to the CW laser. 734
APPLIED OPTICS / Vol. 53, No. 4 / 1 February 2014
electrical measurement channel, as opposed to being spread out over GHz in an FCIS measurement. Following [19], the spectrum of acetylene at 225 Torr should be accurately modeled by a Voigt function, as this models both the Doppler and collisional broadening, which are the dominant physical mechanisms at play. Figure 4 shows the result of fitting a Voigt profile to the measured transmittance. The fit residuals are overlaid on both (a) and (b) in order to both show the scale of the errors relative to the measurement and the fine structure of the errors. The widths of the both Lorentzian and the Gaussian as well as the absorption depth and the constant baseline level are free parameters of the fit. The fine structure of the residuals reveals that systematic errors dominate this measurement, as the standard deviation of each spectral bin is approximately 10 times smaller than the residuals themselves. Note that this standard deviation is the sample standard deviation, estimated from our data set of 11 data points, and so this estimate is somewhat noisy, but the same conclusion can be inferred from the tight scatter of the measurements at each frequency versus the scatter of the measurement across the spectrum. We attribute those errors to temporal aliasing of the calibration and measurement traces, as the scale of these errors is too large to come from the sample’s spectrum. The fact that systematic errors dominate our measurements even for such short measurement times and careful calibration validates our hypothesis that the calibration process is critical for an FCIS system to yield high-accuracy spectra. An improved calibration process could potentially yield even higher quality data, although this is still to be demonstrated. Nevertheless, the signal-to-noise ratio of the obtained
Fig. 4. Measured transmittance profile and Voigt fit. In (a), the solid dots are the measured transmittance values of the 11 measurements; the red solid trace is the Voigt profile fit; and the x’s are the residual, offset for clarity. In (b), each point corresponds to the average residual of the 11 measurements, and the errors bars denote the four standard deviations range on each spectral bin. The frequency axis has an offset corresponding to the frequency of the comb mode closest to the CW laser, which was not measured in this work.
transmittance compares favorably with spectra obtained by previous multiheterodyne spectroscopy work [20], especially considering the short measurement time used here, although there are still worse systematic errors in the FCIS technique. Compared with previous work on FCIS [9,10], we can state with confidence that the random errors of our results are much lower, as their beat spectra show 13 dB of peak SNR, while we can see in Fig. 2 that our noise floor is 55 dB below our peak signal, a factor of 16,000 improvement. It is difficult to conclude on the amount of systematic errors present in the only other known work on FCIS, as the random noise is too high. Considering the fact that the authors do not mention any calibration of the frequency response, we can assume that such errors are likely present, but it is difficult to conclude on their level. 5. Conclusion
We have demonstrated two improvements that we believe to be critical for high-accuracy FCIS measurements. A phase-correction algorithm relaxes the constraints on the LO CW laser and enables unlimited coherent integration times, while a frequency-response normalization procedure flattens the response of the electronics. These two improvements were paired with parallel acquisition of all the beating modes using a high-speed oscilloscope to yield measurements with low random noise, and little distortion of the line shape. The full potential of the FCIS technique becomes available once we use a fully self-referenced comb with a 1f–2f setup. In this case, the comb is essentially perfect and probes the sample at accurate frequency points. The CW laser then simply down-mixes the electric field to measurable frequencies. The phase-noise cancellation technique that we presented allows using a simple CW laser with moderate noise specifications, rather than needing a state-of-the-art CW laser to match the self-referenced comb. The residual errors in this technique were shown to be systematic calibration issues, which highlights the importance of careful calibration of FCIS spectra. Thus the next step in the improvement of the technique would be to further improve the calibration step to reduce the systematic errors. The authors would like to thank the Natural Sciences and Engineering Council of Canada and the Fonds Québécois de la Nature et des Technologies for their financial support. References 1. J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. 172, 59–68 (1999).
2. S. A. Diddams, L. Hollberg, L. S. Ma, and L. Robertsson, “Femtosecond-laser-based optical clockwork with instability less than or equal to 6.3×10−16 in 1 s,” Opt. Lett. 27, 58–60 (2002). 3. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 13902 (2008). 4. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, and N. Picqu, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2009). 5. 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). 6. M. C. Stowe, M. J. Thorpe, A. Peer, J. Ye, J. E. Stalnaker, V. Gerginov, and S. A. Diddams, “Direct frequency comb spectroscopy,” Adv. At. Mol. Opt. Phys. 55, 1–60 (2008). 7. A. M. Zolot, F. R. Giorgetta, E. Baumann, J. W. Nicholson, W. C. Swann, I. Coddington, and N. R. Newbury, “Direct-comb molecular spectroscopy with accurate, resolved comb teeth over 43 THz,” Opt. Lett. 37, 638–640 (2012). 8. E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, and N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011). 9. K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and singlewavelength laser,” Appl. Phys. Lett. 101, 051105 (2012). 10. K. Urabe and O. Sakai, “Multiheterodyne interference spectroscopy using a probing optical frequency comb and a reference single-frequency laser,” Phys. Rev. A 88, 023856 (2013). 11. C. Dorrer, D. C. Kilper, H. R. Stuart, G. Raybon, and M. G. Raymer, “Linear optical sampling,” IEEE Photon. Technol. Lett. 15, 1746–1748 (2003). 12. P. A. Williams, T. Dennis, I. Coddington, W. C. Swann, and N. R. Newbury, “Vector signal characterization of high-speed optical components by use of linear optical sampling with milliradian resolution,” IEEE Photon. Technol. Lett. 20, 2007–2009 (2008). 13. J. D. Deschênes, P. Giaccarri, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express 18, 23358–23370 (2010). 14. P. Giaccari, J. D. Deschênes, P. Saucier, J. Genest, and P. Tremblay, “Active Fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method,” Opt. Express 16, 4347–4365 (2008). 15. J. McClellan, T. W. Parks, and L. Rabiner, “A computer program for designing optimum FIR linear phase digital filters,” IEEE Trans. Audio Electroacoust. 21, 506–526 (1973). 16. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. 34, 2153–2155 (2009). 17. A. Czajkowski, J. E. Bernard, A. A. Madej, and R. S. Windeler, “Absolute frequency measurement of acetylene transitions in the region of 1540 nm,” Appl. Phys. B, 79, 45–50 (2004). 18. M. Nikodem and G. Wysocki, “Molecular dispersion spectroscopy—new capabilities in laser chemical sensing,” Ann. NY Acad. Sci. 1260, 101–111 (2012). 19. W. C. Swann and S. L. Gilbert, “Pressure-induced shift and broadening of 1510–1540-nm acetylene wavelength calibration lines,” J. Opt. Soc. Am. B 17, 1263–1270 (2000). 20. N. R. Newbury, I. Coddington, and W. Swann, “Sensitivity of coherent dual-comb spectroscopy,” Opt. Express 18, 7929–7945 (2010).
1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
735
