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Two frequency-doubled combs are generated by nonlinear frequency conversion to realize spectroscopic measurements around 775 nm. Frequency-doubled interferograms are corrected in real-time by monitoring the relative instabilities between the combs at their fundamental frequency. Rubidium absorption lines are used to demonstrate the technique’s accuracy and serve as absolute references to calibrate the frequency grid of computed spectra. The method allows frequency-doubled interferograms to be averaged without distortion during long periods of time. The calibrated frequency grid is validated by the measurement of the oxygen A-band. Moreover, the measurement analysis of the acetylene ν1 + 3ν3 overtone band has revealed some discrepancies with previous publications.
© 2013 Optical Society of America
1. Introduction
Dual-comb spectrometers combine advantages of Fourier transform and laser spectroscopy; they perform fast, sensitive and high-resolution spectroscopic measurements over bandwidths of several tens of nanometers [1–6]. Despite this, the use of dual-comb spectrometers for real-world spectroscopic applications is limited by the unavailability of combs in some optical regions [7]. This problem can be circumvented by shifting combs from their fundamental optical band using nonlinear processes such as second harmonic generation (SHG), difference frequency generation and sum frequency generation [7]. No matter the optical band in which combs are used, the same fundamental problem arises; variations of the repetition rate and the carrier-to-envelope offset difference between the two combs distort the measured spectrum. To prevent this undesirable effect, the variations must either be stabilized [4], phased corrected and sampled without distortion [6], or monitored and corrected in post-processing [3,5]. Successfully correcting the combs generated by nonlinear processes enables extending the use of dual-comb spectrometers to other spectral regions. Having the references at the fundamental frequency of the combs may be useful in terms of system complexity since the same referencing signals can then be used to correct several non-linearly generated combs.
Recently, frequency-doubled combs were generated in the green region using SHG to probe an iodine cell [8]. A 12 ms interferogram (IGM) was successfully sampled without distortion, but phase fluctuations were not corrected prior to the sampling since only one reference signal at the fundamental frequency of the comb was used. This limits the frequency range over which the correction is effective [9]. Sensing of oxygen in air in the 775 nm region using frequency-doubled combs was also reported [10]. This time, combs parameters were monitored and the interferogram corrected in post processing using reference signals at the fundamental frequency of the comb. To retrieve interesting measurements, IGMs were averaged over 9 hours because of the weakness of these oxygen lines combined with a short detection path. However, the referencing technique used in [10] only corrects relative fluctuations between both combs and it does not account for drifts in the absolute position of the teeth of each comb. This problem occurs when the parameters of the each comb (repetition rate and/or carrier-to-envelope offset) drift together in the same direction. Even if these drifts are generally very slow, the outcome is a spectrum frequency grid that varies from one measurement to another and averaging many IGMs under these conditions leads to a deterioration of the spectral resolution. Fortunately, in the case presented in [10], the resulting instrument line shape was much smaller than the 3 GHz line width of the oxygen lines. The problem linked to the frequency grid variations when co-adding for long period of time has also never been addressed for IGMs at the fundamental frequency of the combs when using an a posteriori correction approach [3,5].
In this paper, we use frequency-doubled combs at 775 nm and a Rubidium (Rb) cell to retrieve spectra with a calibrated frequency grid. The narrowness and the position of Rb absorption lines make it the perfect candidate for the calibration task. First, performances of the real-time correction technique that uses references at the fundamental frequency of the combs to correct frequency-doubled IGMs a posteriori are evaluated by analyzing narrow and strong Rb absorption lines. Then, the frequency grid accuracy is validated by comparing oxygen (O2) measurements of the A-band to the HITRAN database. We conclude by the measurement of acetylene (C2H2) which has many overtone bands in the 775 nm region.
2. Accuracy of frequency-doubled measurements
The technique that uses two continuous wave (CW) lasers as intermediate oscillators to monitor the IGM instabilities and correct them in post-processing is known to retrieve accurate measurements for dual-comb setups [3]. This approach can be used to achieve the same result on frequency-doubled IGMs. In fact, in addition to using the same technique, it uses the same referencing signals. The references at the fundamental frequency of the combs are used to correct the frequency-doubled IGM. To better understand the relation between the referencing signals and the instabilities of the frequency-doubled IGM, let us start from the IGM expression of a spectroscopic dual-comb measurement. Indeed, as demonstrated in [3], the IGM Sm sampled once per incident optical pulse pair k is expressed as:
where fm is the optical carrier frequency, Δφ is the IGM phase variation, ΔT is the IGM optical path difference (OPD), which includes sampling variations with respect to a uniform grid and Am contains the sample response. More precisely, in a signal processing meaning, Am contains the autocorrelation of the sample’s impulse response convolved with the cross-correlation of the pulses of each combs. To correct the IGM described in Eq. (1), Δφ must be removed prior to resample the IGM on an equidistant OPD grid. Since the digitally extracted phases ϕ of both references are:subtracting ϕ1 from the IGM phase removes the Δφ dependence, leaving only a non-uniform OPD sampling grid. To complete the correction, the measured IGM sampling grid is found by subtracting ϕ1 from ϕ2. The measured IGM is then resampled on a uniform OPD grid. Here, a linear interpolation is sufficient as demonstrated in [11].Following the same formalism leading to Eq. (1), a frequency-double IGM SSH can be expressed as:
where ASH is the autocorrelation of the sample’s impulse response convolved with the cross-correlation of frequency-doubled pulses. In the doubling operation, the pulse rate is of course maintained, but the optical carrier frequency is doubled. This is correct as long as fm is not defined in terms of the modes of the original spectrum. Note that from a correction point of view, the only difference between Eq. (1) and Eq. (3) is that the argument of SSH is doubled. Therefore, the extracted reference phases of Eq. (2) need only to be multiplied by two numerically to successfully correct the frequency-doubled IGM and this can be easily applied to the correction algorithm previously presented in [3,5].After the IGM correction, the frequency grid can be computed using the absolute optical frequencies of the two CW lasers. In this case, the grid accuracy is directly linked to the knowledge of these frequencies and any frequency drift from one measurement to the other can be problematic when averaging many IGMs. This implies that to compute a highly accurate frequency grid the CW lasers must be locked to known high-precision references or measured with a high-resolution instrument. When possible, a simpler way to reconstruct the frequency grid consists of using the center of two narrow and strong absorption lines as two absolute frequencies. In this case, the accuracy depends on the absorption lines strengths and widths combined with the measurement SNR [12]. To minimize scaling errors, the ideal scenario would be to have at least one line at each extremity of the spectrum.
3. Experimental setup
To demonstrate the referencing technique for frequency-doubled IGMs, we use the setup presented in Fig. 1. The combs are Menlo Systems mode-locked erbium fiber lasers with a 100 MHz repetition rate and an optical bandwidth of approximately 100 nm around 1550 nm. The system’s theoretical spectral resolution is thus 100 MHz, which corresponds to an OPD of 10 ns. The SHG modules are also from Menlo Systems; each one consists of a temperature-controlled periodically poled lithium niobate (PPLN) crystal matched for 1550 nm sources. To maximize the generated frequency-doubled combs, we optimize the peak intensity of the pulses at the PPLN crystal by using the dispersion-compensated comb output of both Menlo lasers. We also use a polarization controller (PC) before each SHG module since the SHG is sensitive to polarization. Starting with two 15 mW combs, we managed to generate two frequency-doubled combs of approximately 500 μW. Both combs are combined in a 780 nm fiber coupler and one output of the coupler is sent through gas cells prior to detection.
Fig. 1 Experimental setup. Solid black lines represent optical fiber, dashed red lines represent electrical cable and dotted blue lines represent free-space optical path. WDM, wavelength division multiplexer; PC, polarization controller; D, detector; CW, continuous wave laser; SHG, second harmonic generation; FPC, fiber port collimator; FPGA, field-programmable gate array; Rb, rubidium.
To monitor the beating between the combs, we use two external cavity CW lasers from Redfern Integrated Optics (RIO) at 1549.351 nm and 1562.236 nm. Wavelength division multiplexers (WDM) filter the combs around the frequency of the RIO lasers to reduce the comb shot noise contribution and PCs are used to maximize the beating amplitude. Note that for the CW laser reference technique to be valid, the accumulated phase noise of the two CW lasers over one IGM must be small enough to be neglected, which has already been demonstrated with these lasers [3]. Also, as mentioned earlier any unaccounted deviation in the frequency of one CW laser from one IGM to another will change the frequency scale such that successive IGMs will need to be rescaled after the correction before being co-added without error.
To compute the optical frequency grid associated with the electrical spectrum measured and corrected by the setup presented in Fig. 1, we use a 75 mm long Thorlabs quartz reference cell filled with the natural isotope ratio of Rb, which is 72.17% 85Rb and 27.83% 87Rb [13,14]. The aim is to use the well-known Rb absorption lines around 780 nm and 795 nm as absolute frequency references. In addition of being strong, these lines have the potential to be very narrow. Indeed, the vapor pressure of Rb at room temperature is less than 10−6 torr [13,14]. In this case, the Doppler broadening limits the line widths and the predicted Gaussian profile has a full-width at half maximum (FWHM) of approximately 515 MHz.
The second gas cell in Fig. 1 is a 135 cm long homemade gas cell always used in combination with the Rb cell made to perform spectroscopic measurements of O2 and C2H2. The temperature of these gases will follow the room temperature which is set to 20.7°C.
The acquisition system consists of a 2U rack-mount PC with a ML605 Xilinx PCI development board and a Nutak MI250 ADC extension card. Real-time correction of IGMs is realized using the algorithms presented in [5] with the only difference being that the measured phases of the referencing signals are doubled before applying the correction to frequency-doubled IGMs. To maintain electrical beating signals away from DC and from the Nyquist frequency, we use feedback loops on one of the comb and the CW lasers. Note that these servo loops do not correct the IGM and are only used to fix operating conditions for the acquisition. In fact, they provide updates every few seconds and are necessary only when averaging IGMs for more than 30 minutes.
The fact that servo loops are required to maintain the beating signals in the right position implies that the combs’ repetition rates and carrier-to-envelope offsets, as well as the frequency of the CW lasers, naturally drift over time. If the frequencies of the CW lasers are used to compute the accurate frequency grid, their evolution over time must be monitored to be able to average many IGMs without distortion. However, since we use the position of the Rb absorption lines to compute the frequency grid, there is no need for extra precise instruments dedicated to monitoring purposes. Instead, the position of the lines can be followed by looking at the spectrum and the appropriate rescaling can be made. Depending on Rb lines SNR and the speed at which system parameters vary, it might be unnecessary to compensate the frequency grid variations at each measurement. Here, we have chosen to correct the frequency grid after each averaging of 50 000 IGMs which is equivalent to 7 minutes at a rate of 120 IGMs per second (these intermediate IGMs are called 50k-IGMs). Due to a low SNR around the 790 nm Rb lines, only an offset, determined by the position of the 780 nm Rb lines, is applied on the frequency grid after each 50k-IGM. The rescaling of the frequency grid is completed at the end of the complete measurement, when the SNR is better, using both sets of Rb lines. The next section will demonstrate that this approach allows accurate measurement of Rb vapor and other gases.
4. Results
4.1 Rubidium
To demonstrate the technique’s accuracy, 1 000 000 frequency-doubled IGMs using the Rb cell are averaged at a rate of 120 IGMs per second. A 10 ns OPD span is used to compute the high-resolution spectrum presented in Fig. 2(a). A polynomial fit of the spectrum is performed to find the baseline used to compute the transmittance. The result is shown in Fig. 2(b). The strongest 85Rb and 87Rb absorption lines are visible around 780.24 nm and come from the Rb D2 transition structure. A closer view is given in Fig. 2(c); each apparent absorption line results from the combination of three transitions of the Rb D2 transition hyperfine structure [13,14]. Weaker absorption lines are also present around 794.98 nm, a zoom on these lines is given at Fig. 2(e). The two strongest lines are from 85Rb atoms and consist of the combination of two transitions of the 85Rb D1 transition hyperfine structure while the four other smaller visible absorption lines come from the 87Rb D1 transition hyperfine structure [13,14].
Fig. 2 (a) The 1 000 000-IGM spectrum of the Rb measurement. (b) Transmittance. (c) and (e) The blue lines are zooms on the transmittance of the Rb D2 and D1 transitions respectively. The red lines are the computed transmittance with an offset to ease the comparison. (d) and (f) Residuals of Rb D2 and D1 transitions; subtraction of the measured transmittance with the calculated transmittance.
Figure 3 shows the grid offset correction applied during the averaging process to reduce the effect caused by the slow variations of setup conditions. In this measurement, the offset was gradually increasing by a few MHz per 50k-IGM to reach a total of 40 MHz after 2 hours. At the end of the measurement, the absolute optical frequency grid of the spectrum is computed using the positions of both sets of Rb lines.
Fig. 3 The frequency grid offset calculated for the 50k-IGMs. These offset values are use for a partial correction of the frequency grid during the averaging process.
Expected transmittances of Rb are plotted in red with an offset for comparison purposes in Fig. 2(c) and Fig. 2(e). The calculated Voigt profile for each transition of the Rb hyperfine structure is made of a Gaussian profile convolved with a Lorentzian profile. The Gaussian FWHM is determined by the Rb mass and the room temperature while for the Lorentzian FWHM we use the natural line width of Rb which is 6.07 MHz for D2 transitions and 5.75 MHz for D1 transitions [13,14]. Using only the natural line width is valid since the pressure broadening is expected to be close to zero for pressure values near 10−6 Torr [15]. The pressure shift is also neglected for the same reason [15]. The relative strengths of the hyperfine transitions of each isotope are taken from [16]. To finalize the computation of the transmittance, we use only the overall strength of the transition lines as a fitting parameter.
Figure 2(d) and 2(f) show differences between measured and calculated transmittances. The fact that the residuals are within than the baseline transmittance noise indicates that Rb absorption lines were measured accurately. From this measurement, it can be concluded that there is no significant instrument line shape that deteriorates the measurement of 515 MHz-wide spectral features when using two reference signals at the fundamental comb frequency to correct frequency-doubled IGMs. The next step will be to validate the frequency grid provided by the Rb lines with calibrated measurements of O2 and C2H2.
4.2 O2
To realize a calibrated measurement of O2, two spectra resulting of 1 000 000 frequency-doubled IGMs co-added at a rate of 120 IGMs per second are taken. The Rb cell is kept in the optical path to produce an accurate frequency grid. Since the lines are expected to have a width of 1.5 GHz, the measurement spectral resolution is reduced to 250 MHz after the frequency grid centering to increase the SNR. The first spectrum is realized with the 135 cm long gas cell filled with O2 at a pressure of 265 torr. Then, the cell is emptied in place and a second spectrum is measured. Even if the two spectra are measured immediately one after the other, we can see in Fig. 4(a) that baselines of both spectra are slightly different. This is not surprising given the long measurement time of more than 4 hours, especially when considering the fact that the SHG process is sensitive to experimental and environmental conditions. It will not cause any problem as long as variations of the spectrum shape are much slower than the features we want to measure. Here, the shape difference is evaluated using a polynomial fit and is compensated in the calculation of the transmittance (spectrum ratio). The result is a flat transmittance baseline and it is shown in Fig. 4(b). Note that the transmittance shows no Rb lines residuals. This indicates that this experimental setup provides repeatable and calibrated measurements.
Fig. 4 (a) Spectra of the O2 experiment. Blue line: gas cell filled with O2 at 265 torr. Red line: gas cell empty. (b) O2 calibrated transmittance. (c) The blue line is a zoom on the O2 A-band absorption lines. The red line is the computed transmittance plotted upside down. (d) Residuals of O2 A-band measurement; subtraction of the measured transmittance with the calculated transmittance.
Figure 4(c) provides a closer look at the O2 A-band absorption lines. The expected transmittance is calculated using measurement parameters and the HITRAN database [17]. Note that the calculated transmittance has not been fitted to our measurement; we only have used the known parameters of the experimental setup. For comparison purposes, the result is plotted upside down in Fig. 4(c) (red line) and Fig. 4(d) presents the difference between the measured and calculated transmittance. The resulting errors merge with the baseline transmittance noise.
4.3 C2H2
The third measurement to demonstrate the potential of this technique is of C2H2. As in the O2 experiment, two spectra of 1 000 000 frequency-doubled IGMs co-added at a rate of 120 IGMs per second are taken while the Rb cell is kept in the optical path to provide an accurate frequency grid. The first spectrum is realized with the 135 cm long gas cell filled with C2H2 at a pressure of 220 torr and the second one when the cell is empty. The measurement spectral resolution is fixed to 250 MHz. The two resulting spectra are plotted in Fig. 5(a). As presented in section 4.2, the baseline difference is compensated before computing the transmittance; the result is presented in Fig. 5(b). Here again, there are no Rb lines residuals, thus validating the frequency grid accuracy and demonstrating the system repeatability.
Fig. 5 (a) Spectra of the C2H2 experiment. Blue line: gas cell filled with C2H2 at 220 torr. Red line: gas cell empty. (b) C2H2 calibrated transmittance. (c) Transmittance of the C2H2 ν1 + 3ν3 band. (d) Transmittance of the C2H2 3ν1 + ν3 band. (e) Strong lines of the R branch of the ν1 + 3ν3 band. The measurement is plotted in blue. The red line shows the expected transmittance. (f) Residuals of the C2H2 ν1 + 3ν3 band measurement; subtraction of the measured transmittance with the calculated transmittance. Several weak absorption lines remaining belong to the P branch of the ν1 + 2ν2 + ν3 + 4ν4 band.
The strongest C2H2 absorption band in this spectral region, the ν1 + 3ν3 band, is shown in Fig. 5(c). The comparison to the literature is not as straightforward as it is with O2 since the C2H2 ν1 + 3ν3 band is not part of the HITRAN database. Instead, Voigt profiles are calculated using the line positions, the line shift coefficients, the self-broadening coefficients and line intensities reported in the works of Valipour et al. [18] and Herregotds et al. [19–21]. To our knowledge, these papers regroup the most exhaustive studies of the line specifications belonging to the C2H2 ν1 + 3ν3 band. However, some line parameters such as the line positions and the self-broadening coefficients differ significantly from one research group to the other, and the differences are listed in Valipour et al. To compute a transmittance that fits our measurement, we have used the line positions of Valipour et al which, compared to Herregotds et al., have a systematic deviation of approximately 750 MHz. For the self-broadening coefficients, even if values in Herregotds et al. are more consistent with our measurement compared to the coefficients of Valipour et al. (who are 20% larger on average), we have had to increase the Herregotds et al. coefficients by 5% to avoid residuals after comparison.
Figure 5(e) illustrates the resulting computed transmittance of the strongest lines of the R branch of the ν1 + 3ν3 band (R(6) to R(11)). The line intensities were only available in Herregotds et al. papers; we used them as relative intensities and fit the overall ν1 + 3ν3 bandstrength. Here again, the values do not fit completely with our measurement, we had to decrease the relative intensity of line R(7) by 2% to compute the appropriate transmittance. The differences between the computed and the measured transmittance are shown in Fig. 4(f). Most features that look like line residuals are in fact lines due to the weak ν1 + 2ν2 + ν3 + 4ν4 and ν2 + 3ν3 + 2ν4 bands [18]. The positions of some lines of the P branch belonging to the ν1 + 2ν2 + ν3 + 4ν4 band are identified on Fig. 5(f).
There is another C2H2 band that can be observed in our measurement, which is the 3ν1 + ν3 band around 768 nm [22]. This band is weak, and to be able to see it, we have to increase the spectral SNR by further reducing the measurement spectral resolution. More precisely, the IGM is apodized to keep a total OPD 15 times smaller than the original IGM. Even then, the lines are barely visible; see Fig. 5(d). To ease the identification, the line positions are plotted in red over the measured lines.
Although the aim of this paper is not to determine or validate C2H2 lines parameters, we believe that this instrument could be used in that manner. To increase the identification precision, it would require a better SNR. This could be possible by increasing the gas cell length (multipass gas cell) and by increasing the power of the frequency-doubled combs.
5. Conclusion
This paper presents advances in the technique of dual comb spectroscopy: this is the first demonstration of fully corrected dual-comb spectroscopy with frequency-doubled combs using reference signals at the combs fundamental frequency. Calibrated spectral measurements with free-running combs are reported using a rubidium reference cell. These advances in the instrumentation allow producing residual-less spectra of molecules in the doubled combs bandwidth. The quality of the results is such that improved spectroscopic information on some acetylene lines is hinted.
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