## Abstract

Frequency fluctuations of an optical frequency standard at 1.39 µm have been measured by means of a highly-sensitive optical frequency discriminator based on the fringe-side transmission of a high finesse optical resonator. Built on a Zerodur spacer, the optical resonator exhibits a finesse of 5500 and a cavity-mode width of about 120 kHz. The optical frequency standard consists of an extended-cavity diode laser that is tightly stabilized against the center of a sub-Doppler H_{2}^{18}O line, this latter being detected by means of noise-immune cavity-enhanced optical heterodyne molecular spectroscopy. The emission linewidth has been carefully determined from the frequency-noise power spectral density by using a rather simple approximation, known as β-line approach, as well as the exact method based on the autocorrelation function of the laser light field. It turns out that the linewidth of the optical frequency standard amounts to about 7 kHz (full width at half maximum) for an observation time of 1 ms. Compared to the free-running laser, the measured width corresponds to a line narrowing by a factor of ~220.

© 2015 Optical Society of America

## 1. Introduction

Noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) is the most sensitive laser-based absorption technique, consisting in the unique combination of frequency modulation (FM) spectroscopy with cavity enhancement of absorption path-length and optical power [1, 2]. It is capable of reaching a detection limit as small as 10^{−14} cm^{−1}, in the detection of ultra-narrow saturated absorption lines in coincidence with weak molecular transitions [3]. Very recently, it has been exploited for the realization of an optical frequency standard at 1.39 µm using an extended-cavity diode laser, frequency stabilized against the center of a sub-Doppler H_{2}^{18}O line [4]. In NICE-OHMS, the phase of the laser light shining the optical cavity is modulated at a frequency that exactly matches the cavity free spectral range (FSR) splitting frequency, so that, when the carrier frequency is locked to a given cavity mode of the n^{th} order, the sidebands are resonant with the modes of order n-1 and n + 1. NICE-OHMS is particularly advantageous for the aims of phase- and frequency-noise reduction of semiconductor diode lasers, since the detection phase can be adjusted so that a dispersion signal can be observed, likely to be employed as error signal. Moreover, if the degree of saturation induced by the two sidebands is much smaller than the one induced by the carrier, the sub-Doppler feature at the line center frequency can be observed only in the dispersion regime of operation [5, 6]. Consequently, an incorrect setting of the phase would affect only the size of the sub-Doppler signal, while the shape would maintain the symmetry that is characteristic of the dispersion profile. This property is very useful for the absolute frequency stabilization of a laser source, since it reduces the risk of locking out of the line center.

The optical frequency standard at 1.39 µm will act as reference laser in a dual-laser absorption spectrometer expressly designed for the spectroscopic determination of the Boltzmann constant, with the ambitious goal of approaching the accuracy of one part over 10^{6} [7]. In any experiment of Doppler-broadening thermometry, one of the most important component of the uncertainty budget is the emission width of the probe laser [8]. This latter, in a scheme based upon the use of a pair of phase-locked lasers, like the one that is being presently developed, will be mostly determined by the width of the reference laser.

This paper deals with the complete characterization of the spectral purity of the optical frequency standard at 1.39 µm, by using an optical frequency discriminator based on the fringe-side transmission of a high finesse optical resonator. A similar characterization is particularly meaningful in our case, since NICE-OHMS has been employed for the first time for the absolute frequency stabilization of an ECDL. In fact, the only existing works dealing with the frequency-standard applications of NICE-OHMS involved solid-state lasers [1,9]. In particular, the frequency-noise power spectral density of the stabilized laser is measured and the emission width is retrieved by using two methods: the so-called β-line approach [10, 11], as well as the general method based upon the autocorrelation function of the laser light field [12].

## 2. Experimental apparatus

The schematics of our setup is reported in Fig. 1. The optical frequency standard based on NICE-OHMS at 1.39 µm has been already described elsewhere [4]. Very briefly, the emission frequency of an extended cavity diode laser (ECDL) is actively stabilized against the center of the sub-Doppler NICE-OHMS signal, in coincidence with the 4_{4,0}→4_{4,1} transition of the H_{2}^{18}O ν_{1} + ν_{3} band at transition at 1389.0840 nm, by using a three-fold locking system. We note for the reader a typing error in [4], in which the rotational assignment of the lower and upper energy levels was inverted. The ECDL (Sacher Lynx TEC120) has a free-running emission frequency (full width at half maximum, FWHM) of about 2 MHz at 1 ms. At the first stage, the ECDL is locked to a resonant mode of a high finesse optical cavity by means of the Pound-Drever-Hall technique [4].

To this purpose, a first pair of sidebands (at a frequency, f_{PDH}, of 30 MHz) is added to the laser beam by means of an electro-optic modulator (EOM1) and heterodyne detection of the cavity reflected beam is performed at the frequency f_{PDH}. This produces an error signal that is employed to actively control the diode laser injection current and the extended-cavity length. Similarly, phase modulation at a frequency f_{PM} close to the cavity FSR splitting frequency (f_{FSR} ≈741 MHz) is performed through the EOM2 device. A second locking loop makes it possible to exactly match f_{PM} with f_{FSR}; this is done by implementing the DeVoe–Brewer approach [13]. In this latter case, the error signal is produced from heterodyne detection of the cavity reflected beam at a frequency given by f_{PM}- f_{PDH}. The bandwidth of the PDH loop acting on the laser current was about 1 MHz, while the bandwidth of the control loop of the sidebands amounted to ~10 kHz. Once the components of the FM-triplet (namely, the carrier plus the two sidebands at f_{PM}) are resonant with the cavity, heterodyne detection is performed on the cavity transmitted beam, while tuning the laser frequency around the center of the absorption line. The center sub-Doppler signal is observed only in the dispersion regime of operation, while it is missing in the absorption regime, because of the fact that NICE-OHMS is not sensitive to the absorption of the carrier. The sub-Doppler signal, resulting from the interaction with those molecules travelling in a plane perpendicular to the cavity axis, acts as an error signal that is used to actively control the length of the high-finesse cavity, so that the selected cavity mode and, consequently, the ECDL frequency is tightly locked to the center of the sub-Doppler line, with a loop bandwidth of ≈350 Hz. The optical cavity has a finesse of about 8900 and it is filled with a 97.7% ^{18}O-enriched water sample, at a pressure of ~3 Pa. An Allan deviation analysis of the residual frequency fluctuations of the stabilized ECDL, as performed on the NICE-OHMS error signal, demonstrated a relative frequency stability of ~5 × 10^{−13} for an integration time of 1 s [4]. This in-loop analysis was not sufficient to quantify the spectral purity of the optical frequency standard.

In this paper, the characterization of the frequency noise of the stabilized laser has been performed by using another optical cavity as frequency discriminator. This latter will be referred to as external resonator, throughout the article. It is made of a Zerodur parallelepiped with a pair of high-quality mirrors (with a reflectivity of 99.95%, built on a UV-grade fused silica substrate) at the two edges, namely, an input flat mirror and a rear concave mirror with a radius of curvature of 1 m. We remind that Zerodur is a homogenous glass-ceramic material characterized by an ultra-low coefficient of thermal expansion (much smaller than 10^{−6} °C^{−1}, in the temperature range from 0 to 50 °C). The cavity length can be controlled through a low-voltage ring piezo-electric transducer (PZT). The PZT and the two mirrors have been carefully glued by using a high-vacuum epoxy so that high vacuum conditions could be established inside the external resonator. The overall resonator length was measured to be 22.30 (1) cm, including the Zerodur block, the PZT length and thickness of the MACOR-holders placed between the input mirror, the PZT and the spacer. The nominal finesse is expected to be larger than 5000, while the cavity FSR is found to be 672.2 (3) MHz. The cavity transmission is monitored by a preamplified InGaAs detector with a bandwidth of 125 MHz. Despite the stability ensured by its spacer, the external resonator is placed in a thermal/acoustic insulation box, to avoid possible perturbations due to the environmental noise. The spacer is placed on a pair of aluminum posts, with vibration isolation pads between the spacer and the post.

A portion of the stabilized ECDL laser beam passes through an acousto-optic modulator (AOM) driven by a radio-frequency (RF) signal at 80 MHz. The 1^{st} diffracted order is injected into the external resonator. By using a threshold detector, in conjunction to a mixer, it is possible to switch off the 1^{st}-order beam with a response time of ~140 ns. Particularly, a periodic scan of the cavity mode frequency around the laser frequency is performed by means of a linear voltage ramp; each time the laser power builds up inside the optical resonator, so that the cavity transmitted signal reaches a given threshold level, the output of the monostable multivibrator commutes from high to low voltage level, thus switching off the AOM.

This makes it possible to easily observe repeated cavity ring-down events, thus leading to an accurate determination of the empty-cavity finesse. Figure 2 shows an example of ring-down event, following the excitation of a TEM_{00} cavity mode. An exponentially decaying function perfectly fits the recorded data, thus giving the value of the decay time (τ) among the output parameters. As a result of repeated measurements (shown in the upper plot of Fig. 2), the decay time can be determined with a statistical uncertainty of 1.5% (corresponding to one standard deviation), the mean value being 1.30 (2) μs. This leads to an empty-cavity finesse of 5500 (80), in good agreement with the expected one, and a cavity-mode width (FWHM) of 122 (2) kHz, this latter being calculated as the ratio between the free-spectral-range and the finesse. It should be noted that the statistical uncertainty is a factor of 5 larger than the internal error on τ resulting from the exponential fit.

Finally, the length of the external resonator can be actively controlled by sending a correction signal to the PZT, so that the reference laser stays at the shoulder of a cavity mode in a weak side-lock configuration. So doing, the external resonator can be employed as a high-sensitivity optical frequency discriminator [14]. In other words, residual frequency fluctuations are effectively converted into intensity noise on the cavity transmission, circumstance that gives precious information on the spectral purity of the stabilized ECDL. To this purpose, the cavity transmitted intensity is measured by means of a fast InGaAs detector, whose output signal is sent to a 1-GHz bandwidth digital oscilloscope, the noise spectra being acquired by using the FFT function of the digital oscilloscope.

## 3. Results and discussions

Figure 3 shows the transmitted signal as recorded at the output of the external resonator, while scanning the cavity mode around the frequency of the ECDL, the scan rate being equal to about 750 MHz/s. The upper graph was obtained with the ECDL operating in the free-running conditions, while the lower peak represents the cavity mode with the ECDL stabilized against the sub-Doppler NICE-OHMS signal. The comparison between the two plots clearly shows the benefits of the frequency stabilization loops. The upper plot is typical of a laser whose frequency fluctuations are much larger than the cavity-mode width and allows one to estimate the emission width of the free-running laser, amounting to about 1.5 MHz for an observation time of 1 ms. Such an estimate results from a Lorentzian fit of the data in the upper graph of Fig. 3.

A Lorentzian profile is also used to fit the spectrum in the lower part of Fig. 3, giving a mode width of 122 (1) kHz, in perfect agreement with the previously determined value. This is a first experimental evidence of the fact that the emission width of the stabilized ECDL laser is much smaller than the full-width at half-maximum of the cavity mode. The two small peaks observed on the sides of the cavity mode give information on the bandwidth of the Pound-Drever-Hall locking loop (~900 kHz). In fact, they are due to the oscillation of the feed-back loop that starts to be observed while increasing the gain. The complete picture of the laser frequency noise is given by the frequency-noise power spectral density, *S _{δν}*, that is directly measured by using the external resonator as a frequency discriminator. To this purpose, the cavity is weakly side-locked to the stabilized ECDL laser by means of a proportional-integrator (PI) servo amplifier (New Focus, model LB1005), the loop bandwidth being about 700 Hz. Figure 4 reports the frequency-noise power spectral density of the cavity transmitted signal, acquired with a resolution bandwidth of 20 Hz. Due to the photons average lifetime in the cavity, the latter acts as a second order low-pass filter with a 122 kHz cutoff frequency. The spectrum given by the blue trace of Fig. 4 is compensated for this cutoff.

The β-separation line is also drawn, defined as ${S}_{\delta \nu}(f)=8f\mathrm{ln}2/{\pi}^{2}$, $f$ being the Fourier frequency. This line geometrically separates the frequency-noise power spectral density in two types of regions: those for which ${S}_{\delta \nu}(f)>8f\mathrm{ln}2/{\pi}^{2}$ significantly contribute to the laser linewidth; differently, in the frequency intervals where ${S}_{\delta \nu}(f)<8f\mathrm{ln}2/{\pi}^{2}$ the frequency fluctuations are too fast to influence the linewidth and only contribute to the wings of the laser emission profile [10]. Therefore, the laser linewidth (FWHM), *Δν*, can be calculated by using this simple formula:

*A*is the area under the frequency-noise curve, calculated in those intervals where ${S}_{\delta \nu}(f)>8f\mathrm{ln}2/{\pi}^{2}$, with a cut-off frequency (namely, the lower limit of the integral) given by 1/T, T being the observation time. As a result, we found

*Δν*= 9.0 kHz for an observation time of 1 ms. Moreover, the width increases up to 21.2 kHz for T = 10 ms. It should be noted that, at low frequencies (smaller than ~700 Hz), the measured frequency noise can be reduced by increasing the low-frequency gain limit (LFGL) of the PI servo amplifier. As a consequence, the laser width retrieved at T = 10 ms cannot be reliable, since it results to be dependent on the LFGL-value. More precisely, an increase of 20% in the gain leads to a reduction of the measured width from 21.2 kHz to 17.2 kHz, for an observation time of 10 ms; instead, the width is unaltered when T is set to 1 ms.

The intensity noise (IN) of the stabilized ECDL was also measured by recording the noise spectrum of the signal produced by the fast detector (PD3 of Fig. 1), when it was illuminated by a portion of the laser beam with a power at the same level of that measured on the cavity transmission. This noise spectrum, together with the photodetector noise floor, is shown in Fig. 5. Here, for a useful comparison, we have also added the raw data of the intensity noise on the cavity transmitted beam, leading to those of Fig. 4. The conversion from intensity noise (measured in dBm) to frequency-noise power spectral density (Hz^{2}/Hz) is performed taking into account the slope of the shoulder of the selected cavity mode (4.06 x 10^{−6} V/Hz) as well as the resolution bandwidth (20 Hz). For instance, −80 dBm corresponds to ~1.52 Hz^{2}/Hz, while −40 dBm translates into a frequency-noise of 1.52 x 10^{4} Hz^{2}/Hz. Therefore, we can exclude possible contributions to the noise spectrum of Fig. 4 resulting from either the intensity noise or the detector noise, as clearly shown in Fig. 5.

As it is well known, the β-line approach is a quite simple approximation allowing one to rapidly retrieve the laser width [10, 11]. Its accuracy depends on the product between the observation time and the frequency ${f}_{m}$, at which ${S}_{\delta \nu}(f)$ intersects the β-separation line. On the other hand, the exact calculation of the laser line shape and, thus, the laser emission width from the frequency-noise power spectral density, over any desired timescale, requires a two-step numerical integration, following the approach proposed by Elliott et al. [12]. In particular, if ${S}_{\delta \nu}(f)$ is the frequency-noise power spectral density associated to a laser light field given by $E(t)={E}_{0}\mathrm{exp}\left[i\left(2\pi {\nu}_{0}t+\varphi (t)\right)\right]$, oscillating at a frequency ν_{0} and characterized by a phase noise *ϕ(t)*, according to the Wiener–Khintchine theorem [15], the laser line shape is given by the Fourier transform of the autocorrelation function ${\Gamma}_{E}(\tau )$, as it follows:

^{2}/Hz), below a cutoff frequency ${f}_{c}$, and then drops to zero above this threshold. If ${f}_{c}\to \infty $, the integration of Eq. (2) gives a laser line shape that is a Lorentzian function with an full-width at half-maximum (FWHM) of $\pi {H}_{0}$. When ${f}_{c}\to 0$ the laser line shape is Gaussian and the FWHM depends on ${f}_{c}$ and becomes equal to $\sqrt{8\mathrm{ln}2{H}_{0}{\text{f}}_{\text{c}}}$ [16].

In our case, since the observation time of interest is 1 ms, the lower limit in the integral of Eq. (3) has to be set at 1 kHz. The resulting line shape is shown in Fig. 6, along with the absolute residuals from a Voigt fit (bottom graph of Fig. 6). It turns out that the laser profile exhibits a FWHM of 6.89 kHz, with a dominant Gaussian component of 6.45 kHz, the Lorentzian contribution being as small as 0.81 kHz. The width is significantly smaller than the one calculated from the β-separation line. This is not surprising, since the quantity $T\times {f}_{m}$ results to be ~2. Previous studies have shown that the relative deviation between the results of the two methods is smaller than 10% as long as $T\times {f}_{m}$ is larger than 5 [10]. In fact, when moving to an observation time of 10 ms so as the product $T\times {f}_{m}$ amounts to ~20, numerical integration yields a laser width of 20.62 kHz, in good agreement with the β-line approach (21.2 kHz).

## 4. Conclusions

In conclusion, we have reported on the spectral purity of an optical frequency standard at 1.39 μm, based on the active frequency stabilization of an ECDL against the center of a sub-Doppler H_{2}^{18}O line, as detected by means of NICE-OHMS under the dispersion regime of operation. A high finesse optical resonator has been employed as a highly-efficient frequency discriminator, so as to measure the frequency-noise power spectral density of the stabilized laser. An emission width of 6.89 kHz (FWHM) has been determined, for an observation time of 1 ms. This corresponds to a line narrowing by a factor of ~220, as compared to the free-running operation condition of the ECDL. To our knowledge, this is the best result obtained so far, for an ECDL when stabilized against the center of a molecular spectral line. In particular, our stabilized ECDL is narrower than the Doppler-free acetylene-stabilized diode laser standards developed at 1.5 μm [17]. Our study confirms that the approach based on the β-separation line overestimates the laser width for small values of the product between the observation time and the Fourier frequency at which the frequency-noise power spectral density intersects the β-separation line.

The stabilized laser will be used in a third-generation experiment to accurately measure the Boltzmann constant (*k*_{B}) by means of Doppler-broadening thermometry, replacing the one that was previously employed, whose emission linewidth was a factor of 4.3 larger [18]. This improvement, in conjunction to the fact that no dithering is added to the stabilized laser, differently from that of [18], will allow us to remove one of the main components of the uncertainty budget related to the spectroscopic determination of *k*_{B} [8].

## Acknowledgments

This research was carried out in the framework of the EMRP Project No. SIB01-REG3 within the InK project (Implementing the new Kelvin, coordinated by Graham Machin, NPL, UK). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The authors are grateful to Gianluca Galzerano and Maurizio De Rosa for stimulating discussions.

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