Electro-optic frequency combs generated via direct digital synthesis applied to sub-Doppler spectroscopy

Direct digital synthesis in concert with an electro-optic phase modulator was employed to generate optical frequency combs with tooth spacings as low as 100 Hz. These combs were utilized to probe electromagnetically induced transparency (EIT) and hyperfine pumping in potassium vapor cells. Long-term coherent averaging was demonstrated with performance similar to that achieved with a vastly more expensive arbitrary waveform generator. From the potassium EIT transition we were able to determine the ground state hyperfine splitting with a fit uncertainty of 80 Hz. Importantly, because of the mutual coherence between the control and probe beams, which originate from a single laser, features with linewidths several orders-of-magnitude narrower than the laser linewidth could be observed in a multiplexed fashion. This approach removes the need for slow scanning of a traditional cw laser or mode-locked-laser-based optical frequency comb. Electro-optic frequency combs [1–3] allow for an unprecedented degree of control over their parameters, providing for digital control over their span, comb tooth spacing, and central frequency (e.g., see Ref. [4–7] and the references contained therein). Critically, with the development of waveguide-based electro-optic modulators, the comb tooth spacing is independent of the laser cavity length and can therefore be made essentially arbitrarily narrow. As a result, combs can readily be produced with mode spacings more than six orders of magnitude denser than with a traditional mode-locked-laser-based optical frequency comb. This ultradense tooth spacing is ideally suited to the study of narrow sub-Doppler features in atomic and molecular systems [8–12].

synthesizers (DDSs) have been shown to be powerful alternatives to AWGs for chirped pulse Fourier transform microwave spectroscopy [20]. Here we apply this approach to generate electro-optic frequency combs with a DDS and demonstrate long-term coherent averaging of those combs for spectroscopic applications.
These combs allow for multiplexed, sub-Doppler spectroscopy whereby the entire spectrum can be acquired in a single interferogram, removing the need for slow spectral interleaving [21,22]. We have applied this platform to potassium transitions in the near-infrared region, probing ultranarrow electromagnetically induced transparency (EIT) features in vacuum and buffer gas cells (see Fig. 1 for an energy level diagram). This approach enables the interrogation of features that are as narrow as 1 kHz, significantly narrower than the initial laser linewidth, and therefore, the width of an individual comb line.
The optical setup was similar to that found in Ref. [9] in which a self-heterodyne configuration was utilized [10,13] (see Fig. 1). The laser source was a fiber-coupled, external-cavity diode laser (ECDL) with an estimated linewidth of 50 kHz to 200 kHz at 5 μs and a wavelength range of 758.9 nm to 790.5 nm. For the present measurements the wavelength was set near 770.1 nm to probe D 1 transition of 39 K. The laser was split into probe and local oscillator legs, with the probe path passed through a waveguide-based electro-optic phase modulator (PM) for comb generation. The PM was driven by repeated, linear frequency chirps generated by either an AWG or a DDS. The AWG had a maximum bandwidth of 5 GHz and operated at 12-bit vertical resolution. AWG sampling rates between 1.5×10 9 samples per second and 1×10 10 samples per second were employed for these measurements.
Our DDS electronics consisted of an evaluation board containing a high-speed, 12-bit DDS chip (Analog Devices AD9914) [24]. Communication with the DDS chip was performed using a microcontroller (μ-controller) and a serial peripheral interface (SPI) (see Fig. 1). We programmed the DDS to start a new frequency chirp upon receiving a trigger from an external delay generator. A separate phase-locked-loop (PLL) circuit provided the clock signal for the DDS at either 3.6 GHz or 3.75 GHz. The PLL and delay generator were both referenced to the same 10 MHz clock signal that was used to stabilize the oscilloscope, ensuring that each piece of equipment shared a common timebase. The DDS output was filtered to avoid aliased signals above the Nyquist frequency and the amplitude was adjusted to match that of the AWG. See the appendix for more experimental details relating to the DDS.
A comparison of the radiofrequency outputs generated by the AWG and DDS can be found in Fig. 2. Both devices produce exceptionally flat radiofrequency combs with transformlimited comb tooth widths. In the central region of the combs between 250 MHz and 600 MHz, the relative standard deviations of the comb teeth amplitudes are just 6.4% for the DDS and 1.9% for the AWG. The DDS comb flatness is slightly degraded by the external RF amplifier that is necessary to match the amplitude of the AWG. Removing this amplifier and one of the low pass filters improves the relative standard deviation of the DDS comb to 4.1%. While the AWG produces a flatter comb, as we shall see in the results discussion, normalization leads to similar spectroscopic signal-to-noise ratios.
An acousto-optic modulator (AOM) was employed to ensure that the positive-and negativeorder comb teeth occurred at unique frequencies in the RF spectrum of the photodetector. After being launched into free space, the probe beam was expanded to a 1/e 2 diameter which was varied between 2 mm and 6 mm over the course of these measurements and made circularly polarized (which can lead to larger cross sections [25]) before being sent into an atomic vapor absorption cell.
Two separate absorption cells were used during these measurements. The first was an evacuated cell 25 cm long with a diameter of 2.54 cm and contained potassium vapor at natural isotopic abundance. The second cell was 7.5 cm in length with a diameter of 2.54 cm and contained potassium vapor at natural isotopic abundance as well as 2.67 kPa of argon. Millions of collisions of the absorber with a buffer gas can occur before loss of coherence, thus leading to a dramatic reduction of transit time broadening [26][27][28]. Stray magnetic fields that affect each cell (measured to be ≤ 86 μT) were reduced by roughly three orders of magnitude by a triply shielded nickel-iron alloy chamber. An insulated box was then placed around this chamber to allow for heating to near 36 °C.
After the probe beam passed through the cell it was returned to linear polarization and was then relaunched into an optical fiber where it was combined with the local oscillator beam and sampled using a 1-GHz bandwidth detector with a noise-equivalent power of 31 pW/Hz 1/2 . The detector signal was amplified and then split into two legs; one of which went to a 4 GHz, 8-bit oscilloscope for data acquisition and the other to a phase-locking servo. The latter signal was fed back to the voltage-controlled oscillator which drove the AOM to allow for long-term coherent averaging by removing phase noise in the laser beam caused by thermal and mechanical fiber fluctuations [14]. The laser wavelength was stabilized with a loop bandwidth of 2 Hz using a high precision wavelength meter with 15 MHz resolution. This servo did not have a large effect on the resulting spectra due to the common-mode nature of the pump and probe beams.
The resulting spectra were then normalized against spectra recorded when the laser was detuned 4 GHz from the relevant absorption features. These normalized transmission spectra (i.e., I/I 0 ) were converted to an intensity-based absorbance scale according to A=−2ln(I/I 0 ), where the factor of 2 accounts for the heterodyne nature of the measurement in which the time-averaged signal intensity is proportional to the product of the field amplitudes of the local oscillator and transmitted probe beams [29].
An optical frequency comb spanning 200 MHz to 650 MHz from the carrier with a comb tooth spacing of 200 kHz (i.e., containing 2 250 positive-order comb teeth) was used to probe the D 1 transitions of 39 K and 41 K near 770.1 nm (see Fig. 3). The unmodulated, carrier frequency of the comb served as the pump (i.e., control) beam to initiate the sub-Doppler spectroscopy. This allows for multiplexed observations of hyperfine pumping and EIT features. Due to the high signal-to-noise ratios, which are enabled via coherent time domain averaging, we are able to also observe the EIT feature for the rare 41 K isotope in our natural isotopic abundance sample. Despite the higher radiofrequency flatness observed when using the AWG (see Fig. 2), the resulting spectra recorded when using the DDS and AWG are similar, indicating that the normalization efficiently accounts for these variations.
Due to the digitally controlled nature of the comb repetition rate, we are able to produce combs with tooth spacings as low as 100 Hz using both the DDS and AWG. The resulting ultrahigh resolution frequency combs can be seen in Fig. 4. The DDS and AWG lead to almost identical optical frequency combs, with the DDS producing a slightly flatter comb.
We utilized these optical frequency combs to probe an EIT feature which is dramatically narrower than the laser linewidth (and therefore the comb linewidths). To produce these ultranarrow EIT feature we employed a potassium cell with an argon buffer gas. The presence of the buffer gas leads to a reduction of the transit time broadening, leading to dramatically narrower EIT features [26][27][28]. As can be seen in Fig. 5, the combs can clearly resolve an EIT feature with a width of only 1.0 kHz. Importantly, this width is a factor of 50 to 200 narrower than the estimated laser linewidth. This sub-laser-linewidth resolution is enabled by the mutual coherence between the pump and probe beams, thus allowing for a significant reduction of the apparent frequency jitter [30][31][32]. The ten DDS-based spectra which were averaged in Fig. 5 were then individually fit to a Lorentzian with a linear baseline. These fits led to a fitted center frequency and standard uncertainty for the EIT feature of 461.71980 (8) MHz. This value agrees with the microwave value of 461.7197201(6) MHz [33] at the 1σ level.
The current self-heterodyne configuration of electro-optic frequency combs enables multiplexed direct frequency comb spectroscopy with unprecedented resolution. Unlike measurement techniques that employ traditional mode-locked-laser-based optical frequency combs, no slow interleaving of combs with relatively coarse tooth spacing is required to achieve high resolution. Further, in comparison to traditional cw laser approaches, the spectra can be recorded in a single shot, thus removing the need for slow spectral scanning. The present approach exploits the high mutual coherence of the pump (control) and probe fields to enable the efficient preparation and highly resolved observation of dressed states.
We have demonstrated a new approach for electro-optic frequency comb generation which overcomes the high cost, size, and power consumption of arbitrary waveform generators while offering higher frequency agility. Further, this approach allows for ultraflat optical frequency combs and coherent time domain averaging, leading to high signal-to-noise ratio spectra which are comparable to that achievable with an arbitrary waveform generator. While the bandwidth of available direct digital synthesizers is lower than that commercially available with high-end arbitrary waveform generators, these type of waveforms can readily be multiplied up to higher frequencies [34][35][36]. Further, a variety of techniques have been demonstrated to extend the bandwidth of electro-optic frequency combs through nonlinear broadening (e.g., [6,7,19]) or cascaded modulators (e.g., [7,17,28]). These attributes make the present method attractive for high-precision frequency metrology in atomic and molecular systems as well as for comb generation in areas such as distance metrology and communications.
is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose Funding National Institute of Standards and Technology; National Research Council (NRC Research Associateship Award).

Appendix
We used the AD9914's built-in digital ramp generator (DRG) to produce the linear frequency chirps using only a handful of parameters. Although the DDS can be programmed to continuously repeat these ramps based upon internal timing settings, in this configuration the initial phase at the beginning of each chirp is not controlled and therefore multiple ramps do not coherently average. Thus, we set the DDS to start a new ramp with a consistent phase upon receiving a digital trigger on the IOUPDATE pin by setting the "Autoclear digital ramp accumulator" and "Autoclear phase accumulator" flags high [24]. The repetition rate (f rep ) and therefore comb tooth spacing were then controlled by a series of digital triggers that were generated externally using a digital delay generator (Berkley Nucleonics Corporation Model 745). We find that excess timing jitter on these digital triggers leads to noise in the comb spectrum, so a high-quality timing source is important. Although the nominal clock frequency (f clock ) of the AD9914 is 3.5 GHz, we have slightly overclocked the device up to 4.0 GHz without issue. Since the DRG updates at a rate of f clock /24, we found that it is important to use a clock frequency that satisfied the condition f clock /24=mf rep , where m is an integer. Violating this condition led to unexpected comb teeth spacing and offsets. The DDS can be programmed to output a trigger indicating that the completion of a frequency ramp, which can optionally be used to trigger data acquisition. (a) Energy level diagram for 39 K. The upper state, 4 2 P 1/2 , splitting is 55.5MHz while the lower state, 4 2 S 1/2 , splitting is 461.7MHz [23]. (b) Direct digital synthesizer (DDS) architecture for chirp generation. A low jitter digital gate delay generator is used to control the chirp repetition rate (f rep ). (c) Experimental schematic. The probe laser is divided into probe and local oscillator legs in a self-heterodyne configuration. The probe comb is generated by an electro-optic phase modulator (PM) driven with an RF source (either the arbitrary waveform generator or a direct digital synthesizer). An acousto-optic modulator (AOM) is used to separate the positive and negative order comb teeth in the resulting interferogram and as an actuator for the phase lock. This phase lock (using a phasefrequency detector, PFD, and voltage-controlled oscillator, VCO) cancels out thermal and mechanical fluctuations between the probe and local oscillator legs and allows for coherent averaging.  Electromagnetically induced transparency (EIT) and hyperfine pumping spectra for the 39 K and 41 K D 1 transitions in the evacuated gas cell using the arbitrary waveform generator (AWG) and direct digital synthesizer (DDS) for comb generation. The x-axis is given as detuning relative to the carrier. The cell was held near 37 °C and contained potassium at natural isotopic abundance. The shown spectra were recorded with a comb tooth spacing of 200 kHz. One thousand interferograms were coherently averaged in the time domain before being Fourier transformed and normalized to produce a spectrum. Each interferogram contained 5×10 6 samples recorded at 5×10 9 samples per second. Ten of these spectra were then averaged to produce the shown traces. Typical ultrahigh resolution optical frequency combs containing 400 comb teeth spaced at 100Hz. The x-axis is given as detuning relative to the center of the frequency comb which is located 461.73MHz from the carrier. The top comb was generated via direct digital synthesis while the bottom comb was generated with the arbitrary waveform generator. The shown self-heterodyne spectra are the average of five off-resonance frequency domain traces. Each component trace was the magnitude of the Fourier transform of one thousand coherently averaged interferograms which contained 5×10 7 samples recorded at 1×10 9 samples per second. The inset shows a zoomed-in-view of the optical frequency combs. Note that the comb teeth are resolution bandwidth limited. The relative standard deviation of the magnitudes of the twenty-one comb teeth shown in the inset are 1.2% (direct digital synthesizer, top) and 1.7% (arbitrary waveform generator, bottom). Electromagnetically induced transparency (EIT) spectrum for the 39 K D 1 transition in the buffer gas cell using the arbitrary waveform generator (AWG) and direct digital synthesizer (DDS) for comb generation. The x-axis is given as detuning relative to the carrier. The DDS trace was shifted down vertically one division for clarity. The cell contained 2.67 kPa of argon buffer gas and was held near 36 °C. The shown spectra were recorded with a comb tooth spacing of 100Hz. One thousand interferograms were coherently averaged in the time domain before being Fourier transformed and normalized to produce a spectrum. Each interferogram contained 5×10 7 samples recorded at 1×10 9 samples per second. Ten of these spectra were then averaged to produce the shown traces. The differences in the baseline curvature are likely due to temporal variations of the normalization procedure rather than due to inherent differences between the AWG and DDS.