Optical frequency combs provide the clockwork to bridge between optical ($\sim 100\mathrm{THz}$) and radiofrequency (RF) ($\sim 100\mathrm{MHz}$) frequencies [1], allowing arbitrary optical frequencies to be measured to an accuracy only limited by the RF reference signal. This property is crucial in fields like dual comb spectroscopy [2], RF and terahertz generation [3], precise length measurements [4], and astronomy [5]. Other applications, for example high-harmonic generation [6] and above-threshold ionization [7], additionally require phase stability in the time-domain. A comb spectrum generated from a short-pulse mode-locked laser is completely defined by two parameters: the repetition rate $f_{\text{rep}}$, and the carrier envelope offset frequency $f_{\text{ceo}}$. The repetition rate corresponds to the inverse roundtrip time of the oscillator. The origin of the carrier-offset frequency is the pulse-to-pulse difference between envelope phase and carrier phase. The performance of a frequency comb is hence given by the intrinsic noise properties of the oscillator and the ability to detect and stabilize $f_{\text{rep}}$ and $f_{\text{ceo}}$.

The repetition rate is typically monitored by a photodiode and phase-locked to a low noise reference oscillator. Several practical methods for stabilizing $f_{\text{ceo}}$ have been demonstrated. A well-established scheme is based on using f-2f interferometry for detection, and closing the feed-back loop by acting on the group-velocity dispersion in the oscillator, e.g., by pump current modulation [8]. Alternatively the $f_{\text{ceo}}$ can be stabilized with an external acousto-optic modulator (AOM), which is used in a feed-forward scheme that avoids loop delay bandwidth limitations [9].

Instead of stabilizing the $f_{\text{ceo}}$ to a finite frequency, the $f_{\text{ceo}}$ can be fundamentally removed from the comb output using difference frequency generation (DFG). This method is used to remove the $f_{\text{ceo}}$ from the comb characterized in this Letter. A DFG comb has previously been realized in pulsed Ti:Sa lasers [10] resulting in an output at 946 nm. Broadening an Erbium (Er)-doped fiber comb such that the DFG output is at 1550 nm allows for a technologically elegant solution [11,12] based on reliable telecommunications fiber components. The residual noise of the comb spectrum at different wavelengths is an essential specification for a multitude of applications. In optical frequency metrology applications, the noise properties of a single comb line are of particular interest. Moreover, measuring the noise properties of comb lines over a large spectral range allows verification of the elastic tape model (ETM) [13]. However, direct measurement of the phase noise is hampered by the minuscule power of a single comb line, typically of the order of 100 nW.

In this Letter we present a generic method to characterize single comb teeth in the output spectra at different wavelengths without relying on the availability of narrow linewidth continuous-wave (cw) lasers. This allows characterization of the phase noise of several discrete comb lines over a broad range of frequencies. The results are in agreement with the ETM of a fiber-based DFG frequency comb. Confirmation of the model then allows the phase noise (and hence, the spectral linewidth) of the comb at any optical frequency to be estimated based on the appropriate scaling of only one phase noise measurement.

The comb teeth frequencies of a frequency comb based on a mode locked oscillator are given by an integer multiple $n$ of $f_{\text{rep}}$ plus $f_{\text{ceo}}$

The linewidth of the comb teeth at different wavelengths is given by the phase noise resulting from the noise of the stabilized oscillator and the corresponding frequency conversion processes. If the influence of a noise source described by a parameter N on ($f_{\text{rep}}$, $f_{\text{ceo}}$) can be approximated to be linearly correlated, then a fix point is given by tooth index

However, the simple scaling predicted by the DFG-ETM can be observed in practice with the setup of the frequency comb shown in Fig. 1. It is based on a temperature-stabilized stretched-pulse mode-locked Er:fiber oscillator [15,16] with a free-space piezo as a length actuator. The output of the Er:fiber oscillator is amplified and compressed to generate a super-continuum spanning more than an octave in a highly-nonlinear fiber (HNF). The amplification and super-continuum generation is obtained from an all-fiber implementation providing a stable, hands-off output. The DFG of parts of the generated spectrum at 850 nm (352 THz) and 1880 nm (159 THz), based on the setup in [12], yields an offset-free comb at 1550 nm (193 THz) with a bandwidth of 100 nm. The output power of the DFG exceeds 1 mW, and an $f_{\text{ceo}}$ suppression of $>90\mathrm{dB}$ is achieved. This spectrum is amplified again using Er:fibers to provide several identical output ports, which can be converted to different wavelengths, e.g., by nonlinear shifting and frequency doubling. In this work, we used three outputs (see Fig. 2): the first at 1557 nm (directly obtained from the DFG preamplifier), and outputs at 1162 nm and 852 nm obtained using fiber amplifiers with nonlinear fiber outputs. The repetition rate is 80 MHz and is detected by a photodiode. For the lock to a GPS disciplined RF oscillator, the 10th harmonic of $f_{\text{rep}}$ is filtered from the photodiode signal and phase-locked via the piezo. The pump diode current can be used for stabilization with a bandwidth exceeding 400 kHz, which is an advantage of DFG combs that do not require this actuator for $f_{\text{ceo}}$ control. In a separate experiment a comb line is optically locked to a 1.5 Hz cavity stabilized cw laser at 1162 nm using the oscillator pump current.

 

Fig. 1. Setup. The laser system is based on a repetition rate stabilized mode-locked Er:fiber oscillator (Osc). A super continuum (SC) spanning more than an octave is generated in a highly-nonlinear bulk fiber (HNF) after amplification (Amp) and compression. The difference frequency of the SC spectrum at 850 nm and 1880 nm generates an $f_{\text{ceo}}$-free output at 1550 nm, which is pre-amplified (PAmp) to provide multiple ports to be converted to different wavelengths.

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Fig. 2. Repetition rate stabilization and outputs: The mode-locked Er:fiber oscillator (Osc) is either locked to a low noise RF reference (1) using the piezos voltage ($U_p$), or alternatively to a 1.5 Hz cavity-locked cw diode laser (2) via the pump current ($I_p$). The characterization is done by a beat with an independent second 1.5 Hz laser at 1162 nm and by DSH detection of two clean-up cw lasers at 1557 and 852 nm. The DSH characterization setup is shown in Fig. 3.

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The phase noise characterization of a frequency comb at different wavelengths can be done by beating the frequency comb with cw lasers exhibiting phase noise significantly smaller than the comb tooth to be measured. However, viable narrow-linewidth lasers only exist at specific wavelengths. In contrast, external cavity diode lasers (ECDLs) are available over a broad spectral range ($630\mathrm{nm}-1.8\mathrm{\mu m}$). Whereas ECDLs typically have free-running linewidths in the order of 100 kHz, they can be modulated up to several MHz. A generic method is, hence, to transfer the noise properties of the comb line to a cw laser via a phase-locked loop and characterize the cw laser by delayed self-heterodyning (DSH) technique [17]. The cw laser then acts as a single comb-line cleanup filter while also providing a much stronger signal.

The characterization setup for one comb output is shown in Fig. 3. The upper part shows the phase-locked loop between a comb line optically filtered with a bandwidth of $\sim 10\mathrm{GHz}$ and a cw laser. The signal-to-noise ratio (SNR) of the beats are typically $\sim 30\mathrm{dB}$ at 100 kHz resolution bandwidth. The lower part shows the DSH measurement. The DSH beat is mixed down to a few MHz and sampled for 1 s with an oscilloscope. The phase noise is then calculated from the recorded trace with software that performs the following operations: a finite impulse response (FIR)-bandpass filter first removes signals further away from the carrier than the frequency band of interest; next, the phase excursions of the signal with respect to the tracked center frequency are extracted. The phase noise spectrum is obtained from the time-sequence of phase excursions by splitting the time-series into several blocks and averaging the resulting Fourier transform of each block. The spectra measured using the DSH technique are corrected for the frequency response of the DSH measurement [18]. The phase noise $S_{\varphi }^{\text{laser}}$ of the laser is given by the measured RF phase noise $S_{\varphi }^{\mathrm{RF}}$ according to

 

Fig. 3. Characterization setup consists of an optical phase-locked loop (OPPL) of a cw laser to one comb line with a fixed RF offset and fiber DSH of the cw laser with a frequency offset given by the AOM frequency.

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The phase noise spectra for the comb locked to the RF reference measured via DSH at 1557 and 852 nm are shown in Fig. 4 together with a direct beat measurement with a narrow linewidth laser at 1162 nm. The 1557 nm DSH setup consists of a fiber corresponding to a delay of ${\tau }_{1557}=98\mathrm{\mu s}$, whereas the 852 nm DHS has a shorter fiber corresponding to a delay of ${\tau }_{852}=5\mathrm{\mu s}$. Well above the locking bandwidth of 10 kHz, the spectra follow the typical noise observed for the free-running Er:fiber oscillator. The beta separation line is a convenient way to distinguish between phase noise spectrum contributing to the linewidths (above) and to the pedestal (below) [19]. The linewidths can be calculated from the phase-noise spectra by numerical integration.

 

Fig. 4. Phase noise spectra (1) corrected for self-heterodyne response at 852 nm (blue, top) and 1557 nm (red, bottom) as well as the beat at 1162 nm (green, middle). The beta separation line (dashed dotted, A) is shown for reference. The inset shows the square root of the integrated phase noise from 1 to 100 kHz.

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According to the ETM, the noise spectra scale quadratically with the carrier frequency ${\nu }_0=c/{\lambda }_0$:

The results in Fig. 5 are rescaled to a fixed carrier frequency of ${\nu }_0^{\prime }\equiv f_{\text{rep}}=80\mathrm{MHz}$. The scaled noise spectra measured simultaneously are correlated to within the detailed random noise structure (see inset in Fig. 5), thus showing excellent agreement with the elastic tape model with a fix point at zero frequency and the predicted quadratic scaling [Eq. (6)].

 

Fig. 5. Phase noise spectra scaled to $f_{\text{rep}}=80\mathrm{MHz}$ carrier according to Eq. (6): The phase noise spectra are shown in two groups: 1. for the comb locked to an 800 MHz RF-reference with zoom (inset), and 2. for the lock to the optical oscillator. The noise of the RF reference oscillator (black, D), the in-loop error of the 1557 nm optical phase-locked loop (OPLL) (black, B) and the out-of-loop beat between the cavity locked lasers (gray, C) are shown for reference.

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Note, that the observed phase noise performance is limited by the detection of $f_{\text{rep}}$ and the RF oscillator noise. We demonstrate this limitation by locking the comb to an optical reference, which allows for significantly lower noise [20]. The optical reference is provided by an external-cavity diode laser that is locked to a high-finesse ultra-low expansion cavity at 1162 nm providing a $<1.5\mathrm{Hz}$ short-time ($\sim 1\mathrm{s}$) linewidth. It shows significantly lower phase noise over the complete frequency range up to 1 MHz as demonstrated by the beat with a second similar system (gray). Hence, the second set ② of phase noise measurements utilizing a weak lock to the optical reference exhibits significantly lower phase noise within the locking bandwidth of about 200 kHz using the oscillator pump laser current as the control element. The deviation of the 1557 nm phase noise at higher frequencies can be attributed to the limited locking bandwidth as shown by the in-loop error signal (black). Figure 6 shows the phase noise for the comb tightly locked to the optical reference. An upper bound for the comb performance as a clockwork is established by measuring the out-of-loop beat with a second identically performing optical reference. The measurement at 1557 nm shows good agreement with the out-of-loop beat at $<300\mathrm{kHz}$ below the servo bump. The 852 nm beat is omitted due to the limited SNR of the DSH at the shorter fiber delay.

 

Fig. 6. Phase noise spectra (1) of comb teeth at 1557 nm (red, top) and 1162 nm (green, bottom) for locking to an optical reference laser scaled to 80 MHz. The in-loop error of the 1557 nm OPLL (black, B) and the out-of-loop beat between the cavity locked lasers (gray, C) as well as the beta separation lines (dashed dotted, A) are shown for reference.

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In conclusion, we have characterized a passively carrier envelope phase stabilized Er:fiber frequency comb based on DFG at several wavelengths. We report the phase noise characterization of several single discrete comb lines over nearly one optical octave. Transferring the properties of a single comb line to a cw diode laser allows for a generic characterization of the phase noise and linewidth via a delayed self-heterodyne measurement. Furthermore, it has been shown that substantial phase noise improvements can be realized by locking the repetition rate of the frequency comb to an optical reference. The results are in good agreement with the quadratic scaling with the carrier frequency as predicted by the elastic tape model. Because the fix point of the DFG is strictly at the frequency origin, it is particularly well-matched for RF generation by locking $f_{\text{rep}}$ to an optical reference oscillator and is suited for future practical applications of optical clocks. In particular, time domain applications will benefit from the advantages of DFG-based combs [6,7].