Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator

The frequency noise of an optical frequency comb is routinely measured through the hetherodyne beat of one comb tooth against a stable continuous-wave laser. After frequency-to-voltage conversion, the beatnote is sent to a spectrum analyzer to retrive the power spectral density of the frequency noise. Because narrow-linewidth continuous-wave lasers are available only at certain wavelengths, heterodyning the comb tooth can be challenging. We present a new technique for direct characterization of the frequency noise of an optical frequency comb, requiring no supplementary reference lasers and easily applicable in all spectral regions from the terahertz to the ultraviolet. The technique is based on the combination of a low finesse Fabry-Perot resonator and the so-called “fringe-side locking” method, usually adopted to characterize the spectral purity of single-frequency lasers, here generalized to optical frequency combs. The effectiveness of this technique is demonstrated with an Er-fiber comb source across the wavelength range from 1 to 2 μm.

Ultra-Low-Expansion cavity, or optical clock) and fast servo electronics can further narrow the comb teeth relative linewidth down to the mHz-level across the full spectral range of the OFC 30,31 . Considering the growing impact of OFCs on current research and their application as frequency rulers, it is apparent that the characterization of OFCs in terms of FN is a fundamental prerequisite.
A time-domain approach is typically adopted to measure the FN of an OFC, based on hetherodyning one comb tooth against a high-spectral purity reference laser. As a necessary condition, the FN of the reference laser has to be well below the FN of the comb tooth under test. The frequency fluctuations of the hetherodyne beat are then converted into voltage fluctuations by a phase-locked voltage-controlled oscillator, providing a signal proportional to the FN of the comb tooth 32 . The limitations of this approach are apparent, as the measurement of the FN across the broad spectral region typically covered by the comb requires a set of reference lasers operating around the different wavelengths of interest 33 ; this limitation becomes even stronger when considering the recent progress in comb technology, that extended the spectral coverage into the UV, mid-IR and THz regions, where narrow-linewidth reference lasers are not easily available. This approach has been also extended to simultaneously measure the FN of many teeth over the emission spectrum of the comb under test, using a second and identical comb operating at a slightly detuned repetition rate for parallel hetherodyning of the comb teeth 34 . The main drawback of this technique is the need for two OFC systems.
By contrast, frequency domain approaches, based on optical frequency discriminators, are more general and do not require supplementary reference lasers. The FN is directly converted and acquired as a voltage fluctuation by a low-noise photodiode, without the need for a phase-locked oscillator. Fabry-Perot (FP) resonators, with a proper linewidth as compared to the oscillator noise bandwidth, can be employed to perform high-sensitivity measurements of the FN of CW lasers regardless of the emission spectral region 35,36 .
In this paper, we report on a direct method for the measurement of the FN of OFCs based on a low-finesse FP resonator operating as an array of optical frequency discriminators. The method is a generalization of the fringe-side locking technique 35 , as commonly used for the characterization of single-mode CW lasers, and allows for FN measurement across the entire emission spectrum of the OFC without using supplementary reference lasers. Although the combined use of OFCs and passive cavities has been widely investigated 7,9,25 , to the best of the Authors' knowledge, this approach has never been used for the FN of OFCs. In principle, the proposed technique allows for FN measurements in any spectral region from the UV to the mid-IR by simply selecting proper FP mirrors; extension to the THz would be also feasible provided that mirrors with specified reflectivities would be available in this wavelength region. As the spectral coverage of OFCs can be very wide, reaching even an octave, band-pass filtering of the OFC can be necessary for the application of the proposed technique to reduce the band under investigation to a span compatible with the FP mirror reflectivity. The measurement of the FN within relatively narrow sub-bands allows for exploration of the FN behavior across the much broader comb bandwidth. The proof of concept given in this paper is based on the characterization of the FN of a commercial Er-fiber frequency comb in the spectral range from 1 to 2 µm, demonstrating the great potential and flexibility of the proposed method.

Theory
The operating principle of the proposed technique is shown in Fig. 1. The comb under investigation, whose output spectrum is constituted by the frequency components ν = + pf f p r o , is coupled to a FP cavity with proper width of the resonances (10-100× the expected linewidth of the comb tooth). In general, the free spectral range (FSR) of a FP cavity is frequency-dependent because of the dispersion originating from cavity mirror coatings 37,38 . The effect of dispersion is particularly relevant when working with high-finesse cavities or near the edges of the mirror reflectivity. For the sake of simplicity, we will neglect the effect of dispersion in the calculation that follow. However, this can be considered a good approximation when working with a FP cavity with low finesse of ~100, and mirror reflectivity broader than the comb bandwidth. Under these assumptions, the FP resonant frequencies are simply given by where c is the speed of light in vacuum, L is the FP length, n ≃ 1 is refractive index of air, and m is the fringe order.
A maximum of the FP transmission is obtained in the "optimal resonance condition", represented by the equation where ν is the frequency of the comb tooth nearest to the center of the comb spectrum and ≅ / N FSR f r the filtering ratio ( Fig. 1 shows an example for = N 4). In the resonance condition, the FP cavity selects a subset of comb teeth (one every N ), however, because the offset frequency f o is typically non-zero, this implies < Nf FSR r , and hence each subsequent FP resonance away from the center frequency ν has an additional detuning δ = / f f m o with respect to the corresponding comb mode. All the comb modes out of the FP resonances, characterized by orders < < ( + ) mN p m N 1 , are sufficiently rejected even using cavity finesse of ∼100 (attenuation > 20 dB for N = 4), so that their contribution to the FN measurement is negligible. The FP acts as a frequency discriminator when its transmission is tuned at around 75% of the maximum, that is in the operating region of highest linearity of the FP transmission fringe (Airy function). Under these circumstances, the comb tooth at ν operates at the nominal 75% transmission point, whereas the comb teeth at lower (higher) frequencies operate at transmission decreasing (increasing) proportionally to the frequency detuning with respect to the corresponding FP resonance. On the other hand, the fringe slopes probed by each comb tooth are very similar, provided that all the teeth up to the edges of the comb spectrum operate within a certain limited region around the 75% transmission point. To this purpose, the number of comb teeth transmitted by the FP cavity is assumed to be = where ν ∆ is the FWHM of the filtered comb spectrum. The cumulative frequency detuning of the FP resonances over the comb band is The FP fringe has high slope (sensitivity) and good linearity in the region from approximately 55% to 85% of the maximum transmission, corresponding to nearly a quarter of the FP resonance bandwidth ν ∆ FP (FWHM). Operation within this desirable region requires Once this condition is met, the frequency fluctuations of each comb tooth filtered by the FP cavity are converted into amplitude fluctuations that can be measured by a low noise photodetector (see Fig. 1), representing a generalization of the standard fringe side-locking technique adopted for the characterization of the FN of single-frequency CW lasers 35 .
The comb intensity noise contribution can be efficiently rejected using a balanced detection scheme 39 , yielding a highly-sensitive FN measurement where all the filtered comb teeth coherently combine with similar frequency-to-voltage conversion coefficients. Assuming that each filtered comb tooth gives a voltage contribution on the photodiode at the output of the FP cavity α = V T P m m m , where α is the power-to-voltage conversion coefficient of the photodiode, T m is the FP transmission at the optical frequency ν = + mNf f m r o and P m is the power of the mN -th comb tooth, the voltage fluctuations due to intensity and frequency noise can be expressed as where the first and second term in the sum represent the comb intensity and frequency noise, respectively. By neglecting the intensity noise that is rejected using a balanced detection scheme and assuming that each comb tooth within the band of interest is characterized by the same FN, the resulting balanced voltage fluctuations are S f in unit of Hz 2 /Hz, can be easily obtained from the PSD of the signal at the output of the balanced detector using the relation The selection of appropriate FP mirrors is governed by few simple guidelines. In general, FP mirrors with lower finesse are preferable to fulfill Eq. (3), that is necessary to get correct FN measurement. However, lower finesse implies lower sensitivity. Hence, the general rule is to select the lowest finesse compatible with the required noise floor of the setup. The noise floor ( )

Results and Discussion
Frequency noise measurement of an Er:fiber OFC. The multiple fringe-side locking technique has been used to characterize the FN of an octave spanning Er:fiber OFC with power of 250 mW, repetition frequency f r = 250 MHz, and spectral coverage from 1 to 2 µm. Figure 2 shows a schematic of the experimental setup. The OFC spectrum is narrowed by bandpass filters with central wavelength of 1060, 1180, 1420, 1560, 1820, or 1980 nm, and optical bandwidth of ~40 nm. The six resulting spectra, each with power content around 10 mW, are shown in Fig. 3. After spectral filtering, the comb passes through an optical isolator and a FP interferometer placed in a polyurethane enclosure to reduce acoustic perturbations and the effect of air turbulence. The frame of the FP (Burleigh RC-110) is made of Super-Invar (thermal expansion coefficient < . × − 0 36 10 6 ) to guarantee an improved stability; previous tests to characterize high spectral-purity lasers have shown an upper limit on the linewidth contribution, due to the FP cavity, of 4 and 17 kHz over observation times of 1 and 100 ms, respectively 40 . When using the filter at 1560 nm, the FP cavity has a free-spectral-range (FSR) of 1 GHz and mirrors with nominal reflectivity of 97%, corresponding to a finesse of ~100 and a linewidth of ~10 MHz; similar FP characteristics have been used with the other bandpass filters. A balanced detection scheme is used to cancel the laser intensity noise contribution to the FN. The FP cavity transmission is kept at 75% of its maximum value by a servo-loop acting on the PZT actuator of the FP interferometer with a slow control bandwidth of ~1 Hz. The PSD of the signal at the output of the balanced photodetector corresponds to the comb frequency fluctuations for Fourier frequencies larger than the control bandwidth. Figure 4 shows the transmission of the scanning FP that is illuminated by either a single-frequency laser at 1560 nm (blue line) or a comb at 1560 ± 20 nm (red line). In the case of the single-frequency laser, the FP transmission is characterized by a classical Airy function with a periodicity of 4.4 ms, corresponding to the FSR of 1 GHz. A more complex structure is observed with the comb: as a first point, because the comb tooth frequency spacing f r is 250 MHz, the FP transmission has 4 resonances within each FSR; as a second point, the envelope of Scientific RepoRts | 5:16338 | DOI: 10.1038/srep16338 the resonances has a maximum when the optimal resonance condition is reached (highest resonance), and an amplitude decreasing rapidly away from the optimal resonance, as the number of comb teeth coupled into the FP cavity is proportionally decreased; finally, the width of the resonances increases away from the optimal resonance, due to the increasing mismatch between FP resonances and comb teeth periodicity 9 . A detailed view of the highest FP resonance is reported in Fig. 4c). In the case of single-frequency laser, the cavity resonance has a linewidth of 43 µs (corresponding to 10 MHz) and a slope of 0.18 V/MHz at 75% of the maximum transmission. On the other hand, the 40-nm wide OFC radiation shows a slightly increased linewidth of 58 µs and a reduced slope of 0.13 V/MHz, representing the overall frequency-to-voltage coefficient, ascribed to the non-zero comb offset frequency and hence to the detuning between cavity resonances and comb teeth. Under this condition, the slope d m of the resonances probed by each comb tooth away from the 75% operating point is slightly reduced, as is the resulting overall slope D of the highest resonance. The minor increase of the FP linewidth when analyzing the OFC is already an indication that almost all teeth within the observed band are coupled into the FP cavity close to the 75% operating point. More specifically, the detuning is acceptable if the condition (3)   Hz 2 /Hz at Fourier frequencies below 100 kHz, whereas the excess noise generated during supercontinuum formation 43 gives an f 2 trend at Fourier frequencies above 100 kHz.
The FN PSD of the FP cavity has been measured using a high-spectral purity Er:fiber laser stabilized against a monolithic Ultra-Low-Expansion cavity. The resulting linewidth of the stabilized laser is below 100 Hz over 1-s observation time, i.e. much less than the linewidth of the FP cavity. Under this condition, the measurement of the FN of the stabilized laser by locking to the side of a FP fringe represents essentially a measurement of the FN PSD of the FP cavity at Fourier frequencies below 10 kHz; at higher Fourier frequencies, the servo bump centered at 180 kHz of the cavity-stabilized laser dominate the measurement of the FN PSD, however the resulting FP linewidth is not affected by this contribution because it lies below the β-line 41 . As shown in Fig. 5(a,b), the FN PSD contribution of the FP cavity is lower than that of the comb in all the frequency range of acoustic vibrations below 10 kHz; however, the resonances below 1 kHz in the FN PSDs of the filtered comb are actually ascribed to the FP itself, and originates from vibration-induced cavity length changes.   44 and reach the highest value of ~47 kHz at 1980 nm. The contribution from the FP cavity amounts to 2.6 kHz, that is sufficiently lower than the comb under investigation.
Validation of the technique. The technique proposed for the measurement of FN of OFCs has been validated using a high-spectral purity single-frequency Er:fiber laser (SFL) at 1560 nm. A comb tooth has been phase-locked to the SFL using the setup shown in Fig. 6. In particular, the SFL and comb beams are combined in a beat-unit and focused on a low noise InGaAs photodetector. The instantaneous frequency of the resulting beating is divided by 12 using a prescaler, and then compared in a phase detector with a low-phase noise synthesizer generating a reference signal at 5 MHz. The output of the phase detector is sent to a fast proportional-integral-derivative (PID) controller acting on an intracavity electrooptic modulator and hence on the comb repetition frequency (actuating bandwidth of ~250 kHz). Figure 7b) shows the beating in locked condition, characterized by a coherent peak with SNR higher than 60 dB at 1-Hz resolution bandwidth and 99.7% of the beatnote power (250-kHz span). The performance of the phase-lock was further characterized by measuring the PSD of the error signal at the output of the phase detector; the result is reported in Fig. 7c) together with the integrated residual phase noise. The noise floor in the phase noise measurement can be evaluated by feeding both inputs of the phase detector with the same 5-MHz reference signal from the synthesizer: in this configuration, the signals compared by the phase detector are inherently phase-coherent, hence any detected phase difference at the output of the phase detector is due to noise and represents the noise floor. The bump in the phase-noise PSD indicates a control loop bandwidth of ~250 kHz. A residual phase noise of 0.17 rad is obtained over an integration bandwidth from 1 Hz to 10 MHz, indicating that more than 97% of the RF power is concentrated in the coherence peak of the beat note signal. It's worth to note that the PSD of the beatnote in Fig. 7b)    however, the phase noise power resulting from integration over the larger span from 1 Hz to 10 MHz is higher and hence the percentage of RF power in the coherence peak retrieved from this measurement amounts to 97%, that is lower than the 99.7% obtained from the beatnote measurement. Under this condition, the coherence properties of the SFL are cloned on the comb teeth around 1560 nm, meaning that FN measurements of the SFL and comb, using the proposed technique, must give same results. Figure 7a) reports the measured comb FN at around 1560 nm in both the free-running (orange line) and phase-locked (blue line) conditions. The FN PSD of the SFL is also reported (green line), as measured using the same FP cavity adopted for the OFC. In locked condition, the FN PSD of the comb closely resembles that of the SFL up to 10 kHz; moreover, a strong noise reduction is observed with respect to the free-running condition up to Fourier frequencies of ~100 kHz. At frequencies larger than 200 kHz, due to the control bandwidth limitation, the FN PSD is higher than that in free-running but still below the beta-line, and does not contribute to the laser linewidth, according to the analysis reported in 41 . It's worth noting that the bump at 180 kHz observed in the FN PSD of the FP cavity (see Fig. 5(a,b)) is not present in the FN PSD of the SFL (green line) as it is only ascribed to the locking servo acting in the cavity-stabilized Er:fiber laser. As a final test, the FN PSD of the residual phase noise has been calculated starting from the data in Fig. 7(c), and added to the FN PSD of the SFL; the result (red line) is in close agreement with the direct measurement of the FN PSD of the comb in locked condition. This means that the discrepancy between the FN PSD of the comb and the SFL in locked condition is due to the quality of the control loop; however, apart from this point, our results prove the validity of the proposed technique to measure the FN of an OFC. The limit to the proposed technique is represented by the FN PSD of the FP cavity adopted in the setup, and more specifically, the linewidth of the FP cavity has to be well below that of the OFC over the observation time considered. The limit imposed by the FP cavity adopted in our setup is evident from Fig. 5: the linewidth of the FP cavity over 1-ms is well below that of the OFC (at all central wavelengths), hence the measurement of the FN PSD above 1 kHz can be ascribed totally to the OFC; on the other hand, the linewidth of the FP cavity over observation times longer than 1 ms increases rapidly, reaching levels comparable to that of the OFC, hence the measurement of the FN PSD at frequency below 1 kHz has some contribution (acoustic resonances) from the FP cavity, and is not accurate. The adoption of more stable FP cavity, made of monolithic low-expansion glass, would allow for the characterization of OFC with even higher spectral purity and reduced linewidth.

Conclusion
We proposed a flexible and high-sensitivity method for the measurement of the frequency noise properties of OFCs that allows for full characterization of the OFC across its entire optical spectrum. The method represents an extension of the fringe-side locking method to OFC sources, and is based on a FP resonator used as an array of optical frequency discriminators. In principle, this approach provides high-sensitivity measurements in any spectral region from the THz to the UV by choosing proper FP mirrors for each spectral region, without using any supplementary reference laser. The capabilities of the method were tested by an accurate characterization of the FN of an Er:fiber OFC in the spectral region from 1 to 2 µm using a FP cavity with a Finesse of ~100. We believe that this measurement technique can be easily and largely adopted for the characterization of OFC sources.