We demonstrate an optical frequency comb with fractional frequency instability of ≳2×10-14 at measurement times near 1 s, when the 10th harmonic of the comb spacing is controlled by a liquid helium cooled microwave sapphire oscillator. The frequency instability of the comb is estimated by comparing it to a cavity-stabilized optical oscillator. The less conventional approach of synthesizing low-noise optical signals from a microwave source is relevant when a laboratory has microwave sources with frequency stability superior to their optical counterparts. We describe the influence of high frequency environmental noise and how it impacts the phase-stabilized frequency comb performance at integration times less than 1 s.
© 2006 Optical Society of America
The development of low phase noise links between the optical and microwave frequency domains is now common practice in national time and frequency institutions around the world. Most applications relate to optical spectroscopy, where frequency combs are used for optical frequency calibration [1–9], or direct femtosecond laser spectroscopy [10–13]. Other applications include synthesizing low noise microwave signals from optical oscillators [14–16] or generating highly stable microwave signals from optical sources [16–18], where the difference is the time scale over which signals are examined. It is well recognized that the performance of frequency standards is expected to gain superiority in the optical domain because the squareroot Allan variance of a periodic signal is inversely related to the quality factor, Q, of its associated resonance, and the Q (=ν/Δν) of an optical transition or optical cavity can be significantly higher than that achieved at longer wavelengths [19,20]. For this reason, the expected approach of frequency synthesis is to transfer the exceptional low-phase noise and/or instability of an optical signal into the microwave domain. For example, Bartels et al.  have demonstrated the synthesis of 10 GHz signals from separate optical references with a fractional frequency instability of ≤3.5×10-15. However, for laboratories whose strength lies in the development of ultra-low noise microwave oscillators, but have the need to generate clean optical signals, the idea of synthesizing an optical signal from a microwave source is not unreasonable. Furthermore, microwave sources with 10-15 level fractional frequency instability are available as commercial devices (e.g. a good quartz oscillator referenced to a hydrogen maser and granted sufficient integration time), yet the equivalent is not yet available in the optical domain. As an example of the demands placed on optical oscillators; to reach the quantum limit in a calcium optical frequency standard, a laser frequency instability below 3×10-16 for the millisecond duration of the atom interferometry is required . To reach such a level it is prudent that a range of approaches be taken to attempt to produce such highly stable optical sources.
The UWA group have a record of developing cryogenic microwave sapphire oscillators with exceptional levels of frequency instability [22,23]; the best recorded frequency instability being ≤3×10-16 between 2 s and 100 s. With these high quality microwave sources, it is possible for us to investigate the limitations of transferring the microwave instability into the optical domain. The approach presents inherent challenges because the phase comparisons used to control the linkage are made in the microwave domain where low level additive noise can have more deleterious effects than would be the case for an optical phase comparison.
The work presented here complements previous optical frequency synthesis work where the frequency of a strong optical output is continuously tuneable and simultaneously linked to an absolute frequency standard through a frequency comb . Here we demonstrate how stable such a synthesizer can be when the spectral purity is derived from a low-noise microwave source.
2. Microwave-to-optical (MtO) frequency bridge
The optical to microwave frequency comparison is composed of three parts: 1) a cavity stabilized Nd:YAG laser with a wavelength of 1064 nm (referred to as the optical oscillator), 2) a fibre and femtosecond-laser-based frequency comb with a comb spacing fR=1 GHz, linking the optical and microwave oscillators, henceforth, referred to as the optical frequency synthesizer or frequency comb, and 3) a liquid helium cooled microwave sapphire oscillator (MSO) with carrier frequency fMSO=10.0 GHz. Each in turn will be described below.
To impose the MSO’s low noise signature onto the repetition frequency of the femtosecond laser, a phase comparison between fR and fMSO/10 could be made, or between the 10th harmonic of fR and fMSO itself. Since the former implies a further multiplication of 100 of the phase noise power from the microwave to optical domains, any noise floors or spurious noise signals in the repetition frequency control system will more adversely affect the signal purity in the optical domain. For this reason only the latter scheme will be described in this paper.
The layout of the experiment is described in Fig. 1. The central graph is an example power spectrum of the frequency comb generated by the femtosecond laser and micro-structured fibre. The frequency instability comparison between the optical and microwave oscillators occurs in the optical domain when ~2mW of the 1064 nm signal is combined with the IR portion of the optical comb (indicated by the photodetector adjacent to the comb in Fig. 1). Heterodyning between the 1064 nm signal and the nearest member of the comb produces an optical beat with sufficient signal-to-noise (~30 dB in 300 kHz) for frequency counting. The light signal is delivered to the optical frequency synthesizer over 20m of single mode optical fibre. As we will describe in more detail below, we have a second stabilized Nd:YAG laser for comparisons with the first. This second laser has an output at both 532 nm and 1064 nm. This enables us to repeat the experiment in the green part of the spectrum.
3. Microwave and optical oscillators
The heart of the microwave oscillator is a cylindrical sapphire crystal resonator , with a height and diameter of 3 cm. It is excited in a wispering gallery E8,1,δ mode (or in alternate nomenclature WGH8,0,0) at 10.00 GHz (Q=2×108), acting as the frequency discrminating element of a loop oscillator circuit. The temperature of the resonator is controlled to within 50mK of 5.9 K, where its frequency becomes independent of temperature to first order . The oscillation frequency is stabilized with a Pound locking scheme, while unwanted amplitude modulation is actively suppressed by an additional control loop. Further details on microwave sapphire oscillators are described in Ref. .
When making a frequency instability comparison between oscillators of widely differing carrier frequencies, there arises the possibility that the synthesis chain linking the signals can impose a measurement limit. In light of this eventuality it is prudent to have independent tests of the each oscillator. The MSO was constructed with a second MSO aligned orthogonally and housed in the same dewar (for reasons not related to this experiment, but fortuitous nonetheless). In the optical domain, two 1064 nm Nd:YAG lasers have their frequencies locked to TE00 resonant modes of separate thermally stabilized resonators. The resonators are all-sapphire Fabry-Perot cavities with finesses of 1250 and 5600. The lasers locked to these respective cavities will be denoted OOSC1 and OOSC2. The cavities are maintained slightly above room-temperature (301 K), so avoiding cryogenic fluids and their corresponding periodic disturbances. The resonators consist of sapphire super-mirrors clamped by spring-loading onto sapphire spacers with a central bore to accommodate the optical beam. The use of sapphire confers excellent intrinsic length stability and vibration immunity (Young’s modulus=345 GPa, coefficient of thermal expansion=3×10-6 °C-1). The cavities are housed in a vacuum cham- ber held below 10-5 Torr by 20 L/s ion pumps. Within the chamber are two copper radiation shields each held to a specific temperature (Outer shield: ~28.3 °C, inner shield: ~29.3 °C, cavity: ~28.0 °C). This two-stage approach has reduced the temperature variation of the sapphire cavity to below 20 nK for time-scales less than 10 s. However, thermal drift problems have not been fully rectified with laser frequency drift rates of up to ~1kHz/s occurring on both cavity/laser systems.
The laser frequency is locked to the cavity mode by phase-modulating the light with the laser’s internal piezo element and feeding back to this same element according to the Pound- Drever-Hall (PDH) scheme . The modulation frequencies for OOSC1 and OOSC2 are 473 kHz and 211 kHz respectively, both with a modulation depth of 0.55 rad. A downside of this approach is that the piezo modulation also imparts a modulation on the direction of the beam which gives rise to a spurious signal in the PDH lock. Just as the phase modulation probes the dependence of the resonance on frequency, the pointing modulation probes the dependence on spatial alignment, hence generating an error signal in addition to the intended PDH signal. This effect, which is principally in the horizontal direction, was suppressed by implementing a novel alignment lock (to be described in a separate publication). The performance of the optical oscillators is detailed in section 5.
4. Optical frequency synthesizer
The UWA optical frequency synthesiser is a femtosecond laser and micro-structured fibre (MSF) based system using the f-2f non-linear interferometric scheme to detect the offset frequency, fCEO. The synthesizer has been described in detail previously . However, since modifications have been made a short description is appropriate here. A 35 cm, long microstructured fibre (MSF) with a central core diameter of 2.0 µm and zero-group velocity dispersion wavelength of 740 nm (NL-2.0-740, Crystal-fibre, Denmark) is used for octave-comb generation. The input of the MSF is collapsed to form a single core fibre onto which an angled ferrule connector is placed. The length of the collapsed region is a few millimetres (the fibre end treatment is carried out by Crystal-Fibre). The output end of the MSF is spliced to ~40 cm of single mode LMA-5 fibre, to which another angled connector is placed.
The collapsed solid core at the input (NA~0.27 at λ=780 nm) implies that the incident (IR) light does not require the high levels of focusing needed to couple light into bare MSFs, thus reducing the risk of fusing foreign material onto the front facet of the fibre. In the event of foreign material accumulating on the face of the fibre, the fibre is easily wiped clean and repositioned. The angled face connectors have been effective in reducing the level of optical feedback into the fs-laser and making mode-locking more robust. Initially, micro-structured fibre with FC/PC connectors was used in the optical synthesizer, but optical retro-reflections from the facets were observed to cause power fluctuations in the fs-laser output power and consequently amplitude variations of fCEO, and of optical beat signals between light sources and the comb. On occasions the optical feedback was observed to annul the mode-locking of the fs-laser altogether.
Generating 12kW peak-power pulses of light entering the MSF is a 30 cm length 6-element Ti:sapphire ring laser (GigaJet 20, GigaOptics ), pumped by 5.5–6.0W of 532 nm TE00 light. This laser, when mode-locked, produces ~55 fs wide pulses at a center-wavelength of 805 nm. No pulse compression is carried out between the laser and the micro-structured fibre. The femtosecond laser emits between 650 and 750mW. Often the pump power is adjusted to tune fCEO to a suitable frequency (between 70MHz and 350 MHz) for division and servo operation.
The offset frequency, obtained from the f-2f non-linear interferometer, undergoes frequency division (by 10 or 20 depending on its frequency) and phase comparison with a synthesized rf signal in a digital phase detector before the correction signal is sent to an AOM in the pump laser beam path to form a closed servo loop. Dividing the signal increases the locking range of the offset frequency. Details about the offset frequency servo may be found in Ref. .
A phase comparison between fR and the MSO is carried out by combining the MSO signal with the 10th harmonic of the repetition frequency (fR,10GHz) in a double balanced mixer. Sufficient fR,10GHz power was obtained from a high speed photodetector followed by an X-band amplifier. A tunable cavity filter between the photodetector and amplifier selects the relevant harmonic for amplification. With low pass filtering, the difference frequency of ~11MHz is employed in a second mixing stage, whose output provides the correction signal for feeding back to the PZT element in the femtosecond laser (via a high voltage driver). A layout of the signal mixing and servo operation is shown in Fig. 2. The commercial rf synthesizer is necessary in our scheme because the repetition rate is not easily tuneable over the range of MHz. The synthesizer also provides flexibility in the locking scheme, since it allows repeated beat frequency measurements to occur at the same frequency. The rf synthesizer, referenced to a hydrogen maser, has a frequency instability of ~1.5×10-13/√τ out to 1 hr of integration time, τ. Its contribution to the instability of the comb is therefore approximately 1.5×10-16 at 1 s. The frequency of the MSO is not freely tunable since its frequency vs temperature turning point dictates the precise MSO frequency.
The two-sided (single sideband) phase spectral density of the free fluctuations of the repetition rate is ~2/f 4 rad2Hz-1 for the 1GHz carrier (or 200/f 4 rad2Hz-1 at 10 GHz), with some additional noise across acoustic frequencies (see Fig. 7). This rapid fall in noise with frequency necessitates imposing high gain at low frequencies while limiting the servo bandwidth to ~7 kHz to maintain the intrinsic low-noise of fR at frequencies beyond this bandwidth. The loop filter in the repetition frequency servo contains two integration stages on top of the integration stage inherent in a phase detection scheme that feeds back to frequency control. The corner frequency of both higher stages is 1.6 kHz, while the full bandwidth can reach ~9 kHz, limited by a resonance in the PZT/mirror mount combination in the femtosecond laser. The stabilised carrier-envelope offset frequency will not impact frequency comparisons until a frequency instability of 10-17 at 1 s is reached by the oscillators. Thus, the ability of the frequency comb to transfer frequency stability is limited by the control of the repetition rate.
5. Results and discussion
A series of heterodyne frequency measurements have been made pertaining to elements in the microwave-to-optical synthesis chain. These are illustrated in Fig. 3. The beat between the optical oscillator and the comb has been carried out for both OOSC1 and OOSC2 in separate experiments. This provides a consistency check or can inform us of possible differences in the behaviour of the two cavity stabilized lasers.
In the first case, an optical beat signal was generated by combining 2mW of 1064 nm radiation from OOSC1 with the appropriate part of the supercontinuum. The beat signal-to-noise ratio was ~30 dB in 300 kHz resolution bandwidth. A measure of the typical fractional frequency instability (square root Allan variance, SRAV) is shown as the upper green trace in Fig. 4, while the best recorded measurement is displayed as the dashed green trace. The red trace is the SRAV of the difference frequency between two MSOs (two separate sapphire resonators in the same dewar) and the grey trace is that for the beat signal between two optical oscillators (the blue trace is commented on below). Each is normalized by the corresponding carrier frequency. It is often the case that an assumption is made when mixing two oscillators that the performance is identical and hence that the single oscillator noise is 1/√2 times smaller (for uncorrelated noises). We have not applied any such factor because the weight of noise contributions to the microwave-to-optical (MtO) beat from the various oscillators can be different at different gate times. For the optical beats we subtract a quadratic fit from the raw data so that the short term behaviour can be properly examined without influence from the thermal drift at longer times. The performance of the cryogenic oscillators does not reach that of previously constructed oscillators  because of a lower Q of the sapphire crystal (the unloaded Q factors were 5.5×109 and 1.1×109 in the case of the best reported frequency instability measurements, compared to Q=2×108 here).
For integration times below 0.5 s the beat stability seems likely to be limited by the frequency comb. The femtosecond-laser suffers from considerable environmental noise between 300 Hz and 1 kHz (further discussion below). Between 0.5 s and 2 s the beat stability is limited by a combination of MSO instability and optical source instability. For integration times longer than a few seconds the rising instability is dominated by frequency variations of the optical oscillator, stemming from temperature fluctuations of the optical cavity. The kink in the OOSC1 vs OOSC2 trace at 0.2 s is due to an optical table resonance at 2.5 Hz. Note, the counter data for the MtO beat frequency and the beat between OOSC1 and OOSC2 were not recorded simultaneously in Fig. 4. The size of frequency variations over the longer term may change through the course of the day, hence the difference between the SRAV traces at the longer gate times (also see comments with respect to Fig. 5 below). To test whether the optical fibre link was imposing a limit on the frequency stability of the MtO beat note, light emitted from OOSC1 was sent through the fibre, retro-reflected and heterodyned with OOSC2. An SRAV comparison of the this beat signal and that of the direct beat between OOSC1 and OOSC2 showed negligible influence of the optical fibre link.
The size of the frequency fluctuations of both the microwave and optical oscillators is not entirely stationary, i.e., we see some variation in the instability of the MtO beat frequency over the day and from day-to-day. The reasons for this are not fully understood. The best measured MtO result dips under the microwave vs microwave and optical vs optical SRAVs. This is plausible if that the behaviour of the two optical oscillators and also that of the two microwave oscillators is not identical, i.e., to attain this measurement both the superior optical and superior microwave oscillator needed to be involved. Part of this conlusion is supported by the following. Another microwave-to-optical beat measurement was taken with the second optical oscillator (OOSC2). This second oscillator has a frequency doubling system incorporated, thus enabling heterodyning at 532 nm. The results of this experiment can be seen in Fig. 5.
A comparison between the results of Figs 4 and 5 show similar levels of microwave-to-optical beat instability, although not quite reaching the 1.3×10-14 level seen with OOSC1. Here there is a larger difference between the Allan deviation of the OOSC1 vs OOSC2 beat (grey) and that of the MtO beat signal (green) for gate times greater than a few seconds. OOSC1 is not involved in the MtO beat here, so this makes it evident that OOSC1 is not as well thermally isolated from its surroundings as OOSC2. Further evidence that the optical oscillator is limiting the SRAV is displayed in the time trace of beat signal frequencies of Fig. 6 (for integration times ≥0.5 s). Here, the mean frequency of each trace has been removed (apart from a small offset) leaving the residual frequency variations. A strong correspondence between the time traces of the optical vs optical beat frequency and the MtO beat frequency is seen, while the variations of the microwave oscillator are significantly less except at the shortest integration time, where the MSO could be having an influence on the frequency variations of the MtO beat signal.
To further investigate factors affecting the SRAV at short gate times, the phase spectral density (PSD) has been recorded for two of the oscillators involved. Fig. 7 shows the closed-loop and open-loop phase noise (traces a and b respectively) of fR,10GHz, while Fig. 8 displays the measurement arrangement. The closed loop PSD is a measure of the in-loop phase noise, ℒϕIL; essentially a measure of the failure of the repetition rate to follow the reference frequency. Trace c shows the phase noise of the cryogenic sapphire oscillator. The spikes at 39 kHz and 43 kHz are residual signals of the modulation used in the Pound servo on the sapphire oscillators; the cause of the remaining spikes is unknown. For frequencies up to 250 Hz the comb repetition rate will have the phase noise of the MSO imposed on it, while for frequencies beyond this, fR,10GHz will follow the in-loop phase noise of the blue trace. The plot is instructive, for it indicates that there is a not a great deal to be gained by increasing the bandwidth of the fR servo. Apart from avoiding the noise spikes in the MSO PSD, the repetition rate phase noise is known to continue falling ∝1/f 4, so it is better to retain the low intrinsic noise of the fs-laser at these higher Fourier frequencies. There is a slight discrepancy between the in-loop phase noise and that of the MSO at frequencies beyond 10 kHz. This may be attributed to the noise floor of the spectrum analyser used in the measurement of ℒϕIL.
To confirm the behaviour of the repetition rate servo, out-of-loop phase noise, ℒϕOL, has been measured and compared to the in-loop PSD. ℒϕIL is a measure of the phase noise at the mixer output preceding the loop filter (refer to Fig. 8), while ℒϕOL is the noise measured when the inline signal is held in 90° phase quadrature with another low noise microwave synthesizer using a separate mixer. ℒϕOL includes the effect of any spurious noise sources in the loop, such as AM to PM conversion in the photodetector or offsets in the mixer, and provides a more complete characterization of the noise floors of the repetition rate lock. The measurements, combined with the noise floor of the ℒϕOL measurement system, are shown in Fig. 9. The out-of-loop PSD (green trace) is masked by the noise floor of the out-of-loop noise measurement system for frequencies below 100 Hz. Beyond 100 Hz the in-loop and out-of loop phase noise are seen to be almost identical. Since ℒϕOL is below the phase noise of the MSO for f<200 Hz, we can be certain that the MSO phase noise is mapped onto fR in this frequency range.
In Figs. 4 and 5 the microwave-to-optical Allan deviation sometimes lies above that of the optical oscillators for short gate times. A possible reason for this is the residual noise at the acoustic frequencies (0.2 to 1.2 kHz) in fR. If we take ℒϕIL (blue trace in Figs. 7 and 9) to be the phase noise of the phase-locked 10th harmonic of the repetition rate (ignoring the MSO phase noise in this instance), then this can be transformed into an equivalent SRAV through the relationship:
where σy(τ) is the square root Allan variance, ℒϕ(f) is the phase spectral density on one side of the carrier, and fc is the carrier frequency, in this case 10GHz. The phase noise is scaled by the fourth power of sin(πfτ) as described by Barnes et al. . Eq.1 is implemented using a discrete summation method across each bin in the frequency range of the phase noise. The SRAV corresponding to ℒϕIL is shown in Figs. 4 and 5 sloping down at 1.2×10-14/τ. Hence, the noise at higher frequencies is seen to influence the signal stability at integration times near 1 s, since the phase noise in the 1-to-10 Hz region corresponds to σ<10-16. To reduce the acoustic noise two approaches can be taken: 1) modify the femtosecond laser to allow faster PZT actuation, but as previously discussed this will likely degrade the phase noise of fR at frequencies ≳10 kHz, or 2) passively reduce the acoustic vibrations coupling into the femtosecond laser.
We report fractional frequency instability measurements of 2.0×10-14 at 1 s for a beat signal between an optical oscillator and a frequency comb stabilised by a cryogenic microwave sapphire oscillator. As far as the authors are aware this is the lowest reported frequency instability of an optical signal whose stability is derived from an microwave source. Microwave signals synthesised from optical frequency references have attained instabilities of 3.5×10-15 at 1 s  (for a single oscillator), thus far superior to the microwave-to-optical approach. In our measurements the frequency instability appears to be dominated by a combination of optical oscillator instability and unsuppressed acoustic frequency noise in the comb for integration times up to 1 s. Beyond 1 s the frequency variations follow those of the optical oscillator. Continuing with the approach here, improvements in all three aspects of the measurement: optical oscillator stability, frequency comb stabilisation and microwave oscillator performance, are needed before a demonstration of 10-15 level microwave-to-optical frequency synthesis can be produced. It is worthwhile noting though, that the difficulties associated with generating an ultra-stable optical oscillator can be avoided by comparing the frequency instabilities of two frequency combs (in different spectral regions) controlled by independent cryogenic sapphire oscillators. This approach will also enable further investigations regarding the deliverance of microwave signal purity into the optical domain.
We have produced an optical frequency comb with frequency instabilities approaching those of ultra-stable cw laser systems. Such optical frequency synthesis provides an alternative means of carrying out precision spectroscopy measurements, in particular direct femtosecond laser comb spectroscopy.
The work at UWA has been supported by Australian Research Council. The authors thank G. Light’s team of technicians in the School of Physics mechanical workshop for their indespensible expertise. We thank other members of the FSM team: E. Ivanov, J. Anstie, A. Fowler, M. Marić and L. Nenadović for the loan of equipment and reviews of the manuscript. We are also grateful for the instructive comments of the reviewers.
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