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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 10 — May. 15, 2006
  • pp: 4316–4327
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Optical frequency synthesis from a cryogenic microwave sapphire oscillator

J. J. McFerran, S. T. Dawkins, P. L. Stanwix, M. E. Tobar, and A. N. Luiten  »View Author Affiliations


Optics Express, Vol. 14, Issue 10, pp. 4316-4327 (2006)
http://dx.doi.org/10.1364/OE.14.004316


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Abstract

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

1. Introduction

The UWA group have a record of developing cryogenic microwave sapphire oscillators with exceptional levels of frequency instability [22

22. A. N. Luiten, A. G. Mann, M. Costa, and D. G. Blair, “Power Stabilized Exceptionally High Stability Cryogenic Sapphire Resonator Oscillator,” IEEE Trans. Instrum. Meas. 44, 132–135 (1995). [CrossRef]

,23

23. S. Chang, A. G. Mann, and A. N. Luiten, “Improved cryogenic sapphire oscillator with exceptionally high frequency stability,” Electron. Lett. 36, 480–481 (2000). [CrossRef]

]; 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.

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.

Fig. 1. Frequency instability comparison between an optical and microwave oscillator. The repetition frequency, fR, of a mode-locked laser is controlled by a highly stable microwave sapphire oscillator (MSO). A Nd:YAG laser is locked to a high finesse optical cavity and a portion of its signal is combined with light from a MSF-broadened comb to produce a beat signal that is frequency counted. The offset frequency of the mode-locked laser is also stabilized. PD, photodetector.

3. Microwave and optical oscillators

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 [27

27. M. Bregant, G. Cantatore, F Della Valle, G Ruoso, and G Zavattini, “Frequency locking to a high-finesse Fabry-Perot cavity of a frequency doubled Nd:YAG laser used as the optical phase modulator,” Rev. Sci. Instrum. 73, 4142–4 (2002). [CrossRef]

]. 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 [28

28. J. J. McFerran, M. Maric, and A. N. Luiten, “Efficient detection and control of the offset frequency in a self-referencing optical frequency synthesizer,” App. Phys. B 79, 39–44, (2004). [CrossRef]

]. 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 [29

29. A. Bartels, T. Dekorsy, and H. Kurz, “Femtosecond Ti:sapphire ring laser with a 2 GHz repetition rate and its application in time-resolved spectroscopy,” Opt. Lett. 24, 996–8 (1999). [CrossRef]

]), 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. [28

28. J. J. McFerran, M. Maric, and A. N. Luiten, “Efficient detection and control of the offset frequency in a self-referencing optical frequency synthesizer,” App. Phys. B 79, 39–44, (2004). [CrossRef]

].

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.

Fig. 2. Control of the repetition frequency of the mode-locked laser by phase locking to a microwave sapphire oscillator. LF, loop filter; LPF, low pass filter.

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.

Fig. 3. Schematic of the oscillators involved in the frequency counting measurements. The mixer and photodetector colours correspond to the trace colours of Figs. 4 and 5. The dotted line is used to indicate that the optical mixing between the cw lasers and comb occur in separate experiments. C, counter.

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 [23

23. S. Chang, A. G. Mann, and A. N. Luiten, “Improved cryogenic sapphire oscillator with exceptionally high frequency stability,” Electron. Lett. 36, 480–481 (2000). [CrossRef]

] 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).

Fig. 4. Square-root Allan variance (SRAV) measurements. Green: optical beat between a cavity stabilized Nd:YAG laser (OOSC1) and an octave spanning comb whose repetition frequency is controlled by a cryogenic microwave sapphire oscillator (typical measurement). Red: beat between two MSOs. Grey: beat between two optical oscillators. Green dashed: best measurement of the microwave-to-optical frequency comparison. Blue: closed loop phase noise of fR,10GHz converted to a SRAV. Lines between data points are a guide to the eye only.

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.

Fig. 5. Square-root Allan variance measurements. Green: optical beat between the second harmonic of a cavity stabilized Nd:YAG laser (OOSC2) and the MSO stabilized frequency comb. Red: beat between two MSOs. Grey: beat between two optical oscillators. Blue: closed loop phase noise of fR ,10GHz converted to square root Allan variance.
Fig. 6. Time traces of three beat signal frequencies recorded simultaneously: (a) between the frequency comb and OOSC2, (b) between two optical oscillators, (c) between two microwave oscillators. The frequency variations for traces (b) and (c) have been scaled to 564 THz. The traces of the residual frequency variations have been suitably offset for reasons of clarity. The gate time is 0.5 s.

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.

Fig. 7. Single-sideband measures of phase noise of the 10th harmonic of the repetition rate and of the microwave sapphire oscillator (two MSOs): (a), fR,10GHz open loop ; (b), fR,10GHz in-loop; (c), MSO.

To confirm the behaviour of the repetition rate servo, out-of-loop phase noise, ϕOL, has been measured and compared to the in-loop PSD. ϕIL 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. ϕOL 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 ϕOL 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 ϕOL 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 ϕIL (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:

σy(τ)=2πfcτ0ϕ(f)sin4(πfτ)df
(1)

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. [30

30. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. 20, 105–120 (1971). [CrossRef]

]. 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.

Fig. 8. Points of measurement for in-loop, ℒϕIL, and out-of-loop, ℒϕOL, phase spectral densities.
Fig. 9. Phase noise measurement of the 10th harmonic of the repetition rate: (a), in-loop; (b), out-of-loop; (c), measurement noise floor for the out-of-loop phase-spectral density.

6. Conclusions

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 [16

16. A. Bartels, S.A. Diddams, C.W. Oates, G. Wilpers, J.C. Bergquist, W.H. Oskay, and L. Hollberg, “Femtosecond-laser-based synthesis of ultrastable microwave signals from optical frequency references,” Opt. Lett. 30, 667–9 (2005). [CrossRef] [PubMed]

] (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.

Acknowledgements

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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A.G. MannA.N. Luiten, “Ultrastable cryogenic microwave oscillators” in Frequency Measurement and Control: Advanced Techniques and Future Trends,ed. (Springer, Berlin, 2001), pp.37–64

27.

M. Bregant, G. Cantatore, F Della Valle, G Ruoso, and G Zavattini, “Frequency locking to a high-finesse Fabry-Perot cavity of a frequency doubled Nd:YAG laser used as the optical phase modulator,” Rev. Sci. Instrum. 73, 4142–4 (2002). [CrossRef]

28.

J. J. McFerran, M. Maric, and A. N. Luiten, “Efficient detection and control of the offset frequency in a self-referencing optical frequency synthesizer,” App. Phys. B 79, 39–44, (2004). [CrossRef]

29.

A. Bartels, T. Dekorsy, and H. Kurz, “Femtosecond Ti:sapphire ring laser with a 2 GHz repetition rate and its application in time-resolved spectroscopy,” Opt. Lett. 24, 996–8 (1999). [CrossRef]

30.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans. Instrum. Meas. 20, 105–120 (1971). [CrossRef]

OCIS Codes
(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing
(140.4050) Lasers and laser optics : Mode-locked lasers

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: February 24, 2006
Revised Manuscript: April 27, 2006
Manuscript Accepted: May 5, 2006
Published: May 15, 2006

Citation
J. J. McFerran, S. T. Dawkins, P. L. Stanwix, M. E. Tobar, and A. N. Luiten, "Optical frequency synthesis from a cryogenic microwave sapphire oscillator," Opt. Express 14, 4316-4327 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4316


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References

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  26. A.G. Mann, "Ultrastable cryogenic microwave oscillators" in Frequency Measurement and Control: Advanced Techniques and Future Trends, A.N. Luiten, ed. (Springer, Berlin, 2001), pp.37-64
  27. M. Bregant, G. Cantatore, F, Della Valle, G, Ruoso and G, Zavattini, "Frequency locking to a high-finesse Fabry-Perot cavity of a frequency doubled Nd:YAG laser used as the optical phase modulator," Rev. Sci. Instrum. 73, 4142-4 (2002). [CrossRef]
  28. J. J. McFerran, M. Maric and A. N. Luiten, "Efficient detection and control of the offset frequency in a selfreferencing optical frequency synthesizer," App. Phys. B 79, 39-44, (2004). [CrossRef]
  29. A. Bartels, T. Dekorsy and H. Kurz, "Femtosecond Ti:sapphire ring laser with a 2 GHz repetition rate and its application in time-resolved spectroscopy," Opt. Lett. 24, 996-8 (1999). [CrossRef]
  30. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, Jr.,W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971). [CrossRef]

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