1. Introduction

Mode-locked fibre-based lasers are relative newcomers to the field of metrological frequency combs. In many applications their key advantages of lower cost, portability and long term operation are of far higher importance than the potential for superior performance that is offered by solid-state based frequency combs [1]. This is especially evident when only commercial frequency references are available, in which case frequency comb precision beyond the level of 10^-13√τ is unnecessary (where τ is the averaging time of the measurement). In these cases fibre laser solutions can absolutely excel. Reviews of developments in laser frequency combs can be found in [2 , 3, 4, 5, 6], while examples of recent advances in fibre based systems can be found in [7, 8]. In this paper we present a new simplified technique for stabilising the frequency comb generated by a standard commercial mode-locked laser fibre laser [9]. The stabilisation performance is sufficiently good that it is capable of providing frequency measurement and calibration that is beyond the performance of any commercial microwave or laser frequency standard. In other words, in all but a few very exceptional measurements in national standards laboratories, the limitations imposed by any measurement will not be those associated with the fibre laser system, but will instead arise from the limitations of the frequency references used by the system.

2. Stabilization scheme

The stabilization scheme stabilises both the repetition rate and offset frequencies and is based on mixing the tripled comb of a mode-locked Erbium doped fiber laser with another comb generated via solitonic process in a specialised fiber [10]. The fibre laser delivers a 100 fs duration pulse at 1560 nm with 94 MHz repetition rate at an average power of approximately 250 mW. The repetition rate has been stabilised by detecting its 100th harmonic on a very fast photodiode and referencing to a stable microwave signal derived from a Hydrogen maser. The 1560 nm output of the laser is then passed through a periodically-poled lithium niobate crystal which generates a second harmonic comb of up to 90 mW at 780 nm. The nonlinear crystal also incidentally produces a sum frequency product between the 780 nm light and the original 1560 nm, giving rise to a 200μW third harmonic green comb at 520 nm. The approach we have taken to stabilise the offset frequency of the comb is to combine this incidental third harmonic comb with another comb of the same wavelength generated by frequency shifting the second harmonic comb in a specialised optical fibre. This 2f – 3f approach simplifies the offset frequency stabilisation by eliminating a non-linear crystal after the optical fibre and can also operate with a much smaller length of fibre and lower power frequency combs.

 

Fig. 1. A micrograph of the Kagome fibre in cross section. In this experiment, we couple 780 nm light into any of the six equivalent intersections adjacent to the core (one of which has been labelled A), resulting in a comb with maximum intensity at 520 nm (green). If we couple into any of the intersection points B, we obtain a comb with a maximum at 570 nm (yellow). Similarly, launching into any of the points C results in 480 nm light (blue).

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Frequency shifting of the second harmonic comb is based around the generation of a resonant dispersive wave in special intersection points in a Kagome microstructured fibre [10]. A photograph of the end face of the Kagome structure is shown in Fig. 1 where we have marked the three independent intersection sites between the struts in the photonic crystal surrounding the hollow-core. For light propagating in these intersections the peculiar dispersion of the intersection results in the generation of two radiative waves that are symmetrically spaced either side of the pump wave in frequency (similar to degenerate four-wave mixing). In this particular experiment we make use of the intersections closest to the core which generate radiation at 520 nm together with conjugate radiation around 1620nm [10]. This 520 nm radiation fortuitously overlaps the wavelength of the third harmonic, as is shown in Fig. 2. All intersections around the core produce the same spectrum, which is characteristic of the dispersion of this particular intersection [10]. The exact wavelength of the output radiation is only weakly dependent on the input power (it can be shifted 5 nm when changing the input power from 30 mW to 60 mW due to a small nonlinear contribution to the dispersion) as is to be expected [10]. Shifts of around 10 nm in the output spectrum can also be achieved by manipulating the polarisation of the input light. We are currently investigating ways to manufacture tailored fibre that will suit applications requiring different output wavelengths.

Fig. 3 shows the layout of our experiment. The third harmonic comb is separated from the fundamental and second harmonic comb and then temporally delayed so that its pulses will arrive at the photodetector at the same time as the lower wavelength component of the spectrally shifted pulses emitted by the Kagome fibre. The interference of these two pulses on the detector produces a beat note at the offset frequency of the comb, which follows from the following analysis. If the frequency of the n^th mode of the fundamental 1560 nm comb is written as

then the modes of the 780 nm second harmonic and 520 nm third harmonic comb modes can be likewise written as:

A theoretical treatment of femtosecond pulse propagation in media with dominant fourth-order dispersion predicts that beyond a certain critical intensity the pump soliton will be unstable. In this situation the soliton will radiate two dispersive waves which are generated by coherent amplification of two narrow wavelength ranges in the spectral wings of the pump soliton [11, 12]. The dispersive waves are symmetrically spaced from the pump wavelength by an amount determined by the magnitude of the fourth-order dispersion [10, 11, 12]. In our case we can thus express the frequencies of the modes of the two dispersive waves (at 520 nm and 1620 nm) in a similar form to that of the input 780 nm pump soliton as;

 

Fig. 2. Three spectra showing the second harmonic comb of the laser (grey-heavy solid) with a peak at 780 nm; the spectrum from the Kagome fibre (red-dashed) when light is launched into the intersection adjacent to the core (labelled A in Fig. 1); the third harmonic comb (blue-solid) generated incidentally in the nonlinear crystal.

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The difference frequency generated by interference of the 520 nm third harmonic comb and the frequency shifted 780 nm comb on an avalanche photo-diode will contain the following frequencies;

After stabilising the repetition rate detected on a separate photodiode, the beat note on the avalanche photodiode will be stabilised by holding f_ceo constant. This stabilises all four frequency combs (ie. the 520 nm third harmonic, 780 nm second harmonic, 1550 nm fundamental and 1620 nm congugate wave comb) providing a useful source for a variety of metrological purposes. We shall detail the locking procedures for both the repetition rate and offset in the next two sections.

3. Repetition rate stabilisation

To measure fluctuations of the pulse repetition rate, a small portion of the output of the mode-locked laser falls upon a fast detector, producing a large number of harmonics of the repetition rate of 94 MHz. This signal is mixed with a high performance synthesiser (Agilent E8257D locked to a H-maser) such that a signal proportional to the relative phase can be generated. We select out the highest possible harmonic consistent with a high signal to noise, which in our case is one hundred times (9.4 GHz) that of the repetition rate, limited by the bandwidth of the detector.

We used the PZT element internal to the commercial laser unit to vary the length of the free space section to control the repetition rate. In order to optimise the control system we must characterise its complex transfer function. Our approach is to tightly phase-lock a microwave synthesiser to the repetition-rate of the laser using the frequency-modulation port of the synthesiser. We monitor the phase and amplitude of the correction signal fed back to the synthesiser with a Spectrum Analyser as we apply a swept-sine input to the PZT port of the laser.

 

Fig. 3. The experimental layout showing how the third harmonic comb from the laser is separated from the fundamental and second harmonic comb with dichroic mirrrors and temporally delayed so that its pulses will arrive at the photodetector at the same time as the lower wavelength component of the spectrally shifted pulses emitted by the Kagome fibre. The interference of these two pulses on the detector produces a beat note at the offset frequency of the comb. The inset shows the beat-note as it appears at the photodiode. The feedback paths for the repetition rate and offset frequency detection are also shown.

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Fig. 4. Amplitude transfer functions of (left) the PZT used to control the repetition rate, and (right) the pump laser current modulation used to control the offset frequency.

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The resulting transfer function is flat up to a few hundred Hertz, with resonances in the few kilohertz region, as shown in Fig. 4. We design a low pass filter to avoid these resonances yet maximising the bandwidth of the control system. Along with suitable amplification stage (including an integrator) this system was used to stabilise the repetition rate relative to the H-maser (which has a frequency stability of the order of ~ 2 ×10^-13/√τ). The effective bandwidth of the repetition rate control loop is 1 kHz with a breakpoint at 100 Hz to give a net gain at 1 Hz of 100 dB.

To measure the frequency stability of the stabilised comb we mix the photodiode output with a second synthesiser (also phase locked to the H-maser), generating a difference signal in the optimal range for a frequency counter (around 5 kHz). We then log the counter at a range of integration times and calculate the Square Root Allan Variance (SRAV). It is neccessary to use a range of different gate times because in this circumstance the residual noise of the phase-lock is dominated by white phase-noise and it is inappropriate to use a juxtaposition of shorter gate times to synthesise the longer gate times [13]. Using this technique we display both the free-running repetition rate and the stabilised repetition rate on Fig. 5.

 

Fig. 5. Square root Allan variance of repetition rate. Crosses represent the free running repetition rate and circles the noise floor of the measurement system. The measured repetition rate was indistinguishable from this noise floor; triangles represent the inferred repetition rate stability using the uncertainty values in the measurement system noise floor.

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The locked repetition rate SRAV is indistinguishable from the measurement system noise floor (Fig. 5.) Using uncertainty values in the measurement system noise floor we calculate an upper limit for the repetition rate of 10^-14√τ, which is better than that presented in [14] where two combs locked to a single RF reference are compared. To our knowledge this is the best reported stabilisation of a fibre comb to an RF frequency reference. The superb frequency stabilisation reported in [15] of 6 × 10^-17 τ ^0.6 was based on the stabilisation of the comb to an optical frequency reference. The quality of the frequency stabilisation can be much higher as an optical phase comparison is less subject to the deleterious effects of electronic noise compared to an RF phase comparison. Nonetheless, in most circumstances an optical frequency standard will not be available and the comb will need to be tightly stabilised to an RF reference as we have demonstrated here.

4. Offset frequency stabilisation

As outlined earlier, we use Kagome micro-structured fibre to create a spectrally shifted portion of the comb for offset frequency detection. This process requires less input power or allows a shorter fibre than conventional octave supercontinuum generation. This is a significant advantage when stabilizing low average output power mode-locked fibre lasers. The laser in this case generates 780 nm pulses with a peak power of 2.4 kW which, using 50 cm of commercial super-continuum fibre, produced a spectrum from 570 nm up to approximately 1050 nm (half-power points). We found that it was not possible to generate a high-level of power over a full octave with this approach meaning that a conventional f – 2f offset frequency stabilisation technique was not easily possible. The use of the dispersive-wave to frequency shift the input soliton, however, concentrates all the optical power into two regions, 520 nm and a conjugate wave at 1620 nm; we obtain 250μW of 520 nm from 60 mW of 780 nm. This means that a good fraction of the output power of the fibre can be usefully deployed in producing a strong beat-note. The prior work of Hong et al [16] made use of 50 cm of commercial fibre to generate just over an octave of broadening (-20 dB level) with a similar level of input power at 780 nm. Examination of their Fig. 1 shows that the useful power in their comb (that which generates the offset frequency beat-note) is about the same as in our 10 cm of fibre.

 

Fig. 6. SRAV of unlocked (filled squares) and locked (hollow circles) offset frequency of the fibre laser, referred to a mode at 500 THz.

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The Kagome fibre temporally broadens the pulse, so the fibre must be kept as short as possible in order to have a good temporal overlap between the third harmonic and the output of the fibre. We found that a fibre length of approximately 100 mm is optimal for the signal-to-noise: this length broadens the 100 fs pulse to 300 fs in duration at the operating input power. At longer lengths (approximately 1 m) the output pulse power is doubled, but is more than 1 ps in duration, resulting in a beat signal that is significantly smaller. The power of broadened light from the Kagome fibre is extremely sensitive to the coupled light; a 2 dB reduction in input power translated into a 50 dB reduction in broadened output for powers less than 25 mW. Above this level the output power was saturated at a near fixed level. Care was taken to maximise incoupled power to the intersections points of the Kagome fibre, requiring a proper filling of an objective lens with a numerical aperture of 0.6 to produce a spot size of 2μm. A half wave plate at the input to the objective aligns the beam’s polarisation relative to the structure of the fibre for optimal incoupling, as the spectral shifting process is only effective for a particular polarisation [10]. With careful adjustments we can achieve excellent spectral overlap between the third harmonic and the 520 nm light from the Kagome fibre.

An optical bandpass filter at 520 nm is then used to select out only those portions of the third harmonic and the spectrally shifted light that overlap in wavelength. When the delay arm is set correctly, we obtained a f_ceo beat signal that is 25 dB above the noise floor, with a measurement system bandwidth of 30 kHz. This is then passed through a sequence of frequency division stages (1/80) to eliminate the amplitude noise and reduce the phase noise, which eases the task of achieving stable phase lock. The existing level of signal-to-noise is limited by the amount of power present in the third harmonic arm of the interferometer. We have measured the free running stability of the offset frequency with a frequency counter and this is shown on Fig. 6.

In order to suppress fluctuations in the offset frequency we modulate the laser pump current. As with the repetition rate control it is important to accurately characterise the transfer function to build an optimal feedback network. We do this in a manner similar to that outlined for the PZT, by phase-locking a synthesiser to the offset frequency and measuring a transfer function as shown in Fig. 4. The transfer function shows a low pass filter characteristic with a breakpoint of order 50 kHz. This value is set by the stimulated transition rates of the heavily saturated gain material as well as other factors relating to the mode-locking pulse parameters of the laser [17].

We amplify the offset signal by 60 dB and mix it with an RF synthesiser to produce a signal proportional to the phase difference. A suitable feedback filter is designed (taking into account the transfer function of the control system) and is then fed into the pump current modulation port of the laser. The locked Allan Frequency Deviation of the offset frequency is stable to 20 Hz over 1 second. This is displayed as a fraction of 500 THz in Fig. 6 to reflect its contribution to optical mode instability. The residual frequency stability after the stabilisation is dominated by discrete steps associated with a fault in an optical amplifier which caused the power of the system to step up and down randomly at approximately 10 seconds intervals. On each occasion that the power changes we observe several cycle slips in the control system. If we remove these unwanted features the SRAV has the same character as that shown for the locked curved Fig. 6 but is 1000 times lower (4 × 10^-17√τ). Nonetheless, even with this unwanted behaviour the performance of the offset frequency stabilisation is still better than the performance of any commercially available microwave or optical frequency standard and thus would not be a limiting feature in almost all circumstances.

A previous attempt at 2f – 3f offset frequency detection was made using 50 cm of commerically available photonic crystal fibre [16]. In that experiment, the high level of frequency noise on the f_ceo signal, and the relatively poor signal to noise of the detected beat note, meant that it was not possible to actively stabilise the f_ceo signal. Furthermore, those authors found that there was a high level of frequency noise generated by the fibre itself. In contrast, our approach does not seem to suffer from this high level of fibre noise and using the technique presented here we were able to obtain a high signal to noise f_ceo signal and thereby actively stabilise the offset frequency. It is perhaps the shorter fibre used in this experiment that allows us to avoid the deleterious fibre noise effects seen in their experiment.

5. Conclusion

Using a novel micro-structured fibre we have generated a coherent, frequency-shifted comb requiring less power than conventional supercontinuum generation. We use this frequency-shifted comb to produce an offset frequency signal, with which we then stabilise the frequency of all modes of the frequency comb. By judicious and careful design of the frequency control networks along with high frequency detection of the repetition rate signal we were able to suppress the frequency noise of the optical modes to a level where any optical frequency measurements will, in all but the most exceptional circumstances, be limited by the performance of the frequency references used by the system. This fibre-based 2f – 3f configuration is able to stay locked for long periods (over one week), enabling long-term, low-noise comparisons between clocks. Crucially, it is relatively inexpensive, robust and transportable, making it useful for fieldwork applications requiring precision in timekeeping, such as that of navigation, astronomy, and telecommunications.