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Remote transfer of an ultralow-jitter microwave frequency reference signal is demonstrated using the pulse trains generated by a mode-locked fiber laser. The timing jitter in a ~ 30-m fiber link is reduced to 38 attoseconds (as) integrated over a bandwidth from 1 Hz to 10 MHz via active stabilization which represents a significant improvement over previously reported jitter performance. Our approach uses an all-optical generation of the synchronization error signal and an accompanying out-of-loop optical detection technique to verify the jitter performance.
©2006 Optical Society of America
1. Introduction
The traditional method for transferring frequency standards uses the Global Positioning System (GPS) [1]. Long averaging times (~days) are necessary in this approach to average out environmental fluctuations in signal paths from satellites. However, these GPS-based systems will not be able to leverage the high-stability, low-averaging times of next generation optical atomic clocks [2],[3]. A good candidate for remote distribution of frequency references derived from optical atomic clocks is an optical fiber link. Such transmission facilitates the comparisons of relative instability and systematic drifts between different frequency standards. There are also a number of applications that would benefit from the transmission of ultra-low jitter timing signals, including gravity wave searches, occultation science [4] and the distribution of timing signals throughout a linear accelerator facility used to produce ultrashort X-ray pulses for time-resolved pump-probe experiments [5]. More specific applications can be found in Ref. [6]. Previous work [7] has utilized techniques based on microwave detection in order to detect and cancel the noise added during transit down the distribution fiber. However, while working in the microwave domain several sources of noise conspire to limit the achievable jitter to approximately 10 fs including Johnson noise, shot noise, amplifier flicker (1/f) noise and amplitude-to-phase conversion in the detection, amplification, and mixing of the signals. A thorough discussion of the technical noise limits can be found in Ref. [6].
Some of the limits inherent in using microwave detection can be circumvented by turning to all-optical detection for generation of the synchronization error signal. This is in part due to the huge lever arm available when optical detection is utilized. A balanced optical crosscorrelation setup was used to derive a dispersion-type error signal for local synchronization of a fs Ti:Sapphire laser with a fs Cr:LiSAF laser [8]. Alternatively, a narrow-linewidth cw laser (stabilized to a high-finesse optical cavity) was employed at NIST to optically synchronize two local fs Ti:Sapphire lasers [9]. While in principle both of these techniques could be used to derive an error signal for remote synchronization, their implementation in this setting would be rather complex. In the former case, near-perfect chirp compensation of the transmitted pulse would be required. The latter case would require stabilized transmission of an optical frequency standard [10].
In this manuscript we present a different approach to all-optical synchronization which is more suitable to applications involving remote synchronization. This technique is also a simple and robust method to synchronize two local femtosecond (fs) lasers as long as there is good spectral overlap between them.
2. Theory
In essence, this approach is a phase locking of two frequencies using an error signal that is generated in the optical domain to obtain extremely high sensitivity. This method is best described in the frequency domain where a fs pulse train can be described as a frequency comb [11]. In this case, when ignoring the spectral phase, the electric fields of two pulse trains (or frequency combs) can be expressed as,
where the index of the comb ranges from n′1 to n′2 (m′1 to m′2) and quantifies its bandwidth, A n (A m) is the amplitude of the corresponding frequency component, f rep1 (f rep2) is the repetition frequency of the pulse train and f o1 (f o2) is the offset frequency. In our case of remote transfer Ẽ 1 is the master comb and Ẽ 2 is the transmitted comb that will be slaved to the master.
In this new all-optical approach, an optical heterodyne beat is formed between these two combs. Following some signal processing, an error signal is generated that has a much higher degree of sensitivity as compared to typical microwave detection schemes. To see how this higher sensitivity is realized, consider the optical heterodyne beat between these two frequency combs as detected by a photodiode. The photo current will be proportional to |Ẽ 1+Ẽ 2|2, which can be expanded using Eq. (1) as,
where, in general, a subset of the optical frequency comb elements (|n 2-n 1| and (|m 2-m 1|) are detected and we have assumed all comb element amplitudes (A n, A m) to be equal as only the generated frequencies are of concern at the moment. In Eq. (2) the first two terms are the standard signals detected when the pulse trains are detected individually; they are frequency combs that are converted down to base-band via the photodetection process. The third term is the interference term which holds the key to the increased sensitivity. Due to propagation through the optical fiber, the second comb has some noise relative to the first comb, both in the comb element separation, f rep2=f rep1+δf r and in the offset frequency f o2=f o1-δf o. In order to transfer an rf frequency standard, it is the repetition rate noise, δf r, that must be detected and corrected. With these substitutions, the photocurrent becomes,
It is helpful to look graphically at the spectrum resulting from these three terms. Referring to Fig. 1, the 1st term of Eq. (3) is a comb of delta functions with elements spaced at f rep1. The 2nd term is a similar comb except that it is shifted from f rep by the noise term δf r times the harmonic number. The 3rd term is a set of sidebands spaced around f rep1 at +[(m+1)δf r-δf o] and -[(m-1)δf r-δf o]. These sidebands scale with the optical comb index m which is on the order of 106 (thus m+1≈m). This characteristic is the optically leveraged noise sidebands that provides much greater sensitivity. As the photocurrent signal has all three of these combs, some processing of signals is necessary to extract the optically leveraged mδf r terms. In this work, two different schemes are pursued for this requisite processing. In both approaches, the heterodyne beat is detected around a harmonic of f rep1 rather than baseband to avoid 1/f and other noise processes at DC.
Fig. 1. A graphical representation of the three terms in Eq. (3) in the frequency domain. (a) 1st term, which shows the local pulse train; (b) 2nd term, which shows the transmitted pulse train (δf r is exaggerated for clarity); (c) 3rd term, which shows the interference between the two pulse trains, where m+1≈m
Fig. 2. Experimental setup of optical heterodyne jitter cancellation and interferometric cross-correlation (ICC) measurement. Scheme 1 uses optical heterodyne detection at spectral extremes, while scheme 2 is single optical heterodyne. EDFL, mode-locked Erbiumdoped fiber laser; BS, beam splitter; DCF, dispersion compensating fiber; SMF-28, single mode fiber; PZT, piezoelectric transducer; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; FPD, free-space photodiode; CPD, fiber-coupled photodiode; BPF, band-pass filter; Amp, radio frequency amplifier; OSA, optical spectrum analyzer; PLL, phase-locked loop; APD, avalanche photodiode.
In scheme 1 shown in Fig. 2, the two fs pulse trains that are to be synchronized are combined collinearly and then two optical heterodyne beats are detected at different optical spectral regions. This arrangement produces two signals of the form of Eq. (3) with the important difference between them that the optical index in the third term is different (denoted by q and p below) and depends on the range of the optical detection wavelength of each signal. Next, the first harmonic of each heterodyne beat is bandpass filtered at f rep1 and electronically mixed together. At the mixer output, (after low-pass filtering), a set of terms at baseband composed of (q-p)δf r, (q+p)δf r, (q)δf r, and pδf r terms are present. As the relative phase of the noise sidebands in Eq. (3) is zero, these terms will add constructively at baseband when mixed. This processing procedure is shown graphically in the frequency domain in Fig. 3(a)–(c). The relative magnitude of these terms depends on a number of factors such as frequency chirp and comb element amplitudes. At a minimum, the noise sidebands are multiplied by a factor of q-p. The difference between the indices q and p can be as high as ~105 for reasonable separation (~50 nm), resulting in a corresponding 105 increase in sensitivity. The output of the mixer is then used as the error signal for a servo (see Fig. 2), which controls a dynamic delay line and actively compensates the noise imprinted on the transmitted comb by propagation down the fiber link. Because the mixer output is used directly as the error signal, a nearly noise-free amplification of 105 of the error signal is achieved. This approach is similar in motivation to using a high microwave harmonic (such as the 100th) when individually detecting each pulse train for conventional synchronization [12]. In this case we are effectively detecting at the ~105 harmonic. An advantage this approach has over any time-domain based correlation technique is that perfect chirp compensation is not necessary as the error signal is not derived from the cross-correlation overall envelope but rather is a frequency domain technique. However, it is helpful to match the chirp (as close as possible) in the two signals as this enables more comb lines to simultaneously contribute to the overall error signal thereby increasing its signal to noise.
Fig. 3. A frequency domain diagram showing the mixing procedures to obtain the optically leveraged noise sidebands by scheme 1 & 2. Optical heterodyne beat detected (a) at the low optical frequency region and (b) at the high optical frequency region in scheme 1; (c) Optically leveraged noise sidebands at baseband produced at the output of mixer in scheme 1; (d) Detected harmonic of repetition rate in scheme 2; (e) Detected optical heterodyne beat in scheme 2; (f) Optically leveraged noise sidebands at baseband produced the output of mixer in scheme 2.
In the application of delivery of stabilized frequency standards to remote locations, the transmitted and reference combs will have the same offset frequencies as long as there is not a rapid, time-varying change in the dispersion of the fiber link – such a condition will cause f o1≠f o2 even though both combs are generated from the same laser. In most experimental setups, including ours, the time-rate of change of the fiber link dispersion will not be an issue (it is too slow to change f o by any noticeable amount. Under this condition, a slightly simpler processing approach can be used. As before, the goal is to extract the optically leveraged noise sideband terms mδf r at baseband for an error signal. In this case a single optical heterodyne beat is formed as shown in Fig. 2 and the resulting photocurrent is given by Eq. (3). With the offset frequencies equal, δf o=0, the noise sidebands are located symmetrically around f rep1. By mixing this heterodyne beat signal with f rep1 and low-pass filtering the result, the signal at baseband will now have terms of pδf r. This mixing procedure is shown in Fig. 3(d)–(f) Similar to scheme 1, when this signal is used as an error signal for a phase-locked loop, the term pδf r provides a increase sensitivity by a factor of 106.
3. Experimental demonstration
We have experimentally demonstrated both of these all-optical error signal generation techniques by detecting and canceling the noise added when a pulse train from a mode-locked fiber laser is propagated through a ~60-m fiber. In an actual implementation of remote synchronization, a portion of the delivered signal would be retro-reflected back and compared with the reference signal. The noise during a one-way transit is assumed to be equal to twice the round-trip and the error signal is simply divided by 2 [10]. For experimental convenience we stabilized the round-trip signal, which is equivalent to a remote transfer of 30-m. For scheme 1 the spectral separation was 50 nm. With this spectral separation (~6 THz) and the repetition rate of the lasers (50 MHz), our implementation generates an error signal that is a weighted sum centered at approximately the 125,000th harmonic. The specific setup is shown in Fig. 2. The rejection port (chirped) output from an mode-locked erbium-doped fiber laser [13] is split into two paths. One portion is saved for the reference signal and passed through a 83 cm of SMF-28 fiber to produce a compressed pulse width of 154 fs at full-width half-maximum (FWHM). The other portion is pre-compensated with 7.9 m of dispersion compensating fiber before being launched into a 52 m spool of standard telecom fiber. Both the reference and transmitted pulse trains were compressed in an effort to match their relative chirp and increase the overall signal-to-noise ratio. These two signals are then collinearly combined and then passed through a 50/50 beam splitter (BS). One output of the beam splitter is used for measurement of the jitter (discussed below). The second output is reflected off a grating (600 lines/mm) and the optical heterodyne beat is detected at 1523 nm and 1573 nm with a bandwidth of ~1.8 nm at each spectral extreme. Each heterodyne beat is filtered at the third harmonic and then mixed together to produce the spectral leveraged error signal for synchronization. A phase-locked loop (PLL) is then utilized to pre-cancel the noise by actuating two free-space PZTs, a long travel (10 µm) PZT with a bandwidth of 100 Hz and a short travel (1 µm) PZT with a bandwidth of 25 kHz.
In the optical heterodyne approach (denoted scheme 2), the collinear signals are directed toward a single photodetector. As indicated in Fig. 2, to generate the high-sensitivity error signal, the resulting optical heterodyne beat is mixed with a f rep1 microwave signal from the reference to avoid noise near DC. Then, similar to scheme 1, with this error signal a PLL is used to pre-cancel the noise.
Figure 4 shows the pulses’ spectra of the reference and transmission paths, the received signals at each spectral extreme and a stable interference pattern of the combined signal (when the PLL is activated for scheme 1). The high frequency interference fringes on all of the traces are due to Fabry-Perot effects within a glass slide (which wasn’t anti-reflection coated) used to sample the various signals.
4. Measurement of jitter
While the stable interference pattern shown in Fig. 4 on the combined signal indicates that the transmitted pulse train is stable with respect to the reference, this measurement technique is not able to quantify the jitter. In practice, one of the methods for characterizing the synchronization stability is to measure the phase noise of the error signal which reports how much the delay between two synchronized signals is jittering in phase [7],[14]. The phase noise is related to the timing jitter through Eqn. (4),
Fig. 4. The green and purple curves show the spectra of reference and transmission paths, respectively. The self-phase modulation due to transmission through the ~60-m fiber results in the slightly wider spectrum in the transmission arm with respect to the reference arm. The blue and red curves are the spectra of the received signals at two spectral extremes. Each spectral slice has a bandwidth of ~1.8 nm. The black curve is the stable interference pattern of combined signal when PLL is activated.
More specifically, the jitter spectral density δT̃(f) of the error signal represents the rms timing jitter at each frequency in a 1 Hz measurement bandwidth and is proportional to the rms phase fluctuation δ(f), where Ã(f) is the measured error signal as a function of frequency, A 0 is the half peak of the unlocked sinusoidal error signal and ν 0 is the nominal center frequency of the reference and transmission signals. In order to calculate the total rms timing jitter (T rms), δT̃(f) is integrated over a frequency range,
However, in our case measuring the in-loop jitter via the error signal is not straight forward. In scheme 1, the error signal contains many different terms (of the weighted sum of k-p) each with its own effective center carrier detection frequency (ν 0 in Eqn. (4)). A worst case estimate of the in-loop jitter can be made using the minimum separation of 48 nm to make ν 0=5.99 Thz. Using this value of ν 0 as the overall carrier frequency and following the prescription of Eqn. (5), a worst case in-loop jitter of 38 as (1 Hz to 10 MHz) is estimated. For scheme 2, the error signal also contains many terms. In this case it is summed over the k index (i.e., the spectral width of the pulses) and as before each term has its own ν 0. A worst case estimate would choose ν 0=197 THz (at 1530 nm, the low end of the pulses’ spectra), resulting in an in-loop jitter of 4.2 as (1 Hz to 10 MHz).
An independent, out-of-loop measurement of the jitter was made to confirm this performance by an optical cross-correlation between the reference and transmitted pulse trains. Previously, a non-collinear intensity cross-correlation between the two pulse trains was made by overlapping the pulse trains in a nonlinear crystal and detecting the sum-frequency signal in a photomultiplier tube (PMT) [12]. However, an interferometric (collinear) cross-correlation measurement can provide an even higher degree of sensitivity. In this case the intensity cross correlation (ICC) oscillates as a function of delay between the two pulses at a frequency equal to the pulses’ optical carrier frequency [15]. Thus in Eqn. (4), ν 0 is on the order of ~200 THz leading to a substantially increased sensitivity for the jitter measurement. A further advantage of this measurement method over the non-collinear cross-correlation is that the two pulses need not have perfect chirp compensation as the overall slope of the correlation envelope is unimportant. From the second port of the polarization insensitive beam splitter used to combine the two signals for error signal generation (see Figs. 2), these two signals (reference and transmitted pulse trains) are polarization de-multiplexed to place an adjustable time delay between them. These signals are then collinearly recombined and an interferometric cross-correlation is performed between them via two photon absorption (TPA) in a Si-APD [16]. The jitter measured by the ICC is caused by two factors. One of them is the jitter in envelope arrival times of two pulse trains [15] while the other is the change in the difference of the carrier-envelope-offset frequencies of the two pulse trains [17]. But in our case where the two combs are derived from the same laser, the latter factor can be neglected as it would have to be caused by a rapid time-varying change in the fiber dispersion (which is not the case in the present experiment).
A few representative time domain traces from the APD as recorded on a digital oscilloscope are shown in Fig. 5. The unlocked signal is shown in red and its amplitude is normalized to 1. At t=0, the pulses are overlapped and then drift apart due to environmental perturbations in the 60-m optical fiber path length. In this signal, a single oscillatory cycle represents a drift of 5.17 fs (one period of an optical cycle at 1550 nm). As the pulses drift away from maximal overlap, the (nonlinear) TPA signal goes down, leading to a reduction in the peak amplitude after every cycle. When the PLL is activated using spectral leveraging (scheme 1), the measured ICC signal is shown by the blue and green traces corresponding to two difference time scales. In both of these traces the initial timing offset is adjusted to approximately the half intensity level of the ICC signal for maximum sensitivity. It is clear from these traces that a tight lock is established.
A frequency domain measurement of the ICC signal followed by the analysis shown by Eqn. (4) and (5) provides the jitter spectral density and total rms jitter measured out-of-loop. The results are shown in Fig. 6 for both scheme 1 and 2. Also shown is the noise floor of the APD. The corresponding total rms timing jitter is shown in Fig. 6 by the identical color but dashed curve. An rms timing jitter of 11.4 as (1 Hz to 10 MHz) is measured when locked by spectral leverage setup (scheme 1), while it is 12.1 as (1 Hz to 10 MHz) when locked by optical heterodyne setup (scheme 2). The noise floor of APD (blue curve in Fig. 6) is 9.7 as (1 Hz to 10 MHz). It’s easy to see both schemes cancel the noise almost to the noise floor of the ICC measurement itself. Moreover, these out-of-loop measurements of the jitter compare well with the measured in-loop measurements reported earlier. As noted previously, the ICC jitter measurement contains both the envelope arrival time jitter and the relative carrier envelope phase jitter. While the latter is most likely zero in our case, it is possible that carrier phase jitter could artificially reduce our jitter measurement. Accordingly, we will quote an upper limit of sub-40 as, estimated from our in-loop measurement for the jitter performance. It is also worth noting that instead of using the ICC to independently measure the jitter, the ICC signal itself could be employed to generate an (all-optical) error signal with a sensitivity and noise floor comparable to schemes 1 and 2.
To illustrate the improvement offered by our all-optical approaches both in the detection and error signal generation, we also locked the same 60-m fiber link with the standard microwave approach. Similar to Ref. [7], the transmitted and reference pulse signals were individually detected at 1.822 GHz (the 37th harmonic). The resulting signals were mixed in the microwave domain to generate an error signal. The jitter spectral density under various conditions is shown in Fig. 7. Note that the upper frequency limit in Fig. 7 is only 100 kHz. The red curve shows the measured, unlocked jitter spectral density, which yields 760 fs (1 Hz to 100 kHz). When this microwave generated error signal is used in conjunction with the PLL, the locked (in-loop) jitter spectral density (purple curve) is measured, giving a rms jitter of 28.5 fs (1 Hz to 100 kHz). The fact that the locked signal is below the noise floor at certain frequencies indicates an outof-loop measurement is required to truly measure the jitter. Nevertheless, when these curves are compared with the noise floor of the ICC signal, (green curve, re-plotted from Fig. 6) the vast improvement in the sensitivity of the jitter measurement by all-optical means is clear. The comparison between the two techniques in Fig. 7 provides quantitative confirmation that, as was suspected in previous investigations [7], a significant portion of the measured jitter was in fact caused by the microwave measurement technique itself.
Fig. 5. Time domain traces of the interferometric cross correlation signal between the reference and transmitted pulses. The green and blue curves show the noise measured respectively for 10 s and 1 ms intervals when it is actively canceled by the approach of spectral leveraging (scheme 1). The equivalent time scale is shown on the right axis. The red curve is the free-running (unlocked) case.
Fig. 6. Power spectral density (solid curves referring to left axis) and integrated jitter (dashed curves referring to right axis) of APD’s noise floor and noise when actively canceled by scheme 1 and 2.
Fig. 7. Power spectral density (solid curve referring to left axis) and integrated jitter (dashed curve referring to right axis) of unlocked noise, noise floor of microwave detection and that of optical detection. Microwave detection is conducted at 1822.46 MHz, the 37th harmonic of the repetition rate of the mode-locked fiber laser.
In addition, when the PLL is locked using these microwave signals, the ICC signal continues to oscillate (similar to the red curve in Fig. 5) with a increased amount of high frequency components, indicating a large amount (> 5 fs) of uncompensated jitter. This observation verifies the independent integrated rms jitter results shown in Fig. 7. More importantly, it also provides another indication of the superiority provided by the all-optical locking schemes shown in Fig. 2.
5. Conclusion
Two different all-optical techniques for distributing an ultra-low jitter frequency standard are experimentally demonstrated. In this work, a high-stability, low jitter rf signal is transmitted with a jitter of 11.4 as (1 Hz to 10 MHz). These performance levels are measured out-of-loop by an interferometric cross correlation. The key for realizing this result is the vast reduction in the noise floor (to 9.7 as) of both the error signal generation and measurement techniques via all-optical methods.
For this first demonstration we have transmitted over a 60-m optical fiber link, corresponding to a remote transfer of 30-m. Future work will entail extending this distance to > 10 km and we are optimistic that a similar level of performance will be achieved for this longer transmission distance for two reasons. When we used the standard microwave phase detection technique, the jitter performance (Fig. 7) of this 60-m link is similar to previously reported results over distances of 9.1 km [7]. This indicates the 9.1-km link jitter performance is most certainly limited by the microwave detection method. Second, the additional noise experienced along a longer transmission distance will be predominated environmental (i.e., low frequency, < 10 kHz) in nature. Such noise can be corrected by common piezoelectric transducers similar to the devices used in this work. However, for the all-optical approach presented herein to be successfully applied to longer transmission distances, the additional accumulated dispersion and loss experienced by the transmitted signal will be need to be properly addressed.
Lastly, while we demonstrated these new all-optical approaches within the context of remote frequency transfer, the spectral leveraging locking technique (scheme 1 in Fig. 2) could be easily and transparently be applied to synchronization of independent (local and remote) fs lasers as long as the difference of the two carrier-envelope-offset frequencies is passively stable enough such that optical heterodyne signal won’t fall out of the BPF’s bandwidth (see Fig. 2). By using this signal or an error signal derived from an ICC measurement we would expect a similar level of jitter performance as we observed.
Acknowledgments
We thank Vivide Chang for constructing the Si-APD used in this work and Kirk Madison and Jun Ye for helpful discussions. This research is supported by Natural Science and Engineering Research Council (NSERC), Canadian Foundation for Innovation, and British Columbia Knowledge Development Fund.
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