Carrier-envelope phase stabilization1, 2 has opened an avenue towards achieving frequency metrology with unprecedented precision3, 4 and optical pulse generation on the previously inaccessible attosecond timescale5. Recently, sub-100-as pulse generation has been demonstrated6, approaching the timescale of the fastest transients in atomic physics. However, further progress in attophysics7 appears to be limited by the performance of the traditional feedback approach used for carrier-envelope phase stabilization8, 9, 10. Here, we demonstrate a conceptually different self-referenced feed-forward approach to phase stabilization. This approach requires no complicated locking electronics, does not compromise laser performance, and is demonstrated with 12-as residual timing jitter, which is below the atomic unit of time. This surpasses the precision of previous methods by more than a factor of five and has potential for resolving even the fastest transients in atomic or molecular physics. Such shot-noise-limited comb synthesis may also simplify progress in current research in frequency metrology11, 12.
Introduction
Measurement of the carrier-envelope phase (CEP) relies on heterodyning different harmonics of broadband femtosecond laser pulses1. This method directly delivers the slippage rate fCE = frepϕCE/2π of the per-roundtrip phase difference ϕCE between carrier and envelope inside a femtosecond oscillator as a radiofrequency (RF) signal, where frep is the laser repetition rate. Establishing a phase-locked loop between fCE and a reference oscillator then enables CEP stabilization. As CEP fluctuations, in particular in free-running Ti:Sapphire oscillators, have turned out to be massive13, with clearly measurable spectral content up to several kilohertz, practical implementations of the negative feedback for closing the phase-locked loop therefore require a fast-acting mechanism for adjusting intracavity CEP. Suitable servo control of the CEP therefore typically relies on an acousto-optic modulator (AOM), which adjusts the intracavity peak power and nonlinear phase shift by means of the oscillator pump power14. Despite its proven utility, however, the servo concept has some distinct drawbacks. First, regardless of the method used, feedback into the laser frequently causes detrimental, dynamic side effects on other laser parameters such as output power, pulse duration or round-trip time. Achieving ideal stabilization performance requires a careful balance between short-term phase jitter and stability against drop-outs15, which often corrupts the achievable bandwidth of the servo loop. Additionally, measuring the CEP relies on RF heterodyning. Therefore, generating the very important case of a pulse train with constant electric field structure (fCE = 0) requires further measures. Although a comb with zero offset frequency has been produced before using an AOM2, the AOM is inserted in one of the arms of an f-to-2f interferometer, limiting the design of the interferometer. In the frequency domain, pulses from a zero-offset comb consist of exact harmonics of the laser repetition rate frep, that is, νn = nfrep. Optical parametric processes16 or difference-frequency generation between two combs of identical fCE ≠ 0 have been demonstrated as an alternative method for the generation of zero-offset combs. However, residual phase jitter of self-referenced schemes is found to be similar to servo-loop approaches. This is explained by laser power fluctuations and strong amplitude-to-phase coupling in the optical parametric process16. In the following, we discuss a scheme in which the optical signal wave in the parametric process is directly replaced by the RF signal fCE. Using an acousto-optic frequency shifter (AOFS) for comb synthesis, we avoid amplitude-to-phase coupling as well as the previously mentioned shortcomings of the servo-loop approach.The concept of our scheme is depicted in Fig. 1. The frequency comb of a free-running femtosecond laser oscillator is split into zero-order and first-order beams by diffraction off the index grating inside the AOFS. In a similar way, AOFSs have been used previously to match a frequency comb to the longitudinal modes of an external cavity17 or to stabilize the frequency of a continuous-wave dye laser18. With typical diffraction efficiencies for an AOFS of 60–70%, more than half of the laser power is available for applications, while the remainder in the zero-order output serves to measure the CEP frequency. As this measurement in the zeroth order remains unaffected by the resulting frequency shift in the first order, our method has a feed-forward character. The power ratio between the AOFS orders is nearly identical to that used in the servo-loop approach. Owing to the interaction with the travelling acoustical wave, the frequency comb in the Bragg reflected first-order beam is shifted by exactly the driver frequency fRF. By feeding the independently measured and amplified carrier-envelope frequency into the AOFS, that is, fRF = fCE(t), the comb in the first-order beam is down-shifted to exactly zero offset, corresponding to a pulse train with identical electric field structure of the pulses. Using RF synthesis for generation of the driver signal as explained below, the comb can be stabilized at arbitrary offsets.
Figure 1: Concept of the direct feed-forward method for stabilization of CEP.
For maximum diffraction efficiency into first order (red comb modes), the Bragg condition 2λacsin α = λn has to be fulfilled. α = αin = αout is the Bragg angle, λac the acoustic wavelength, λ the optical wavelength, and n the refractive index. The comb modes in the transmitted beam (zero order) remain unaffected, but each individual frequency of the diffracted beam is shifted by the input frequency of the acousto-optic device fRF, which can be used to shift the comb to zero offset frequency when choosing fRF = fCE.
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Full size image (56 KB)
For experimental verification and measurement of the residual CEP jitter, we used the set-up shown in Fig. 2, which is described in detail in the Methods section. For detection reasons, the signal is shifted out of the baseband to obtain stabilization at fCE = 60 MHz. Figure 3a shows the time series of the residual phase noise, measured over 5 s at a 5 megasamples per second sampling rate. For comparison, we also recorded the signal with f and 2f components in the out-of-loop (OOL) interferometer temporally mismatched to suppress interference. The resulting quantum noise-induced signal allows the detection limit to be estimated. Fourier transforms of the temporal signals (Fig. 3b) indicate that the residual phase noise is essentially limited by detection noise in the range from 30 Hz to the sampling limit of 2.5 MHz, with only relatively few discrete frequencies appearing significantly above the detection limit. Over five decades, therefore, the performance of our stabilization scheme is determined by shot noise. This situation is in strong contrast to previously reported performance8, 10, 13, where a broad acoustic band was consistently reported that exceeded the detection limit by some 25 dB Hz−1/2, peaking in the 100–1,000 Hz region. Moreover, it is also surprising that the spectral analysis does not report any prevalent noise contributions in the 100 kHz range, as have previously been observed in servo-stabilized lasers and attributed to the multimode concept of the pump laser being used9. Given their relatively high frequency above the loop bandwidth, such phase noise contributions are impossible to equalize in traditional servo-loop concepts. This limitation does not play a role in our feed-forward concept, which is only limited by the bandwidth of the AOFS, which we compute to be 1 MHz from manufacturer data. At frequencies below ~10 Hz, slow dephasing between the OOL and in-loop (IL) signal is observed. We attribute this phase drift mainly to a slow drift of the CE frequency of the laser oscillator in combination with the phase lag of the AOFS. To a lesser extent, a relative drift of the two interferometers may also contribute. Note that no means were taken to stabilize the CE frequency of the free-running laser oscillator in any way. Moreover, this drift is relatively minor, amounting to an r.m.s. excursion of 570 mrad in a 35-min observation time (Fig. 3c). We estimate that this drift-like 1/f noise exceeds detection noise at frequencies below ~100 Hz, which also compares favorably with previously reported stabilization behaviour8, 10.
Figure 2: Experimental set-up used to characterize CEP stabilization performance.
MSF, microstructured fibre; AOFS, acousto-optic frequency shifter with numbered output orders; PPLN, periodically poled lithium niobate crystal, designed for efficient second-harmonic generation of a 1,064-nm input wave; IF, interference filter; APD, avalanche photodiode; DSO, digital sampling oscilloscope. Interferometers use a quasi-common path geometry with a split end mirror for compensation of the group-velocity dispersion in the MSF and subsequent optical components. The IL interferometer is used to generate the input signal for the AOFS at frequency fRF. The OOL interferometer is used for optically and electronically independent analysis of the residual phase noise. The set-up shown incorporates an additional frequency synthesis step using an external frequency generator, two double-balanced mixers (R, RF input; L, local oscillator input; I, intermediate frequency output) and suitable RF filters. This step is only required for unambiguous and sensitive phase noise analysis. Directly feeding the IL signal into the AOFS without frequency synthesis or choosing fRF as an integer multiple of the laser repetition rate generates a comb with zero offset.
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Full size image (92 KB)
Figure 3: Residual CEP noise.
a, Measured 5-s time series of the residual phase noise with an r.m.s. of 45 mrad. The inset to the right shows histograms of the occurring phase values for this time series (red) and for a time record of the measurement noise floor (grey) originating from shot noise. This noise floor is measured for non-beating f and 2f components and, therefore, also includes noise sources of electronic origin inside the phase extraction circuit. b, Frequency analysis of the 5-s time series. Shown in red is the phase noise density (PND) and in blue the integrated phase noise (IPN). The grey curve shows the PND of the measurement noise floor. As a guide to the eye, the dotted lines show the shot-noise level as well as 1/f noise characteristics. c, PND and IPN of a longer 35-min time series with an r.m.s. jitter of 570 mrad.
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Full size image (71 KB)
The total integrated phase noise (0.2 Hz to 2.5 MHz) of the data in Fig. 3b amounts to 45 mrad, which corresponds to a temporal jitter of ~20 as. At higher frequencies, the performance of the feed-forward scheme only appears to be limited by detection shot noise, which alone already amounts to 30 mrad. It is striking that residual phase noise exceeds detection noise by a factor of only about Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. Fitting Gaussians to the phase noise distribution in Fig. 3a, it is estimated that the underlying residual phase noise jitter amounts to 30 mrad, corresponding to a timing jitter of only 12 as. Although this value corrects for noise in the OOL interferometer, it does not account for quantum noise in the IL interferometer. The bottleneck of our detection scheme is the frequency-doubling process in the 2f arm. Optical amplification before doubling could serve to further improve the signal-to-noise ratio in our scheme, and promises to push residual jitter into the single-attosecond regime. With or without subtraction of detection noise in the OOL interferometer, the measured residual timing jitters are below the atomic unit of time, which is considered to be a characteristic order of magnitude at which the fastest inner-atomic transient processes may occur. Therefore, the feed-forward CEP stabilization scheme exceeds the previously reported behaviour of feedback servo schemes by nearly an order of magnitude (Fig. 4). In particular, the acousto-optic implementation clearly outperforms the optical self-referencing technique16.
Figure 4: Comparison of demonstrated attosecond pulse duration and timing control.
Numbers refer to references. Residual timing jitters of demonstrated CEP8, 9, 10, 16, as measured in an OOL f-to-2f interferometer, are also shown as phase jitters, where a wavelength of 800 nm was assumed for conversion. Values from refs 8 and 10 were based on an integration over the reported spectral phase noise densities matching the frequency range (0.2 Hz to 2.5 MHz) of our measurement in Fig. 3. This allows for an unbiased comparison.
Other than the superior noise performance, the feed-forward scheme has several additional advantages. First, it does not require any intervention into the free-running oscillator, whereas the servo-loop approach modulates the oscillator pump power for controlling the CEP. Second, combs with arbitrary offset frequencies can be synthesized. The most important case of a comb with zero offset is most easily obtained, directly feeding the IL signal into the AOFS. Finally, our method does not require any locking electronics. To avoid self-oscillation, servo-loop schemes require that a phase margin of at least π over their total gain bandwidth is maintained, which significantly reduces the achievable feedback bandwidth compared to the fundamental limit imposed by the acoustic delay inside the AOFS or AOM. As no lock has to be maintained in the feed-forward technique, stabilization will immediately reestablish after a brief interruption of the input signal. Additionally, several long-term stabilization tests of up to 12 h have shown the robustness of the method. Thus, the scheme is readily applicable to any type of mode-locked laser, with measurability of the CE frequency being the only prerequisite. Compared to servo-loop stabilization, the only disadvantage appears to be the additional dispersion of the AOFS in the beam path. This dispersion can be compensated by suitably designed chirped mirrors, which, however, may cause additional losses. Note that these effects barely play a role when seeding a chirped pulse amplifier that intrinsically requires temporal stretching of the pulse. Therefore, given the numerous advantages of the feed-forward scheme, we expect it to replace the traditional servo-loop approach, in particular for attosecond pulse generation, where this technique promises to enable greatly increased control on the atomic timescale. Following progress from pulses with a duration of a few femtoseconds to a hundred attoseconds over the last decade, our simple feed-forward scheme boosts the limits of attainable temporal control by another order of magnitude, and straightforward technical improvements in the f-to-2f interferometer lie at hand to push this limit towards single attoseconds.
Methods
Experimental set-up
As a laser source, we used a Millennia pumped sub-10-fs oscillator (Femtolasers GmbH, FEMTOSOURCE synergy), which was spectrally broadened in a microstructure fibre (Crystal Fibres, Femtowhite800) to obtain octave coverage. The spectrally broadened beam from the fibre was recollimated and fed into the AOFS. The AOFS was 2 cm thick and made of fused silica. The device was optimized for operation at 70 ± 10 MHz, with a maximum 70% diffraction efficiency at +38 dBm drive power. We minimized the electronic phase lag caused by the travel time of the acoustic wave to the interaction zone. The mechanical construction of the AOFS allowed for a minimum distance between the laser beam and acoustic actuator of ~2 mm, which translates into a feed-forward loop bandwidth of ~1.5 MHz. This is more than one order of magnitude larger than the best loop bandwidths of ~100 kHz reported for optimized feedback servo loops15. Each of the AOFS diffraction orders were fed into independent f-to-2f interferometers. Both interferometers were built in a quasi-common path geometry19 to minimize relative phase drift. The zero-order beam served to synthesize the AOFS drive signal in the IL (right-hand box in Fig. 2) interferometer, while the first-order beam in the OOL (lefthand box in Fig. 2) interferometer was used for noise analysis. The latter interferometer was designed to compensate angular dispersion in the first diffraction order.
Beat-frequency generation
For a high nonlinear conversion efficiency, two 10-mm-long PPLN crystals (Covesion, SHG3-10) were used. These crystals were optimized for second-harmonic generation at a fundamental wavelength of 1,064 nm. Out-of-band signals were rejected from photodetection by interference filters. The beat signals were detected by silicon avalanche photodetectors (Silicon Sensors, AD1500-11). The electrical signals were bandpass-filtered and amplified further. We obtained 30 and 40 dB signal-to-noise ratios (100 kHz resolution bandwidth) in the OOL and IL interferometers, respectively.
Frequency synthesis and phase-noise characterization
Feeding the amplified IL signal directly into the AOFS generated a zero-offset comb in the first diffraction order. Characterization of residual CE phase noise in the vicinity of zero frequency was difficult because of increased 1/f noise. Additionally, phase ambiguities could arise, as positive CE frequency deviations can only be distinguished from negative ones by increasing the complexity of the optical set-up20. To avoid these issues, we decided to shift our signals out of the baseband, using a fixed 60 MHz frequency derived from a quartz-stabilized RF synthesizer.
Heterodyning this reference with the independently measured OOL signal gave access to residual phase jitter, which was then recorded by a digital sampling oscilloscope. To determine the noise background in the detection scheme, timing between f and 2f components was temporally mismatched to suspend beat signal generation, while nevertheless ensuring identical powers on the detector. This procedure allowed for the detection noise levels to be estimated. We estimate that detection is mainly limited by quantum noise and by excess noise in the avalanche process.
Luiggi Escalante
CI. 18878611
CRF
Fuente: http://www.nature.com/nphoton/journal/vaop/ncurrent/full/nphoton.2010.91.html
CI. 18878611
CRF
Fuente: http://www.nature.com/nphoton/journal/vaop/ncurrent/full/nphoton.2010.91.html
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