1. Introduction

Since the first demonstration of the self-referenced optical frequency synthesizer [1, 2], much experimental effort has been devoted to develop more reliable, more stable, and simpler optical clockworks. In particular, long-term operation of optical clockworks was hindered by the microstructure fiber employed to broaden the laser pulse spectrum to span one octave, which is required for convenient f -to-2f self-referencing. Without feedback control it is a difficult task to maintain efficient coupling into the small fiber core and the cleaved faces can be damaged by the high light intensities. These problems are absent in optical clockworks based on octave-spanning Ti:sapphire lasers [3, 4, 5, 6, 7] and carrier-envelope (CE) phase independent clockworks based on sum-/difference-frequency generation [8, 9], e.g., for the HeNe/CH₄ optical molecular clock [10].

High-repetition rate Ti:sapphire lasers bring along many advantages for optical frequency metrology, ultrafast time-domain spectroscopy, and other applications. In optical frequency metrology, they yield a larger power per comb line leading to larger signal-to-noise ratios (SNRs) of the measured heterodyning beat signals. Moreover, for repetition rates above ~150 MHz, the individual comb lines can conveniently be resolved using commercial wavemeters [11]. In ultrafast time-domain spectroscopy, they enable shorter data acquisition times and improved SNRs. CE phase-stabilized systems can be used to study physical processes sensitive to the CE phase, e.g., multipath quantum interference effects [12, 13], carrier-wave Rabi flopping [14], above-threshold ionization [15], and high-harmonic generation [16]. Isolated attosecond soft-X-ray pulses generated by high-harmonic generation using few-cycle driving pulses might furthermore be used to seed next generation X-ray free-electron lasers [17]. For such experiments and applications, the integrated CE phase error should be as low as possible.

Recently, Ti:sapphire laser systems with repetition rates below 100 MHz, that directly produce octave-spanning spectra for frequency comb stabilization using the f -to-2f self-referencing scheme, have been demonstrated [4, 5, 6, 7]. In addition, Ti:sapphire laser systems with repetition rates up to 1 GHz [18] were CE phase stabilized without the need for external broadening using the more complex 2f -to-3f self-referencing technique [19]. Alternatively, the CE frequency of few-cycle Ti:sapphire lasers can also be stabilized using the interference between self-phase modulation (SPM) and second harmonic generated in thin ZnO crystals [20, 21], and analogously, using the interference between SPM and difference frequency generated in a periodically-poled lithium niobate crystal [22]. Also erbium fiber-laser based frequency synthesizers represent an attractive alternative for metrological applications with turnkey operation [23, 24, 25]. However, at present the CE beat note of fiber lasers exhibits ~200 kHz linewidth in a 100 kHz resolution bandwidth [24], indicating increased CE phase fluctuation. The larger CE phase jitter might limit the usefulness of CE-phase stabilized fiber lasers for time-domain spectroscopy in the regime of extreme nonlinear optics. In optical frequency metrology, the larger CE beat linewidth implies decreased short-term stability and longer averaging times to obtain a desired stability.

In this paper we demonstrate CE phase stabilization of a 200MHz octave-spanning Ti:sapphire laser without the need for external spectral broadening. The individual frequency comb lines spaced by 200MHz can conveniently be resolved using commercial wavemeters. The CE beat note is detected using a compact and stable f -to-2f self-referencing scheme without separating and recombining the f and 2f spectral components. The accumulated in-loop CE phase error (integrated from 2.5 mHz to 10 MHz) using a broadband analog mixer as phase detector is 0.117 rad, equivalent to 50 attosecond CE phase jitter at 800 nm. The more than tenfold reduction of the CE phase error compared to our previous result [7] results from the monolithic CE beat detection scheme, improved acoustic vibration isolation and shielding against environmental perturbations (e.g., air currents), as well as improved phase-locking electronics.

2. 200MHz octave-spanning Ti:sapphire frequency comb

The Ti:sapphire laser (see Fig. 1) is similar to the one described in Ref. [7], but it operates at 200MHz repetition rate and emits an average output power of ~270mW. The laser resonator is set up in an astigmatism-compensated Z-folded geometry employing two concave double-chirped mirrors (DCMs) with 75mm radius of curvature. The laser is pumped by focusing ~6.5W (measured in front of the acousto-optic modulator (AOM)) of 532 nm light emitted by a Coherent Verdi V6 pump laser into the gain crystal using a 50mm focal length lens. The dispersion of the 2 mm-thick Ti:sapphire gain crystal and the air within the cavity is precisely compensated for by DCM pairs, a BaF₂ plate, and BaF₂ wedges for dispersion fine-tuning. BaF₂ was chosen as a material because it has the lowest ratio of third- to second-order dispersion in the wavelength range from 600 to 1200 nm. This property allows the design of octave-spanning DCMs with 99.9% reflectivity from 580 to 1200 nm. Moreover, the dispersion of 0.5mm of BaF₂ is similar to that of 1m of air. This enables scaling up the cavity to higher repetition rates by removing air path and correspondingly adding BaF₂ material to maintain the proper dispersion balance. To achieve the ultrabroad spectrum depicted in Fig. 2, which has a Fourier limit of 3.6 fs, the output spectrum is shaped using a broadband quarter-wave ZnSe/MgF₂ output coupling mirror [5] with a transmission of 1% in the center and 78% at 570 nm.

 

Fig. 1. Carrier-envelope phase stabilized 200MHz octave-spanning Ti:sapphire laser. The femtosecond laser itself (located inside the grey area) has a compact 20 cm×30 cm foot-print. AOM, acousto-optical modulator; S, silver end mirror; OC, output coupling mirror; PBS, polarizing beam splitter cube; PMT, photomultiplier tube; PD, digital phase detector; LF, loop filter; VSA, vector signal analyzer. The carrier-envelope frequency is phase locked to 36 MHz.

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Our f -to-2 f self-referencing setup represents a major improvement over previous setups: in previous setups, to generate a CE beat note with sufficient SNR for phase locking (~30 dB in 100 kHz resolution bandwidth), the short- and long-wavelength portions of the laser spectrum were spatially separated (using a dichroic mirror in a Mach-Zehnder type interferometer [1] or a prism in a prism-based interferometer [6]), the long-wavelength portion was frequency-doubled in a second-harmonic generation (SHG) crystal, and the short-wavelength fundamental light and the SHG light were recombined again. Although the wavelength components, that interfere with each other generating the CE beat note, can conveniently be overlapped in space and time, these interferometer setups tend to be bulky and alignment sensitive. Even more important, mechanical fluctuations in the interferometer introduce additional CE phase noise in the frequency range up to ~10 kHz. In our setup (see Fig. 1), in contrast, the time delay between the 570 and 1140 nm spectral components used for f -to-2f self-referencing is produced by 10 bounces on DCMs, which is intrinsically more stable than the above-mentioned interferometric setups. After the DCM-based delay line, the Ti:sapphire output is focused onto a 2 mm-thick BBO crystal cut for type I SHG at 1160 nm. We measured an SHG conversion efficiency on the order of 10^-3. The emitted SHG light and the orthogonally polarized fundamental light are projected onto a common axis using a half-waveplate and a polarizing beam splitter cube, spectrally filtered using a 10 nm wide interference filter centered at 570 nm, spatially filtered using a~1mm diameter aperture, and finally detected using a photomultiplier tube (Hamamatsu H6780-20).

 

Fig. 2. Output spectrum of the Ti:sapphire laser on a linear (black curve) and on a logarithmic scale (red curve). The reflectivity of the ZnSe/MgF₂ output coupler (blue curve) is shown for comparison. The wavelengths 570 and 1140 nm used for f -to-2f self-referencing are indicated by two dashed lines. The Fourier limit of the pulse spectrum is 3.6 fs.

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3. Carrier-envelope phase stabilization results

In the radio-frequency power spectrum shown in Fig. 3, we observe a peak at the CE frequency with a SNR of ~35 dB in a 100 kHz resolution bandwidth. This SNR is sufficient for direct and routine CE phase stabilization. When the CE beat is phase locked, its linewidth is 2 Hz FWHM (measurement limited). Phase locking is achieved by a phase-lock loop (PLL) in feeding an error signal back to an AOM placed into the pump beam which regulates the pump power and thus changes the CE frequency [26]. A bandpass filter is used to select the CE beat signal at 36 MHz. This signal is amplified, divided by 4 in frequency to enhance the locking range of the PLL, and compared with a reference frequency supplied by a signal generator using a digital phase detector. The output signal is amplified in the loop filter, which in our case is a proportional-integral (PI) controller, and fed back to the AOM, closing the loop. The output of the phase detector is proportional to the remaining jitter between the CE phase evolution and the local oscillator reduced by the division ratio of 4.

Since a digital phase detector requires a low pass filter (~1.9MHz in our case) for operation, any signal with a higher frequency than the low-pass filter cutoff will be attenuated yielding a CE phase error which does not represent the true CE phase error of the system. Fundamentally this makes a separate out-of-loop measurement consisting of a separate SHG process, CE beat detection and phase comparison necessary. The Allan deviation of an SHG process using a nonlinear crystal was measured to be on the order of 10^-16 for an averaging time of 1 s [27], thus it is reasonable to assume no significant contribution of the SHG process to the CE phase noise. Furthermore, our monolithic f -to-2f self-referencing setup employing a DCM-based delay line is not expected to introduce additional CE phase noise either, in contrast to the commonly used Mach-Zehnder type or prism-based interferometers. Hence, assuming that our f -to-2f self-referencing setup and the photomultiplier detection truthfully reflect the CE phase dynamics, only a phase detector with a high enough intermediate frequency (IF) bandwidth is necessary to measure the true CE phase error within the loop. In a second measurement, we therefore replaced the digital phase detector with an analog mixer.

 

Fig. 3. Radio-frequency power spectrum of fundamental and frequency-doubled light transmitted through a 10 nm wide interference filter centered at 570 nm, resolution bandwidth (RBW) is 100 kHz. The peak at the carrier-envelope frequency f_ϕ exhibits a signal-to-noise ratio of ~35 dB, sufficient for direct and routine carrier-envelope phase stabilization.

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Fig. 4. Power spectral density (PSD) of the carrier-envelope phase fluctuations Sf (blue and red curves) and integrated carrier-envelope phase error Δ_ϕ (green and orange curves) measured with a digital phase detector and mixer, respectively.

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The power spectral density (PSD) of the CE phase fluctuations S_ϕ for a feedback loop using a digital phase detector or a mixer measured with a vector signal analyzer (VSA) is shown in Fig. 4. The accumulated (root-mean-square) CE phase error Δϕ can be obtained from S_ϕ by integration over frequency according to

resulting in a value of 0.103 rad and 0.117 rad (integrated from 2.5 mHz to 10 MHz) for the digital phase detector and mixer, respectively. This is equivalent to 44 attosecond and 50 attosecond CE phase jitter at 800 nm, respectively. Both measurements are in good agreement with each other considering that they were taken on different days with different loop-filter settings. These results reflect the elaborate acoustic vibration isolation and shielding against environmental perturbations (e.g., air currents) as well as the effectiveness of our PI control loop up to ~20 kHz. At present, the bandwidth of our PI control loop is limited by the AOM used to modulate the pump power. By using an electro-optic modulator, the CE phase fluctuations are expected to be even further suppressed in the future.

4. Conclusion

In conclusion, we have demonstrated CE phase stabilization of a 200MHz octave-spanning Ti:sapphire laser without external spectral broadening. The CE beat note was detected using a compact and stable f -to-2f self-referencing scheme without separating and recombining the f and 2f spectral components. With this setup we achieved a CE beat note with 35 dB signal-to-noise ratio in a 100 kHz resolution bandwidth, the CE beat linewidth is 2 Hz FWHM (measurement limited). The accumulated in-loop CE phase error (integrated from 2.5 mHz to 10 MHz) using a broadband analog mixer as phase detector is 0.117 rad, equivalent to 50 attosecond CE phase jitter at 800 nm.