1. Introduction

There has recently been considerable interest in stabilizing the optical carrier phase relative to the pulse envelope (the carrier-envelope phase) for pulse trains produced by modelocked lasers [1]. Such stabilization has led to significant advances in optical frequency metrology and optical atomic clocks [2,3] and has assisted in the observation of carrier-envelope-phase sensitivity in a variety of physical processes such as above-threshold ionization, [4,5] photoelectron emission from metal surfaces, [6,7] carrier-wave Rabi sideband interference, [8–10] and coherent control of injected photocurrents in semiconductors [11–13].

The most common method of stabilizing the carrier-envelope phase (CEP) is currently ν-to-2ν self-referencing [14–16]. This method involves the use of second harmonic generation (SHG) to interferometrically compare the phase of the low (ν) and high (2ν) frequency wings of an octave-spanning pulse spectrum. The result of this comparison is a heterodyne beat note at a frequency, known as the carrier-envelope offset frequency, corresponding to the rate at which the CEP evolves from one pulse to the next. This beat note can be measured on a photodetector, compared to a stable reference frequency, and used as feedback in a phase-locked loop to stabilize the evolution of the CEP in time.

With the CEP evolution stabilized to a reference rf oscillator it is possible to use a second, independent ν-to-2ν interferometer to determine the residual CEP noise of the locked system. Specifically, by performing phase-sensitive detection between the beat note from the second interferometer and the reference rf oscillator, researchers have measured the out-of-loop residual CEP noise density as a function of frequency [17]. This CEP noise spectrum can be useful in identifying various contributions to the overall phase noise that occur at different frequencies. By integrating the CEP noise density over frequency, one can obtain the overall phase noise accumulated in a given time and estimate the CEP coherence time of the generated pulse train. The CEP noise density is also valuable for determining the efficacy of the stabilization system. In the present work, we show that a semiconductor photocurrent quantum interference effect can be used to directly detect the CEP noise spectrum of a stabilized modelocked laser by simply focusing prepared pulses onto the semiconductor.

Quantum interference control (QIC) of injected photocurrents in semiconductors involves interference between the probability amplitudes for single- and two-photon absorption. This interference causes an imbalance in the otherwise symmetric momentum-space carrier distribution in direct-gap semiconductors. The direction and magnitude of the resulting photocurrent depend on the relative phase of the light fields that drive the absorption processes at frequencies ν and 2ν. The effect has been studied previously by focusing two harmonically related pulse trains onto a region of semiconductor between two metal electrodes (for current collection) [18–20]. It has also recently been investigated using a single pulse train with an octave-spanning spectrum [12,13]. In the latter case, the single- and two-photon absorption pathways are driven by the spectral wings of the same pulse (at ν and 2ν), and the generated photocurrent is sensitive to the CEP. When the CEP evolves in time, an oscillating current is produced at the carrier-envelope offset frequency. One can compare this oscillating current to a reference rf oscillator and measure the CEP noise in direct analogy with the ν-to-2ν technique. Here we demonstrate this compact, solid-state noise measurement method. Additionally, this detection process exhibits no amplitude/phase correlations, and features a built-in balanced-detection advantage over standard photodetection.

Other CEP-sensitive processes are also under investigation for simplifying and improving CEP noise detection and stabilization. Researchers have shown that the spectral broadening and frequency doubling required for the ν-to-2ν interferometric technique can be performed in a single ZnO crystal for very short pulses [21]. The resulting beat note has very recently been used for CEP stabilization [22]. Photoelectron emission from metal surfaces using such ultrashort pulses also appears promising for CEP noise analysis and stabilization [6,7].

2. Experimental setup

Figure 1 shows the setup that we used to measure the CEP noise spectrum via QIC. We generated a train of ~10-fs pulses from a modelocked, Ti:sapphire laser with a repetition rate of 93 MHz, a center wavelength near 840 nm, and an average power of 400-500 mW. The light from this laser was split into two portions. We broadened both portions in 4-cm lengths of microstructure optical fiber to generate sufficient spectral bandwidth [23]. We used the first portion to phase-lock the laser’s CEP evolution to a stable rf oscillator via ν-to-2ν self-referencing. We directed the second portion through a prism pair to spatially disperse the spectrum. This light was then retroreflected using of a pair of independent mirrors to enable adjustments of the time delay between the spectral wings. After reflection back through the prisms, we sent the recombined beam on one of two paths for detection of the CEP noise. The “QIC path” was simply a focusing objective lens and the semiconductor sample. The “SHG path” was composed of a focusing lens, a frequency-doubling crystal, a polarizer, a spectral filter, and a photodiode. The SHG path was equivalent to ν-to-2ν self-referencing.

 

Fig. 1. Experimental setup used to compare the CEP noise measurements via the QIC path and the SHG path: MS, microscructure fiber; P, polarizer; BPF, band pass filter; ×2, SHG crystal.

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For the QIC path, we focused the light (10-μm spot size for both colors) onto the low-temperature-grown gallium arsenide (LT-GaAs) sample in the semiconductor region between two gold electrodes that were separated by 30 μm. LT-GaAs was used to reduce the carrier lifetime (~500 fs), thereby maximizing the effective shunt resistance of the metal-semiconductor-metal structure during illumination. Details of the semiconductor and electrode structure are given in Ref. [11]. For each of the two measurement paths, we amplified the signal and mixed it with the stable frequency reference. The mixer produced voltage fluctuations that were linearly proportional to phase fluctuations relative to the stable reference when the mixer was operated in its linear region. We monitored the fluctuations that were measured by each path, one at a time, using a fast Fourier-transform dynamic-signal analyzer (FFT). Because the dispersion was slightly different for the two paths, we optimized the optical time delay separately for each path.

3. Results and discussion

Figure 2 shows the resulting CEP noise spectral density as a function of sampling frequency. The dotted trace was measured using the SHG path, the solid trace was measured using the QIC path, and the dashed trace was generated from an in-loop phase-noise measurement. The background noise measured when the light is blocked has been subtracted from these traces. To eliminate the possibility of errors in the relative phase-noise levels caused by inaccurate calibration of the mixer output voltage, we modulated the position of one of the split mirrors at a known frequency (100 Hz) to produce a phase-noise spike that was common to both measurements, regardless of path. By matching the spike amplitudes at 100 Hz from the two paths, we minimized calibration errors.

 

Fig. 2. Phase noise density versus sampling frequency measured using QIC (solid line), using SHG (dotted line), and in-loop (dashed line). The traces indicate that the QIC method does not add phase noise relative to the SHG method.

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The traces measured for the QIC and SHG paths are very similar for frequencies greater than about 20 Hz but diverge for lower frequencies. Similar correspondence at high frequencies was common to all data sets taken. However, the divergence at low frequencies was not common to all data sets. For some sets the QIC noise was higher, for some the SHG noise was higher, and for others they were essentially identical. For these low frequencies, the phase noise depends critically on the coupling stability into the microstructure optical fiber. We therefore attribute the observed deviations at these frequencies to amplitude-to-phase noise generated in the fiber [24]. This effect is independent of the measurement path but can vary because the measurements are taken at slightly different times. Our measurements uncovered no reproducible or consistent phase noise increase when measuring with the QIC path versus the SHG path at any of the frequencies we measured. We attribute the discrepancy between the out-of-loop and the in-loop measurements to amplitude/phase coupling in the fibers, physical instabilities in the interferometers, calibration errors, and the non-simultaneity of the measurements, which can lead to differences in laser and locking conditions.

We also tested the sensitivity of the measured signal phase to amplitude fluctuations. We placed a pair of broadband polarizers after the microstructure fiber and before the prism pair. By rotating the first polarizer, we could adjust the optical power at the sample without altering its polarization or phase. For this measurement, we replaced the mixer and FFT in Fig. 1 with a lock-in amplifier. Figure 3 shows a time record of the amplitude and phase of the QIC signal measured with lock-in detection relative to the stable reference as we rotated the first polarizer to attenuate and then unattenuate the light level at 5-s intervals.

 

Fig. 3. Time record of the QIC signal amplitude (solid line) and phase (dotted line) from lock-in detection (100-ms time constant) while the light level was repeatedly attenuated and unattenuated at 5-s intervals. No associated amplitude/phase correlations are observable.

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An overall slow drift of both the amplitude and phase is evident from the data and is correlated. This drift was due to unintentional decoupling of the light into the microstructure fiber, which is known to cause amplitude/phase coupling [24]. The amplitude/phase coupling in the QIC detection process itself should appear as correlated shifts at the five-second intervals. From Fig. 3 it is clear that the optical power reduction caused a drop in the QIC amplitude by roughly a factor of two but caused no observable effect on the QIC phase. We performed a more rigorous search for correlations in the data by removing the slow linear drifts and plotting the signal phase as a function of its amplitude as shown in Fig. 4. Using a linear fit to the data, we calculated a linear correlation coefficient of r = -0.025.

 

Fig. 4. Phase versus amplitude of the measured lock-in signal. The near-zero slope of the linear fit to this data indicates a low probability of correlation between the amplitude and phase (linear correlation coefficient: r = -0.025).

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The SHG phase-noise comparison and the amplitude/phase-coupling measurements therefore both indicate that the solid-state QIC detection method is a suitable candidate to replace the ν-to-2ν technique for laser stabilization. However, our signal strength relative to noise contributions that are unrelated to the phase detection (i.e., thermal noise, amplifier noise, etc.) is presently insufficient for such stabilization. This background noise was subtracted from the traces in Fig. 2, but it leads to the widths of the lines in that plot. The signal-to-noise ratio for our beat note is 30 dB (1 kHz bandwidth), whereas we estimate that we will need an additional 10 dB to stabilize. We are optimizing the electrode structures and compensating for dispersion in the microstructure fiber to increase the signal strength. We are also improving our amplifier design to minimize the noise.

4. Intrinsically balanced detection

One intriguing aspect of the QIC method for CEP detection and stabilization is its built-in mechanism for balanced detection. To explain this, we imagine that the carrier-envelope offset frequency is locked to zero and compare the dc responses of the SHG and QIC methods. The SHG method ultimately generates an electrical current from a standard photodetector that is similar to a traditional interferometer:

where I_v and I _2ν are the irradiances of the ν and 2ν beams, respectively, η₁ accounts for the SHG efficiency, η₂ accounts for imperfect mode matching, and ϕ _CE is the carrier-envelope phase. Here we neglect issues such as dispersion and assume that ϕ _CE can actually be measured without a phase offset. As a function of ϕ _CE, the generated photocurrent is therefore the sum of two offset terms and a sinusoidally oscillating term due to the interference that is typically much smaller than the offsets. We consider a phase measurement when ϕ _CE=0. For this simple case, the electrical current used to determine the phase value of ϕ _CE is critically dependent on the offset irradiances of the two beams [i.e., the first 2 terms on the right side of Eq. (1)]. More to the point, a shift in ϕ _CE cannot be immediately distinguished from a change in irradiance. Without data treatment, this leads to an inherent amplitude/phase coupling fundamental to the detection process when offsets exist. This effect is most prevalent at dc, but also exists at higher frequencies. We were able to confirm a small (<3%) contribution to the SHG-path phase noise shown in Fig. 2 that was due to amplitude fluctuations despite the fact that we performed the detection away from dc.

The solid-state QIC detection method is immune to this drawback. Because the quantum interference breaks an otherwise symmetric carrier flow in the semiconductor, QIC yields current perturbations about a zero mean. This effect serves as a built-in balanced detection mechanism, and the measured QIC current fundamentally represents the CEP. In Eq. (1), this mechanism eliminates the two offset terms, leaving only the interference term. Of course, its magnitude is power dependent, but when measuring (or locking) ϕ _CE near zero, the measurement will be insensitive to amplitude changes in the driving field. For some CEP detection methods that utilize standard photodetectors, it is possible to imagine balanced detection schemes involving additional detectors that would suppress the amplitude sensitivity caused by the offsets. Nevertheless, the simplicity of the solid-state QIC method is attractive.

In summary, we demonstrated solid-state detection of the out-of-loop CEP noise using quantum interference control of injected photocurrent in a semiconductor. The QIC method added no measurable phase noise compared to the standard ν-to-2ν method. The technique displayed no significant amplitude/phase correlations, and we discussed the advantages of its intrinsic balanced detection feature. These measurements and discussions indicate that the solid-state QIC detection method is a suitable, and in some cases preferable, candidate to replace the ν-to-2ν technique for CEP stabilization.