1. Introduction

Control of the relative phase between the carrier wave and the pulse envelope (the carrier-envelope phase) from one pulse to the next is critical for stabilizing mode-locked femtosecond laser systems [1–3]. Many researchers currently use a powerful self-referencing technique known as ν-to-2ν stabilization for this purpose [2,3]. This technique involves interferometrically comparing the optical phase of the low frequency tail of the pulse spectrum at frequency ν with that of the high frequency tail at frequency 2ν through second harmonic generation. The resulting beat note is detected and used as an error signal in a feedback loop to stabilize the relative phase of the two spectral extremes in the frequency domain. This results in stabilization of the carrier-envelope phase in the time domain [4]. The ν-to-2ν technique necessarily requires a source with an octave-spanning (factor of 2) optical emission spectrum [5–7]. With such a source, ν-to-2ν stabilization has led to carrier-envelope phase coherence times exceeding a few minutes or 10¹⁰ pulses [8]. Such coherence properties have enabled fascinating advances in optical waveform synthesis, optical frequency metrology, optical atomic clocks, and extreme nonlinear optics [2,9–13].

Independent of this carrier-envelope phase stabilization work, other researchers have demonstrated that semiconductors can be sensitive to the relative phase between two coherent, harmonically related pulse trains (one at the optical frequency ν and one at 2ν) [14–16]. This sensitivity is due to currents generated by quantum interference between single- and two-photon absorption and has been observed in gallium arsenide and low-temperature-grown gallium arsenide (LT-GaAs). LT-GaAs was preferred because of its shorter carrier lifetime. The quantum interference control (QIC) signal was measured by focusing the two collinear pulse trains onto the semiconductor region of a metal-semiconductor-metal (MSM) structure. With the proper field polarizations and when the pulses were temporally and spatially overlapped, QIC ultimately generated a measurable voltage difference across the two metal electrodes, one on either side of the focused spot. The sign and magnitude of the voltage difference depended on a relative phase parameter between the two pulse trains. In this way, QIC can allow one to detect, and ultimately control, the relative phase between two harmonically related pulse trains.

In the present work, we use two such pulse trains to characterize QIC in LT-GaAs. We demonstrate real-time QIC interference fringes, optimize the QIC signal fidelity, uncover important signal dependences on beam spatial position, establish clear signal dependences on the fundamental and second harmonic average optical powers, and demonstrate the signal dependence on the focused beam spot size. Our motivation for this study is to assess the possibility of using the QIC phenomenon to measure and eventually control the carrier-envelope phase evolution of a single, octave-spanning pulse train. QIC can accomplish this because of its sensitivity to the phase difference between the pulse train spectral extremes (at frequencies ν and 2ν). It may therefore be possible for QIC to replace the present self-referencing ν-to-2ν stabilization technique that uses an interferometer. We anticipate that the QIC method can ultimately provide a simpler and higher performance method of stabilizing the carrier-envelope phase than the ν-to-2ν interferometer. We predict an initial signal-to-noise ratio (SNR) of 5-10 dB (1 kHz bandwidth) for the QIC signal generated by a single octave-spanning pulse train if ~200 µW of useful light is present in each of the two spectral tails.

2. Experimental apparatus

For the majority of the QIC measurements, we used the experimental setup shown in Fig. 1, which is similar to that used in Ref. [16]. We frequency doubled a stretched-pulse mode-locked fiber laser (44 MHz repetition rate) at 1550 nm [18] using periodically poled lithiumniobate (PPLN). We estimated the pulse durations for the fundamental and second harmonic light to be approximately 0.2 ps and 3 ps, respectively, from the QIC signal dependence on the pulse temporal overlap. These estimates were consistent with those inferred from the frequency spectra. The relatively long second harmonic pulse duration was due to the finite phase matching bandwidth of the PPLN crystal. We separated the undoubled portion of the fundamental beam from its harmonic using a dichroic beam splitter. The two resulting beams, effectively representing two arms of an interferometer, were then retroreflected (using mirrors M1 and M2), recombined spatially and temporally, and focused onto the LT-GaAs sample. We used an optical bandpass filter in each interferometer arm to ensure spectral purity and as a beam pickoff to sample a small portion of light for monitoring purposes. The lenses L1 and L2 directly before and after the PPLN crystal were singlets with 5 cm and 7.5 cm focal lengths, respectively. The final lens, L3, was a 10x doublet objective. The sample was held in a motorized three-axis translation stage. In order to minimize spatial interference patterns across the sample (which can wash out the measured temporal QIC signal), we optimized the spatial overlap and colinear alignment of the two beams after propagating them several meters.

 

Fig. 1. Setup used to measure the quantum interference control (QIC) signal. The 775 nm arm length was dithered for lock-in detection and the 1550 nm arm length was ramped slowly over several wavelengths to generate real-time interference fringes. The lower right inset shows an image of the LT-GaAs sample with lithographic striplines. L1-L3, lenses; M1-M3, mirrors.

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The inset in the lower right of Fig. 1 shows a magnified image of the undoped LT-GaAs sample used in this work. The sample consists of a 1.84-µm layer of LT-GaAs grown on a 500-µm substrate of semi-insulating GaAs at 400°C with no buffer layer. We performed photocurrent measurements on the sample to verify that there was no significant below-gap absorption. Two 10-µm wide lithographic striplines separated by 30 µm and a tab region separated by 10 µm were deposited on the sample using e-beam evaporation and are composed of 5 nm of Ni, followed by 100 nm of Au:Ge (88%:12%). The gap between the two striplines and both field polarizations were oriented along the [110] crystal axis (y-direction in the Fig. 1 inset). The sample was electrically unbiased for all measurements shown in this work.

As indicated in Fig. 1, the 775-nm arm of the interferometer was dithered sinusoidally over about λ/4 for lock-in detection, while the 1550-nm arm was ramped linearly over several wavelengths at a rate within the lock-in amplifier bandwidth so that real-time interference fringes could be observed from the amplifier analog output on an oscilloscope. We dithered the length of one interferometer arm rather than chopping one beam for lock-in detection to eliminate phase-insensitive signals. When chopping one of the interferometer arms, phase-insensitive single- or two-photon absorption can easily overwhelm the QIC signal.

3. Results

Using the setup of Fig. 1, a typical QIC signal is shown in Fig. 2 with the associated linear ramp on the 1550-nm interferometer arm. Clear fringes were observed that were due to currents generated by interference between single- and two-photon absorptions. As one might expect from an interferometric measurement, the amplitude of the fringes was very sensitive to the relative alignment of the two beams. Therefore, the amplitudes that we measured often varied drastically with slight beam alignment adjustments even when the average optical powers remained fixed. The largest peak-to-peak fringe amplitude that we obtained was 210 µV (23 dB signal to noise, 0.3 kHz bandwidth) for a 9 µm spot diameter for the 775 nm beam (the 1550 nm spot is assumed √2 times larger) with 3.9 mW of 1550 nm light and 0.60 mW of 775 nm light (average power). For the data shown in Figs. 2–4, we used the same spot sizes, but 3.2 mW fundamental and 0.48 mW second harmonic average optical powers. Note that because we dither the optical path length for lock-in detection, we actually measure the derivative of the QIC voltage. Thus, the amplitude of our measured QIC signal is always smaller (the amount depending on the dither amplitude) than that of the QIC voltage difference across the striplines.

 

Fig. 2. Measured QIC fringes due to interference between single- and two-photon absorption. The slight dc offset is not related to the QIC phenomenon.

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A small dc offset voltage is also evident from the signal shown in Fig. 2. Similar offsets were reported in previous QIC measurements [15,16]. The dominant contribution to our offset was from imperfect axial translation of the stacked PZT element that we used to dither the 775-nm path length. Slight tilting of the beam caused positional changes of the 775-nm focused spot on the sample, which slightly altered the detected single-photon absorption signal because this signal depended on spatial proximity to the striplines. The offset remained when the two pulse trains were not temporally overlapped and when the 1550-nm interferometer arm was blocked. We will show later that the offset can be of some use.

We find that the QIC signal depends strongly on the spatial location of the beams on the sample relative to the striplines. Figure 3(a) shows the QIC signal amplitude as a function of the vertical and lateral positions. To generate the intensity plot, we raster scanned the sample in the x (lateral) and y (vertical) directions, while holding the axial (along the beam direction) position fixed. Figures 3(b) and 3(c) show vertical cross-sections of the signal away from and through the tab region, respectively, as indicated by dotted lines in Fig. 3(a). Gray bars in each cross-section plot are provided for reference purposes and have the dimensions of the stripline metallization on the horizontal axis.

 

Fig. 3. (a) QIC signal amplitude as a function of vertical and lateral position of the sample. The QIC signal is largest in the tab region of the sample. (b) Vertical cross-section away from the tab region. (c) Vertical cross-section through the tab region. The horizontal gray bars have the dimensions of the stripline metallization.

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Several characteristics of the observed spatial behavior deserve mention. First, comparing the measured QIC signal shown in Fig. 3(a) with the inset sample shown in Fig. 1 (with a rotation), or with the help of the inset stripline dimensions (gray bars) in Figs. 3(a) and 3(b), it is clear that the stripline metallization blocks a portion of the beams, and thereby decreases the QIC signal amplitude in the metallization regions, as expected. Second, the QIC signal amplitude is largest when the beam is in the tab region, where the electrode separation is similar to the beam spot diameters. Third, in the region between the striplines, but away from the tabs [see Fig. 3(b)], the signal amplitude is largest near the edges of the striplines and falls by more than an order of magnitude at the center (equidistant from the two striplines). Fourth, we observe signal amplitude near the outside edges of the striplines, and the phase of the sinusoidal behavior shown in Fig. 2 flips by ~180 degrees in these regions relative to the region between the striplines.

The last three characteristics show a strong dependence of the QIC signal amplitude on the spatial proximity of the beams to the striplines. The circuit model introduced by Haché, Sipe, and van Driel [16] assumes uniform illumination between the electrodes and is therefore inappropriate for describing such spatially dependent behavior. The observed dependence may suggest a stronger influence of the carrier momentum, leading to particle current, on the QIC signal generation and detection process than was previously thought. Moreover, because the Schottky barrier imparts momentum to the carriers near the metal-semiconductor interfaces through band bending, it may therefore also pay a critical role. The precise mechanisms that cause the observed spatial behavior are currently under investigation. We also note that, by misaligning the spatial overlap of the two beams, we could observe spatial profiles that were not as symmetric as that shown in Fig. 3. More specifically, we could misalign the spatial overlap so that we observed larger QIC signals near one stripline relative to those near the other stripline.

The dependence of the QIC signal on axial (along the beam direction) location of the sample is also of interest. We positioned the beams laterally in the middle of the stripline tabs. Figure 4 shows the QIC signal (a) dc offset and (b) amplitude while translating the sample toward the incoming beams through the beam foci, and also scanning the sample vertically across the MSM gap. The signal amplitude exhibits a peak near the middle of the axial scan range shown as a result of the beam focusing. However, the peak dc offset and the peak amplitude do not occur at the axial same location. This indicates the presence of chromatic aberrations since the offset is due only to the 775-nm beam, while the QIC amplitude is due to both beams, with a peak very near the 1550-nm focus. Figure 4 shows that the peak QIC signal amplitude is displaced axially by 200 µm from the 775-nm focus. We made no attempt to correct the chromatic aberrations for these initial measurements.

 

Fig. 4. (a) DC offset and (b) amplitude of the QIC signal as the sample is translated through the foci of the beams (axial position) and scanned vertically. The separation of the peak dc offset from the peak amplitude indicates chromatic aberrations. The decrease in the signal amplitude in the location of the dc offset peak suggests the influence of excess carrier generation.

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A different and more subtle effect is also evident from Fig. 4. The data indicates that the QIC signal amplitude decreases in the region near the 775-nm focus (axial position of ~480 µm in the figure). We attribute the diminished signal to an increased carrier generation due to single-photon absorption, which leads to a decrease in the effective resistance in the 775-nm focus region. A circuit model for the MSM device and the measurement apparatus shows that such a drop in resistivity can account for reductions in the QIC signal amplitude [16]. Under these circumstances, more current flows directly back across the gap instead of through the measurement device (the lock-in amplifier).

We also examined the precise signal dependence on each of the average optical powers. We placed a rotary variable attenuator in one of the two interferometer arms at a time to change the optical power and we monitored the signal amplitude. Figure 5 shows the signal amplitude as a function of 1550-nm optical power (bottom axis, circles), and as a function of 775-nm optical power (top axis, squares). The measured dependences on the fundamental and second harmonic average powers match the linear and square root behaviors (shown as solid lines in the figure), respectively, predicted in Ref. [16]. For higher average second harmonic power, the QIC signal amplitude is predicted to depart from the strictly square root behavior due to excess carrier generation. Initial signs of this departure may be evident from the data shown in Fig. 5, but higher second harmonic powers were not available to confirm this. Neither curve in Fig. 5 extends to the maximum available power specified in the figure caption because of losses, slight misalignment, and phase front distortions from the attenuator.

 

Fig. 5. QIC signal amplitude as a function of fundamental (bottom axis, circles, 0.58 mW of second harmonic) and second harmonic (top axis, squares, 3.5 mW of fundamental) optical powers at the sample. The measured dependences match the linear and square root behaviors, respectively, predicted in Ref. [16].

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For the purpose of optimizing the signal amplitude, we also examined the signal’s dependence on the beam spot size for this MSM structure. Because the PPLN doubling crystal shown in Fig. 1 was needed for another experiment, we replaced it for this portion of the experiment with a 1.4 cm LBO crystal cut for type I phase matching at 1064 nm, and we replaced the singlet lens L2 with a doublet 4x objective. We inserted a half-wave plate after lens L2 to rotate the 775 nm polarization to vertical. The wave plate did not substantially alter the polarization of the 1550-nm light. Because the doubling crystal was not cut for the 1550-nm wavelength, we corrected the resulting beam walkoff using an extra mirror reflection in each interferometer arm. We verified that using the doublet for L2 significantly reduced chromatic aberrations, as one might expect.

 

Fig. 6. QIC signal amplitude as a function of vertical and lateral sample position using a beam spot size diameter of (a) 37 µm, (b) 15 µm, and (c) 4 µm. The cross-sections attached to the right of each intensity plot show the relative strengths of the signals on linear scales for the three different spot sizes. We used 5.33 mW fundamental and 0.04 mW second harmonic optical power for these measurements.

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Using this modified setup, Fig. 6 shows the resulting QIC signal amplitudes versus lateral and vertical position of the sample for three different objective lenses L3: 4x, 10x, and 40x (infinity corrected, anti-reflection coated for the visible). These gave 775-nm beam waist diameters (full width at 1/e of the field maximum) at the sample of 37 µm, 15 µm, and 4 µm, respectively. The beam diameters at the location of the sample were extrapolated from waist diameter measurements directly before the objective lens L3. These measurements were performed by chopping the 775-nm beam and measuring the resulting decay time with a somewhat fast detector and an averaging oscilloscope [18]. The 1550-nm spot diameters are assumed √2 times larger. We estimated the measurement uncertainty for the spot sizes at the sample to be ±4% based on the standard deviation of the decay times, the uncertainty in the measured chop period and radial distance to the beam, and the uncertainty in the focal lengths of the objectives. The same optical powers were used for each of the data sets shown in Fig. 6. We adjusted and optimized the temporal overlap of the two pulse trains after replacing each objective to compensate for slight differences in dispersion. For these measurements, we made no adjustments to the spatial overlap alignment of the two beams as we changed the objectives. However, we confirmed that such overlap adjustments did not significantly improve the peak signal values. On the other hand, overlap adjustments did affect the symmetry of the signal relative to the striplines, as mentioned previously.

The intensity plots in Fig. 6 are each normalized to the peaks of the particular data set, but the cross sections attached at the right of each plot (with linear scales) show the relative strengths of the signals for the three plots. Again, the stripline metallization blocks a portion of each beam, the effect of which depends on the beam spot sizes. In Fig. 6(a) we find that with 37-µm spot diameter, the individual striplines are unresolved and no signal minimum occurs anywhere between the striplines because the spot sizes are similar to the stripline separation away from the tabs. The peak signal amplitude reaches 25 µV in the region between the tabs. In Fig. 6(b), which shows data for the 15-µm spot diameter, the striplines are just resolved and a dip in the signal occurs midway between the striplines away from the tabs because the spot sizes are similar to the stripline width rather than the stripline spacing. A peak signal of 140 µV occurs midway between the tabs. In Fig. 6(c) we find that the 4-µm spot diameter results in a peak signal of 78 µV, which does not occur midway between the tabs. Instead, because the spot sizes are about a factor of 2 smaller than the tab spacing, two nearly equal maximum signal amplitudes occur near the edges of the tabs and we observe a local minimum in the signal amplitude near the midway point between the tabs.

The decrease observed in the peak signal amplitude (even outside the striplines) between the 15-µm and the 37-µm cases was unexpected and we are uncertain of its cause. To verify that the deleterious effects of excessive carrier generation for high second harmonic power were not prominent, we decreased the second harmonic power and observed no increase in the signal amplitude for any of these measurements. Differences in phase front distortion and dispersion between the three objectives will affect the relative peak values.

4. Single pulse train QIC measurement

We now propose an initial proof-of-principle experiment to use the QIC signal to measure the carrier-envelope phase of a single, octave-spanning pulse train. In particular, we propose to use the setup shown in Fig. 7, which is similar to that described in Ref. [8].

As the figure shows, the output from a modelocked Ti:sapphire laser system is divided by a beamsplitter and each portion is spectrally broadened in a length of microstructure fiber to obtain two octave-spanning spectra. Two separate microstructure fibers are employed in order to utilize as much of the Ti:sapphire power as possible, while keeping the optical power in each fiber below the level where we find that the fiber characteristics and alignment begin to degrade. The broadened output from one of the fibers is used to actively stabilize the carrier-envelope phase evolution of the laser system to a synthesized local oscillator frequency (f _SYNTH) using the established ν-to-2ν stabilization technique. This ensures phase coherence between the spectral extremes of the pulse train [8]. The broadened output from the second microstructure fiber is used to measure, rather than stabilize, the carrier-envelope phase evolution using the QIC technique. However, instead of simply directing the entire pulse onto the QIC sample, we propose to use a prism-based interferometer to compensate for dispersion in the nonlinear fiber (by adjusting the positions of mirrors M3 and M4) and to preferentially select the spectrally useful portions of the pulse train. As shown in the figure, the majority of the pulse spectrum will be filtered out and discarded in order to prevent the deleterious effects of excessive carrier generation, as discussed above. The two spectral wings of the pulse are retroreflected, recombined, and focused onto the LT-GaAs sample. We can then perform lock-in detection of the QIC signal at the frequency f _SYNTH.

 

Fig. 7. Proposed experimental setup for measuring the QIC signal generated by the two spectral extremes of a single octave-spanning pulse train. In the prism-based QIC interferometer, the spectral centers of the pulses are discarded as shown to avoid excess carrier generation. Dotted lines indicate electrical paths. M1-M5, mirrors; BS, beam splitter; MS, microstructure; P1-P2, prisms.

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We anticipate future improvements to the QIC signal strength from optimization of the spot size, gap spacing, structure shape, and reduction of the detection noise level. The fact that we will use higher energy photons (~1200-nm fundamental) and that we will not need to dither one path length for lock-in detection in the proposed experiment should also improve the signal strength. Nevertheless, without these improvements, a simple extrapolation from our measured signal strength and power dependence yields an estimated single pulse train QIC signal with a 5–10 dB SNR (1 kHz bandwidth) if the microstructure fiber can generate ~200 µW of useable average power at each of the two spectral extremes (ν and 2ν) in high-quality spatial modes. With increased signal fidelity (typically 20 dB SNR in a 100 kHz bandwidth is needed), the roles of the two interferometers can be reversed and the QIC signal can be used to stabilize, rather than just measure the carrier-envelope phase evolution. Furthermore, due to the relatively shallow absorption depth, and hence small dispersion, one can envision using QIC to measure the carrier-envelope phase itself, rather than merely its evolution from pulse to pulse.

5. Conclusion

We demonstrated real-time QIC fringes by focusing a pair of harmonically related pulse trains onto a region of LT-GaAs between two stripline electrodes. We uncovered a critical dependence of the QIC fringe amplitude on spatial location of the focused spot relative to the striplines. This finding suggests the possible influence of carrier momentum and Schottky barriers on the QIC generation and detection process. We also established precise QIC signal dependences on the fundamental and second harmonic optical powers in the low power limit and we investigated the signal dependence on the beam spot size. Using these measurements, we proposed a proof-of-principle experiment to detect the carrier-envelope phase evolution of a single octave-spanning pulse train. We estimated that a SNR of 5–10 dB (1 kHz bandwidth) is possible with ~200 µW of useable average power in each of the two spectral extremes and no further improvements to the measurement process.

Precise investigations of different gap separations and shapes are currently underway to further enhance the signal. Furthermore, due to the strong dependence shown in this work on the spatial location of the focused spot relative to the striplines on the sample, theoretical and experimental investigations of the possible influence of carrier momentum and Schottky barriers on the QIC signal are planned for the near future.