The preservation of carrier envelope phase (CEP) during Cross-Polarized Wave Generation (XPWG) is demonstrated through two independent experiments based on the spatially and spectrally resolved interference fringes formed by the XPW beam and its fundamental. In a first measurement, we found that the vertical fringe position on the spatial detector was maintained over many consecutive laser shots, implying practically no change in relative CEP between the XPW and the fundamental. In a second experiment, we measured the change in relative CEP between the XPW and fundamental beam by systematically varying the amount of material dispersion inside the XPW arm of the interferometer. The recorded rate of relative phase change was in excellent agreement with the theoretical value.
©2009 Optical Society of America
The carrier-envelope phase (CEP) of few-cycle light pulses becomes a critical parameter for intense laser-matter interactions since the pulse envelope can vary almost as rapidly as the laser electric field itself [1–3]. It is therefore essential to stabilize the intrinsic shot-to-shot CEP drift from the laser source. Laser-matter interaction experiments above at intensities 1015W/cm2 require pulses with high temporal contrast. One of the most recent methods to enhance the temporal contrast of high-power femtosecond pulses is cross-polarized wave generation (XPWG) [4–6]. XPWG is a direct achromatic χ(3) process in which a new wave, polarized in the orthogonal direction, is efficiently generated in a cubic crystal . Recent studies have shown that XPWG can also help smooth the input laser spectrum and broaden it by more than a factor of two [8–10]. While XPWG results from four-wave mixing and is therefore expected to preserve CEP, it is important to demonstrate that the process does not deteriorate the CEP stability of the system, especially when implementing a XPWG contrast filter inside a CPA laser chain.
Spectrally and spatially resolved interferometry (SSRI) has been proven to be a powerful technique for dispersion measurement of various materials and optical elements [11–15], characterization of non-linear processes [16,17] as well as for the measurement of laser pulses [18–20]. It typically consists of a two-beam interferometer equipped with an imaging spectrograph. The interference pattern is imaged onto the input slit of the spectrograph, so that spatially (along the slit) and spectrally resolved interference fringes are formed on the 2D detector (CCD camera) located at the output plane of the spectrograph. The slope and the curvature of the interference fringes are characteristic of the dispersion coefficients of the sample as group delay (GD), group delay dispersion (GDD), etc…. The absolute spatial position of the fringes is, however, directly and uniquely related to the relative carrier envelope phase of the sample and reference beams .
In this paper we give clear experimental evidence that the carrier envelope phase is preserved during the XPWG process.
2. Theoretical background of the measurement
To introduce the basic idea of the measurement, let us regard first a spectrally and spatially resolved Mach-Zehnder interferometer illuminated by a femtosecond pulse (Fig. 1a ). After the first beamsplitter, the reference pulse with a spectral intensity IR(ω) propagates undisturbed in one arm of the interferometer while the spectral phase of the sample pulse (IS(ω)) is shifted by a dispersive optical element with respect to the phase of the reference pulse. It is customary to describe the phase shift of a pulse in the form of Taylor series around the center frequency ω0 of the laser pulses, so that the phase shift of the reference and the sample pulses within the interferometer is
Please note that the physical meaning of the first member of the series, that is φR,S(ω0), is the initial absolute phase of the field. Hence φR,S(ω0) is directly connected to the carrier-envelope phase defined usually by the phase difference of the carrier wave and the maximum of the intensity envelope . The difference of CEP and above mentioned initial phase is a constant that is characteristic to the envelope. Thus, the position of the carrier wave relative to the intensity envelope can be uniquely described by φR,S(ω0) as well.
When the pulses are made to temporally overlap at the second beamsplitter with their phase fronts tilted by an angle ε (Fig. 1b), interference fringes are formed along the y axis parallel to the input slit of the spectrograph. The intensity distribution of the spectrally and spatially resolved interference fringes is15].
The relative spectral phase shift Δφ(ω), that is the difference between the phase of the reference and the sample pulses at the output of the interferometer, can be expressed as15].
The leading spirit of our method is the first member of the series since it is basically the carrier-envelope offset phase difference ΔφCE between the reference and sample pulses. One can hence write
Let us assume that the subsequent pulses in the sample arm are provided with different CEP while their higher order dispersion relative to the appropriate reference pulses (D(ω)) is not affected. In this case, the maxima of the otherwise identically shaped spectrally resolved interference fringes generated by subsequent reference and sample pulses at ω0 would be located at y1, y2, y3, ..., each of them being characteristic of respective CEP differences ΔφCE1, ΔφCE2, ΔφCE3, ... . Consequently, from the shift Δy1,2,3 of the individual interference fringes, the amount of CEP change in the sample arm can be deduced. If the detector, i.e. the CCD chip of the imaging spectrograph, is slow or its exposure time is chosen long enough, then it actually captures many subsequent interference patterns, stacked on top of each other. As a result, the visibility of the recorded interference pattern would be substantially degraded. However, if there is no CEP difference between the pulses propagating in the reference and the sample arm, then the position of the fringes should be maintained and the visibility of the integrated fringe system is expected to be independent of the exposure time.
In our experiment (Fig. 2 ) the laser source is based on a commercial 1 kHz CEP-stabilized chirped pulse amplification laser system from Femtolasers GmbH. The energy of the laser pulses is boosted by a home-made multipass amplifier, while the final compression takes place in a hybrid compressor constructed of prisms and chirped mirrors. The system produces 4 mJ, 21 fs, laser pulses at 800 nm . To avoid white light generation but keep the XPWG process saturated, the pulse energy is considerably attenuated prior to the XPWG stage, consisting of a f = 1.0 m focal lens and a holographic cut BaF2 crystal . The energy of the fundamental pulse at the front surface of the crystal is 40 μJ providing an intensity in the crystal less than 1012 W/cm2, whereas the XPWG efficiency reaches 15%. The generated XPW beam and the rest of the fundamental pulses were collimated by a second lens. The orthogonally polarized XPW beam is split off from the fundamental beam by a Glan polarizer. After adjusting their relative polarization and timing, both beams are sent to the input slit of an imaging spectrograph.
Two independent experiments were carried out to measure the relative change in CEP between XPW and fundamental beams. First the fringes were recorded for different exposure times of the CCD camera. Here, both the position and the visibility of the fringes were measured. In a second experiment, the CEP of the XPW beam was varied independently of the fundamental beam with a wedge pair and the relative CEP change was determined.
4. Dependence on the exposure time
In the first experiment, the exposure time of the CCD camera was set to 1 ms, 10 ms, 100 ms and 1000 ms to test the fringe stability and visibility over time. Since the repetition rate of the laser is 1 kHz, the captured images result from the interference of a single pulse, ten pulses, hundred pulses and one thousand pulses, respectively.
Nine interferograms were captured for each exposure time. Figure 3 shows images compiled from slices cut out from each interferogram along the spatial axis (y) at three different wavelengths. As one can see, the position of the fringes is maintained over nine subsequent interferograms captured with single pulses, that is, for an exposure time of 1 ms. There is very little change for 10 pulses (10 ms). The position changes slightly for 100 pulses, due most probably to fluctuations in the interferometric setup. This change, however, is less than π/4 for CEP. The position of the fringes of the interferorgrams for 1000 pulses (1000 ms) is again very stable. As is clear from the composed images at three different wavelengths, the vertical position of the fringes is maintained over the large variation of the exposure time. Moreover, the spatial position of each fringe is maintained during the relatively long experiment, that is, it is the same for the first 1 ms fringe system as for the last 1000 ms one.
The above evaluation of the fringes is spectacular but restricted to certain wavelengths. We can assess the entire interference pattern, that is each spatial point at each wavelength, if we compute the visibility of the fringes for the whole interferogram. The visibility map was calculated for each interferogram of each series and the average of nine maps consisting of each series is displayed on Fig. 4 .
From the captured interferograms, the visibility was also evaluated, varying from 0.883 to 0.823 as the exposure time was increased from 1 ms to 1 s. Moreover, it is worth noting that the average visibility is highest for the 100 ms case even if the position of the fringes changed (see Fig. 3). This small change in visibility is due to inevitable fluctuations of the large and uncovered Mach-Zehnder setup. It may be worth mentioning that similar result would have been obtained if interference fringes of subsequent pulses were recorded, then evaluated. In the lack of ultrafast CCD camera and frame grabber system, however, it was not possible.
5. Control of CEP of the XPW beam
To separately confirm the stability of the relative CEP, in the second measurement we introduced a known change in relative phase difference between the two arms of the interferometer. A pair of fused silica wedges with an apex angle of 10.3° was inserted into the XPW arm of the interferometer. The change in relative CEP between XPW and fundamental beams was measured as a function of the amount of material inserted into the XPW beam by moving one of the wedges. Nine interferograms were captured at each wedge position, controlled with a precision translator in 10 μm steps.
The change in fringe position shown in Fig. 5 is displayed in a similar way to that used in Section 4 with the difference that now only one slice of the first interferogram of each series is shown (Fig. 5). Note, that the spatial position of a fringe system is determined by the relative CEP as well as relative group delay between the interfering pulses. Upon shifting one of the wedges, both the CEP and the GD are changed. As a result, not only does the shape of the fringes (practically their slope) change but also their vertical offset position. Only at the central wavelength, where the interferometer was originally balanced and hence the SSRI fringes were straight, do the fringes stay in the same position. However, they apparently suffer a phase jump of 2π. These phase jumps are clearly indicated and calculated from the shift of the side wavelengths. The reason is that a wedge displacement of 10 μm already causes a phase jump somewhat larger than 2π.
When calculating the relative total phase shift between the fundamental and the XPW beam, the interferograms captured at the initial wedge position were considered as the starting point. The absolute change in CEP for the XPW beam was then calculated for each wedge position. The values of the change in GD as a function of wedge insertion were calculated from the slope of the interference fringes (Fig. 6a ). As one can see, the rate of GD change versus wedge position of 280.5 fs/mm is in very good agreement with the theoretical value of 283.1 fs/mm. In Fig. 6b. the rate of phase change of the XPW beam over 100mm was evaluated at 16.4 rad/mm, which matches well the theoretical value of 18.7 rad/mm.
One can, of course, describe the change of the absolute CEP resulted in both the phase and the GD variation upon shifting the wedge (Fig. 7 ). The slope of the curve is 621 rad/mm, while the theoretical value is 647.0 rad/mm. However, if only the 100-200 μm range is taken into account, then the measured value of 644.6 rad/mm is in an excellent agreement to the theoretical value.
The slight discrepancy between the theoretical and measured values for GD, CEP and the absolute phase change may be due to two reasons. First, the exact type of fused silica (and the temperature of the laboratory environment) differs from that one for which the dispersion equation has been measured . Second, a slight deviation of about 15’ of the manufactured apex angle from the specification can also result in such small discrepancies.
In summary, we have proved through two independent measurements that the CEP is preserved during XPWG. Both measurements rely on the spatially and spectrally resolved interference fringes formed by the XPW beam and its fundamental. In a first experiment, we have shown that the spatial position of the spectrally and spatially resolved interference fringes is determined by the carrier envelope phase difference between the fundamental and the XPW pulses. It was found that their position is maintained over many hundreds of pulses in the laser pulse train and the visibility of the fringes is independent of the number of pulses interfering. In a second experiment, the CEP of the XPW pulses was manipulated independently of the fundamental beam. The measured change in absolute phase of the XPW beam is in excellent agreement with the expected values. Such an accurate and noise-free measurement would not be possible if there were random shot-to-shot changes in the relative CEP between the fundamental and the XPW.
Hence, both experiments above reveal that the CEP of the XPW pulses changes with the same rate as that of the fundamental pulse, i.e. there is no visible loss in CEP stability of the generated XPW beam compared to that of the fundamental beam. These findings might help further spread the use of XPWG in high intensity pulse shaping technology. In particular, with the use of XPWG as a nonlinear filter in a double-chirped pulse amplification scheme , not only could intense few-cycle pulses be generated with high temporal contrast  but also with stable CEP.
This work was supported by the Hungarian Scientific Research Found (OTKA) under grant No K75149.
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