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Abstract
Liquid-crystal spatial light modulators (LCM) are a common tool to tailor femtosecond laser pulses. The phase stability of 1 kHz, sub-20 fs visible shaped and unshaped pulses are investigated. Our results show that the spectral phase after the LCM varies from pulse to pulse leading to strong deviations from the predicted pulse shapes. This phase instability is generated only by LCM and is strongly temperature dependent. Based on the experimental data, a numerical model for the phase was developed that takes the temperature-dependent phase instability as well as pixel coupling across the LCM into account. Phase stability after the LCM can be improved by an order of magnitude by combining the control the temperature of the LCM and by using rapid-scan averaging. Reliable pulse shapes on a pulse-to-pulse basis are crucial, especially in coherent control experiments, where small differences between pulse shape are important.
© 2017 Optical Society of America
1. Introduction
Tailoring of laser electric fields has found widespread applications in several areas of optics and spectroscopy [1–4]. Manipulation of the phase, polarization and amplitude of ultrafast optical fields permits the control of a vast range of phenomena in a broad spectral range [5–9]. Spatial light modulators (SLM), in particular those based on liquid crystal (LC) molecules, have been one of the most used approaches to tailor properties of femtosecond laser pulses. Liquid crystal modulators (LCMs) are commercially available, covering spectral ranges from visible to mid-infrared, with several hundreds of pixels and transmission efficiencies well above the typical transmission efficiencies of other types of modulators [10–19].
In spite of its extensive use, tailoring of laser fields using LCMs is not always straight forward. A well-known artifact is the so-called space-time coupling. It is an inherent optical artifact when SLMs in general are used in a 4f-setup. In such an optical arrangement, the modulation of the temporal shape of a short pulse also affects its transverse spatial distribution. Nevertheless, space-time coupling can be minimized with a suitable choice of geometrical parameters for mirrors and gratings [20, 21].
An additional challenge in the use of LCM is related to the orientation fluctuations of its LC molecules and the effect on the modulation stability in control experiments, which until now has been remained undiscussed. LCMs are based on the orientation of birefringent molecules via external application of a control voltage [22]. The re-orientation of the molecules with the applied electric field leads to a change of the effective index of refraction with respect to a given polarization of the light being transmitted through such a medium. In spite of an applied control voltage, the orientation of LCs is not fixed in space but is only constrained in a libration-like movement. As a consequence, such hindered orientational movement takes place in each pixel of an LCM and therefore can lead to noticeable phase instabilities on the output laser pulse. In this work, we investigate in detail the pulse-to-pulse stability of femtosecond pulses going through commercial LCMs. We initially illustrate the phase instability in a shot-to-shot second-order intensity autocorrelation and its dependence on the temperature of the LCM. This experimental evidence is further expanded by measuring the integrated second-harmonic, overall beam energy and beam directional instability. Since the hindered orientation of the nematic LC molecules is strongly affected by the temperature, it can be experimentally exploited to reduce the phase instability effect on tailored pulses. Moreover, we will address the question why this effect plays a role under some experimental conditions, while in others e.g. signal averaging conditions it is difficult to observe. This point is motivated by the different kinds of measurements performed in time-resolved spectroscopy and coherent control using shot-to-shot data acquisition (e.g. rapid-scan approach [23, 24]) and long term signal averaging (e.g. step scan). Furthermore, the second-order intensity auto-correlation is also numerically simulated to get a deeper insight into the temperature dependent phase instability on the pulse output. The simulation includes pulses with second-order Taylor and sinusoidal phase parameterizations to demonstrate the action of phase noise on typical shaped pulses in coherent control experiments.
2. Material and method
2.1 Experimental details
For all experiments, the output of a Ti:Sa amplifier (Libra, Coherent) at 800 nm with a repetition rate of 1 kHz pumped a non-collinear optical parametric amplifier (NOPA). The NOPA spectrum was centered at 540 nm and the pulse duration was 14 fs. The LCM was placed in the Fourier plane of a 4f-setup with f = 60 cm and gratings with 900 grooves/mm (Master No. 1329, Richardson Gratings). The specifications of the two LCM models utilized are summarized in Table 1. For all measurements, except for the tailored pulses, maximum voltage was applied to all pixels of the LCM. This way, the spectral and voltage calibration effects on the LCM can be excluded. The small additional chirp due to the material of the LCM was manually compensated by the optimization of the second grating position.
Table 1. Specifications of the two LCMs given by the manufacturer.
The temperature of the LC displays was controlled by fixing two aluminum blocks on the housing of the LCM above and below the LC display. The cooling water continuously flowed through these aluminum blocks. For LCM 1 two temperature sensors (according to manufacturer not absolutely calibrated) on both sides of the display monitored the temperature while LCM 2 did not include any temperature sensors. The electronics heated the LCM 1 to 26.0 °C without cooling. Around 30 min after starting the circulation of cooling water, its temperature decreased to 20.0 °C. The measurements with the cooled LCM 1 and LCM 2 were started after at least 45 min in order to ensure stable thermal conditions. The temperature of the lab was kept constant at 21.0 °C ± 0.1 °C.
For the first measurement, the second harmonic generation autocorrelation (SHG-AC), the energy per pulse and beam stability (energy passing through pinhole) were recorded simultaneously. The SHG-AC was measured with a shot-to-shot commercial autocorrelator NOPA-PAL [25]. In this commercial device, the time delay between the two replicas of the pulse is realized by a continuously moving piezo, i.e. shot-to-shot data acquisition. Its position and the signal of the photosensitive detector are acquired as a data pair. This means that each point (delay) of the SHG-AC is generated by one laser shot.
To monitor the energy per pulse shot to shot in the same way as in the SHG-AC, the beam was focused on a photodiode (PDA55, Thorlabs). A spatial portion of the beam was detected by another photodiode (PDA155, Thorlabs) for the additional information about the beam directional stability. In a second series of measurements, the collinear second harmonic (SH) signal was detected with a photodiode (PDA155).
Each of these measurements was performed under several experimental conditions: First, the light directly from the NOPA, i.e. without 4f-setup, was measured as a reference. Second, the beam was sent through the non-cooled LCM and the same measurement as before was acquired. Next, the LCM was cooled and the measurement was performed with the cooled LCM. Finally, the first measurement without 4f-setup was repeated to control and exclude any possible drifts of the laser influencing the results.
2.2 Simulation details
A numerical noise model was implemented using Lab2 [26], an add-on to LabVIEW, to reproduce the experimental shot-to-shot SHG-AC with cooled and non-cooled LCM. Input for the simulation was a Gaussian pulse spectrum (512 points) with the same central wavelength and spectral width as in the experiment. Moreover, the geometry of the 4f-setup was a direct input parameter set. To model the SHG process, non-depleted three wave mixing was implemented with optical parameters like thickness and cutting angle of the nonlinear crystal comparable to the experiment. Noise was set as a direct input parameter on the spectral phase φ(ω) of the laser pulses. Therefore, the spectral electric field E(ω) was modified according to
Two different approaches were used to simulate the experimental noise. The first approach uses different kinds of noise (uniform noise with various magnitudes as well as Brownian (1/f2) noise). The noise was generated independently for each spectral component (pixel). In a second approach, a coupling between the pixels was implemented. The coupling between the pixels was generated by using six supporting spectral components which were distributed evenly among the spectral components of the pulse. An initial phase for each of the six supporting spectral components was generated by Brownian noise, which was then multiplied by a vector containing N samples in time with the same experimental frequency dependence (see section 3). From these data, the noise for all pixels was calculated via spline interpolation.
To distinguish between the cooled and non-cooled case, the magnitude of the noise was varied by a factor of two. Thus, the limits for the phase were set to be for the non-cooled (cooled) case. 150 shot-to-shot SHG-AC traces were generated in a time window of ± 150 fs with a temporal resolution of 2 fs.
3. Experimental results
Due to its straightforward implementation, SHG-AC traces were detected to characterize the optical instabilities due to the LCM on a shot-to-shot basis. In the following, the results for LCM 1 are shown. The shot-to-shot SHG-AC traces are compared in Fig. 1 for the three different cases (without 4f-setup, with cooled LCM and non-cooled LCM). The SHG-AC traces mainly fluctuate in their maximum height, while the shape and width remain nearly the same, without 4f-setup, Fig. 1(a), as well as with 4f-setup but without LCM (not shown). This behavior is very similar for the cooled LCM as shown in Fig. 1(c). For the non-cooled LCM, Fig. 1(b), however, the overall SHG-AC shape strongly differs. For some traces, wings with different shape and intensity appear on both sides of the traces. This instability can be numerically quantified by fitting the traces to determine the pulse duration. The individual traces are fit (not shown) with a Gaussian to determine the pulse duration via the width of the Gaussian. Thus, the relative standard deviation of these values is used as a measure for the stability of the shape of the laser pulses. For the measurement without 4f-setup, the pulse duration has a stability of 0.65%. The non-cooled LCM significantly increases the value to 7.2%; however, the cooling reduces it to 2.0%.
Fig. 1 SHG-AC traces measured (a) without 4f-setup, with the (b) non-cooled and (c) cooled LCM 1. Particular traces are selected and highlighted in color.
The instability is not only present in the SHG-AC traces with maximum voltage at all pixels of the LCM, but can also be clearly observed in SHG-AC traces of shaped pulses as shown in Fig. 2. Second-order Taylor (linear chirp of 100 fs2) and sinusoidal phase parameterizations (called multipulse, interpulse spacing of 100 fs) are chosen which are typical for experiments with LCMs [27–30]. Figure 1(b) shows that the instability of the linearly chirped pulse for the non-cooled LCM is for this example stronger than for the unshaped pulse. For cooling the LCM, as done for Fig. 1(c), a clear improvement is achieved for both parametrizations with a final instability comparable as for the unshaped pulse. Although this clearly shows that the instability as well as the cooling effect is independent of the calibration of the LCM and of the voltage control of the single pixels, the instability has different effects on different parameterizations. For example, for multipulse parameterization, the relative amplitudes of the subpulses are strongly affected (by almost a factor of 3), while the interpulse separation is only slightly disturbed (by less than 5%).
Fig. 2 SHG-AC traces with a non-cooled (a, b) and cooled (c, d) LCM 1 for different phase parametrizations: (a, c) chirped pulses second-order Taylor (100 fs2) and (b, d) multipulses (interpulse spacing 100 fs). Particular traces are selected and highlighted in color.
Optical instabilities in the SHG-AC signal as observed in Figs. 1 and 2 do not necessarily need to be caused only by instabilities of the optical phase, but they can also be caused by instabilities in e.g. the pulse energy, the beam direction after the LCM as well as beam overlap during the SHG-AC process. In order to verify and exclude other origins for the instabilities in the SHG-AC traces, three additional measurements (energy, pointing stability and SH signal) were performed. The absolute energy and pointing of the beam going through LCM were measured shot-to-shot (not shown). The relative standard deviation for both quantities over 20000 pulses is mainly determined by the laser-typical noise (1.1%) and does not change whether a mask is used or whether it is cooled or not. Consequently, these quantities do not cause the instabilities seen in the SHG-AC traces of the non-cooled LCM. The SH signal alone is equally sensitive to the optical phase, but it is not sensitive to other parameters like beam pointing stability and beam overlap in the SHG-AC process. Differences for the cooled and non-cooled LCM are clearly visible in the pure SH signal: Without 4f-setup, a sequence of 20000 pulses has a relative standard deviation of 1.3%. This value jumps to 6.7% for the non-cooled LCM. Cooling the LCM improves the stability to 4.9%. These results clearly show that the instability detected in the SHG-AC and SH measurements have the same source, i.e., the optical phase of the laser pulse.
To gain more insight into the instability’s time dependence and the changes caused by the LCM and temperature dependence, the power spectrum of the SH signal detected shot-to-shot is determined. The power spectrum on a linear scale, Fig. 3(a), mainly shows components with frequencies f < 1 Hz for the measurement without 4f-setup as well as with 4f-setup but without LCM (not shown). Frequency components up to f = 10 Hz contribute clearly to the power spectra for the experiments with LCM. The components for the non-cooled LCM are on average twice as strong as for the cooled LCM. The linear scale in Fig. 3(a) clearly illustrates the impact of the LCM and of its cooling. The same power spectra are plotted on a log-log scale and displayed in Fig. 3(b). This representation shows higher frequency components and the frequency dependence in more detail. Without 4f-setup, the spectrum scales with 1/f2 for f < 27 Hz. Such a frequency dependence is typical of laser systems [24, 30, 31]. In the cases with LCM (cooled and non-cooled), components for f < 3 Hz are constant and most dominant. Up to f = 27 Hz, the spectrum scales with 1/fn with n changing from 2 to 5. The shape of the power spectrum for the cooled and non-cooled LCM is the same but the power of the components differs for f < 27 Hz. In all three cases, the components for f > 27 Hz are distributed uniformly with powers two orders of magnitude smaller than in the low-frequency range.
Fig. 3 Power spectra of the experimental SH signal on a linear (a) and log-log (b) scale without 4f-setup (black squares), after the non-cooled LCM 1 (red circles) and after the cooled LCM 1 (blue triangles).
Moreover, in order to test the generality of the experimental results observed for the LCM 1, the measurements of the SH signal were repeated with an additional commercial LCM. LCM 2 shows a very similar behavior as LCM 1. The relative standard deviation of the SH signal is 6.7% for the non-cooled LCM 2. It can be reduced to 4.3% by cooling. As for LCM 1, the components of low frequencies increase strongly when the LCM 2 is non-cooled and a clear cooling effect is similarly visible (not shown). Although the architecture of the two LCMs is not exactly the same (LCM 1 has two displays while LCM 2 has only one), the experimental results above and the observed temperature dependence undoubtedly shows that the same physical mechanism is behind the experimental phase instability. A more quantitative comparison between a single and double mask is at the moment not possible as there are too many unknown differences between the two LCMs as e.g. the effective thickness of the LCs, their material and properties as well as the modulation ac frequency.
4. Simulation results
The experimental SHG-AC traces with unshaped and tailored pulses, Figs. 1 and 2 show two important features to understand the source of the phase instability. The first one is the strong temperature dependence, which obviously indicates a temperature-dependent physical mechanism. The second one, which is not as obvious as the first one, is especially depicted in Fig. 1(b); not all SHG-AC traces are equally affected, i.e. there are traces which strongly differ (e.g. the red trace) from a Gaussian shape while others (e.g. the blue trace) do not. Since a single SHG-AC trace takes more than 250 ms (250 laser pulses at 1 kHz repetition rate) to acquire, this behavior suggests that a low-frequency noise must be the main source for the experimental phase instability. This is corroborated by the power spectra, Fig. 3, which shows the presence of low-frequency components with higher amplitude when the LCM was used. Perhaps even more interesting than that, the behavior observed in Fig. 1(b) also suggests an additional feature of the phase instability. Complete random phase noise at each pixel of the LCM would lead to a SHG-AC with weaker intensity but not to asymmetric traces as observed in Fig. 1(b). The asymmetric traces in Fig. 1(b) suggests that several spectral components of the laser spectrum passing through different pixels of the LCM must be under the effect of a similar phase instability. Based on these considerations, a numerical noise model was implemented (see section 2.2) to reproduce the experimental shot-to-shot SHG-AC with cooled and non-cooled LCM.
In our numerical simulation, the initial laser pulses (i.e. without 4f-setup) were chosen noise-free since the focus was on the effect of the LCM and the temperature dependence. With regard to the discussion above, two approaches in respect to the noise simulation were implemented. For the first one, completely uncorrelated Brownian noise, the simulated SHG-AC traces were affected only in the intensity as expected and the traces were symmetric (not shown). These results hold for noise with intensities over several orders of magnitudes like 1/f2 noise as well as for other power spectra with higher noise order, e.g. 1/f3. In the second approach, 1/f2 noise with a coupling between pixels is combined with noise whose magnitude is constant per pulse over all pixels. Best agreement with the experiment could be achieved for this type of noise. The magnitude of the noise was chosen to introduce ± 0.6 radian as a maximum additional phase in order to reproduce qualitatively the experimental SHG-AC. In our numerical simulation, the noise magnitude was reduced by a factor of 2 to simulate the cooling effect. Under these assumptions, Fig. 4 shows the simulation results for all three parametrizations (TL, chirp, multipulse) under cooled and non-cooled conditions. In a very good agreement with the traces depicted in Figs. 1(b) and Figs. 2(a)-2(b), Figs. 4(a)-4(c) displays the same asymmetrically distorted traces observed in the experimental traces. Moreover, the SHG-AC distortion is strongly suppressed when the noise magnitude is halved (see Figs. 4(d)-4(f)), which qualitatively replicates the experimental temperature dependence.
Fig. 4 Simulation of shot-to-shot SHG-AC traces for the non-cooled (a)-(c) and cooled (d)-(f) cases for different parametrizations of the phase; (a), (d) TL, (b, e) linear chirp of 100 fs2 and (c), (f) multipulse with interpulse spacing 100 fs. Particular traces are selected and highlighted in color.
5. Discussion
Having numerically simulated the experimental temperature dependence, there is still a central question about under which experimental conditions the LCM phase noise as described above can be observed or even minimized.
In time-resolved spectroscopy and neighboring research fields like coherent control, there are basically two ways of averaging optical signals. One way is based on the N-times acquisition of a given optical signal in synchronization with the laser repetition frequency without changing any parameter (e.g. delay between pump and probe). After these N signals are acquired and averaged, parameters are changed (e.g. a new delay between pump and probe). Since the scanning of parameters takes place step-wise, this is called step scan. Obviously this way of averaging can be very sensitive particularly to low-frequency noise for large N. Another approach for averaging signal often used to overcome low-frequency noise is the so-called rapid-scan technique. In this method, only one single optical signal (i.e. one laser pulse) is acquired before experimental parameters are changed. This way, a single scan is made of only a few number of laser pulses. The averaging over many (typically tens or hundreds) of scans leads to a data set, which is free of low-frequency noise, i.e. like applying a high-pass filter. This technique of averaging optical signals has been exploited in, for example, many multidimensional time resolved spectroscopies [32–34].
The low-frequency noise described for LCMs in the previous sections can be strongly suppressed when rapid-scan averaging is applied. To illustrate the influence of the averaging on the LCM phase noise, the experimental SH data are evaluated with these two averaging schemes, namely rapid and step scan. A sequence of 4900 laser pulses was detected. In the step-scan averaging of the SH signal, each data point is the average of N = 70 laser shots detected successively as shown in Figs. 5(a)-5(c). In the rapid scan, each data point is also the average of N = 70 laser shots, Figs. 5(d)-5(e), but not detected successively (see Fig. 5(g)). The differences between the averaging approaches and the effect on suppression of the LCM phase noise are striking. In all cases (no LCM, non-cooled and cooled LCM), the instability of the SH signal averaged with rapid scan is smaller than with step-scan. Particularly remarkable is the effect of the rapid-scan averaging when the LCM (non-cooled and cooled) is used. The phase noise due to the LCM is suppressed by one order of magnitude. In general, the data shows that using rapid-scan averaging leads to an almost noise-free measurement with, independent of whether the LCM is cooled or not as shown in Figs. 5(d)-5(f). A noteworthy observation concerns the rate of the rapid-scan averaging. Here N = 70 SH signals were chosen, which means that 70 laser pulses, which were 70 ms apart, were averaged as a data point which allowed an almost complete suppression of the LCM phase noise. This parameter must be carefully chosen according to the power spectrum and its main spectral contributions though.
Fig. 5 A comparison between (a)-(c) step-scan and (d)-(f) rapid-scan averaging. For both schemes 70 data points are calculated from the SH signal of 4900 adjacent laser pulses; (a), (d) without 4f-setup (black squares), with (b), (e) non-cooled (red dots) and (c), (f) cooled (blue triangles). The gray dashed line is a guide to the eye for the deviation of the points from the mean over the signals from all 4900 laser pulses. (g) Average scheme using step scan and rapid scan for the same set of data.
The consistent results for the SH signal for both LCMs hints that the LCM phase noise should appear in all kind of experiments, in particular when step-scan averaging is used. However, to our knowledge, this has not been the case. There are several examples of e.g. control experiments using similar LCMs, with reported phase stability orders of magnitudes better than the ones recorded here [35–37]. The explanation for that resides on experimental parameters like the laser repetition rate and the tailored spectral bandwidth. Experiments using sources with repetition rates of several hundreds of kHz up to MHz can easily average well more pulses than what is feasible with kHz sources (like in this work). Moreover, averaging e.g. 70 laser shots with an 80 MHz laser takes less than 1 µs, while for a 1 kHz laser it takes 70 ms. This illustrates very clearly the fact that high repetition sources are well less sensitive to low-frequency noise than low-frequency sources. More important than that is perhaps the tailored spectral bandwidth. Most experiments using LCMs are still performed using spectral sources with a bandwidth of a few tens to a few hundreds of wavenumbers spectrally spread over several hundreds of pixels of the LCM. Very few experimental works have been performed with bandwidths over 1000 cm−1 like in this work. A phase noise of few radians per pixel, as observed and simulated here, has a very small effect on a laser with smaller bandwidth. For example, this is further illustrated in Fig. 6. While a 30 fs pulse is still affected by such phase noise, a spectrally narrow 100 fs pulse remains completely unaffected.
Fig. 6 Simulation of shot-to-shot SHG-AC traces for the non-cooled, TL cases for pulses with pulse durations of (a) 30 fs and (b) 100 fs. Particular traces are selected and highlighted in color. Phase noise was the same as in Fig. 4.
Finally, possible origins of the phase instability still remain to be discussed. The working principle of LCMs is based on the change of the phase by the orientation-dependent refractive index of the birefringent nematic LC molecules. In order to avoid electromigration effects, a high frequency (often several kHz) square wave voltage is applied to control the LC orientation and therefore the phase within the single pixels [22, 38]. The application of a square wave voltage is based on the fact that the orientation of the LC does not depend on the sign of the voltage, but only on the amplitude of the square wave. In spite of that, one cannot exclude without further experimental work, that nematic LCs under this electric field are completely oriented without any additional orientation dynamics. Depending on the angle of the LCs, a small variation of the orientation can cause large changes in the birefringence, and therefore, change in the optical phase. For example, the phase instability experimentally observed and simulated in this work was on the order of one radian. This amplitude corresponds to a variation of about 10° in the LC orientation, according to the LCM manufacturer and calibration curves. Therefore, one plausible cause for the phase instability is the ability of LCs to perform a small amplitude libration-like movement constrained by the applied control voltage. Cooling of the LCM electronics itself may also play a role in the phase instability, if the amplitude of the applied square wave is not stable, affecting the actual phase. However, the fact that the phase instability occurs for two LCMs different manufacturers and electronics, hints strongly at a LC molecular orientation origin. It is interesting to note that although the reduction of the LC orientation mobility via cooling leads to an increase of the phase stability as shown above, cooling of the mask is not always desirable in optical applications. Rapid phase mask changes by overdrive switching, for example, is ultimately limited by the molecular re-orientational movement of the LC [39], and cooling in that case would constrain how fast the phase mask can be changed.
Beside the origin of the phase instability, the coupling between pixels across the LCM, which was required for the simulation of the experimental SHG-AC results, is also surprising. Orientation coupling between nematic LC over several mm even cm has not been reported and is above the scale typically known for LC coupling. A potential candidate for the source of this coupling is the LCM electronic itself. Typically, the applied AC voltage over the pixels is not completely random from pixel to pixel but several chunks of pixels are driven by the same AC wave, i.e. the AC wave has the same phase. This cues why the noise over several pixels show some kind of coupling over long ranges. This effect, is still an open question and will be investigated in a future work.
6. Conclusion
Precise and noise-free tailoring of ultrashort laser pulses with LCMs is a challenging experimental business. In this work, we have reported a low-frequency phase noise generated by LCMs. In combination with kHz laser systems, this phase noise leads to very strong pulse deformations of sub-20 fs pulses, even when no phase parameterization is applied. The experimental power spectrum and the strong temperature dependence of that phase noise hints at a 1/f2 noise component due to the thermally activated libration of the nematic liquid crystal molecules in the mask. This is further corroborated by a numerical simulation of an autocorrelation using a phase noise with similar features. In spite of the large effect of this phase noise on the experimental laser output of the LCM, it is possible to strongly minimize it by a careful choice of experimental parameters. The first parameter is the operating temperature of the LCM, which improves the shot-to-shot stability of the output tailored pulse by a factor of two, when the LCM is externally cooled. The second parameter is the signal averaging approach. A rapid-scan averaging leads to a suppression of the low-frequency phase noise by an order of magnitude. The combination of these two strategies is a central knob in order to apply LCMs to spectrally broad laser pulses with high precision. It offers a promising result for successful coherent control experiments, where only small differences in the shaped pulses are important.
Acknowledgments
A scholarship of the International Max Planck Research School for Quantum Dynamics (IMPRS-QD) to E.B. is gratefully appreciated.
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