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Abstract
We have shown experimentally the successful engineering of femtosecond pulse shaping at a 76 MHz repetition rate input pulse with an acousto-optic modulator (AOM). High repetition rate (HRR) femtosecond laser pulse shaping using an AOM in the Fourier plane was incomprehensible because of its intrinsic 100 kHz acoustic update limit. We demonstrate an effective way of pulse selection and a calibration routine (Fourier shift theorem), which enables generation of ∼10 MHz shaped output pulses from the HRR input pulse train. We have generated a temporally shifted rectangular shaped pulse profile by applying modulation on both the phase and amplitude of the ‘sinc’ RF modulation function.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Numerous laser spectroscopic applications depend on amplitude, phase, and frequency of the applied electromagnetic field. In a number of diverse research areas, such as quantum coherent control, imaging, optical communications [1–3], faithful generation of laser pulses with control in phase and amplitude is of paramount importance. Theoretically, it has been demonstrated that shaped laser pulses can selectively break bonds [4,5], populate selective high vibrational energy levels in anharmonic ladder [6,7], can manipulate curve-crossings between states [8,9], and can suppress intramolecular vibrational relaxation [10]. These studies mainly prompted the beginning of femtosecond optical pulse shaping. Since its first demonstration, the advancement and reliability of femtosecond pulse shaping to generate arbitrarily shaped (amplitude/phase) optical pulses has been used in several complex experiments. Tracing of electronic coherence via multi-dimensional spectroscopy [11–13], coherent control of vibrational and/or electronic excitations to manipulate chemical reactions [2,14–16], coherent control of dark and bright states in atomic and more complex quantum systems [17–18] are few of the well-known experiments performed using pulse shaping technology. In recent times the usefulness of pulse shaping setup has been greatly enhanced by its use in multidimensional electronic spectrometers [11,19] where pulse shaping is commonly used to create two time-delayed pulses to excite the sample. Inherent phase stability of the time delayed pulses created in the pulse shaper simplifies the multi-dimensional experiments.
Optical pulses are defined by their amplitude, phase, and polarization variation with time (and/or frequency). Ultrafast pulse shaping is a powerful tool to precisely manipulate electric field into desired temporal and spectral forms [20]. Fourier spectral filtering in a 4-F geometrical setup [21] is the most commonly used approach [22]. As linear time-invariant filters, liquid crystal modulator (LCM), and acousto-optic modulator (AOM) are extensively used to shape pulses by controlled management of phase and intensity of the laser pulses in Fourier plane of the 4-F setup [23]. Beside 4-F pulse shaper, there are other pulse shaping technologies, which are routinely used for femtosecond pulse shaping. Acousto-optic programmable dispersive filter (AOPDF) [2,24,25] is used for ultrafast pulse shaping, which is based on a collinear acousto-optic interaction that maximizes the interaction length. In the AOPDF, the acoustic wave creates a longitudinal transient grating which maximizes this interaction length. In this technology, an acousto-optic crystal acts as a highly birefringent material for both the acoustic as well as for the optical waves. In the crystal, grating generated by the acoustic wave shapes the optical pulse by diffracting the ordinary incoming optical wave to the diffracted extraordinary optical wave. Thus, optical pulse shaping both in phase and amplitude is achieved through anisotropic interaction. For this reason, the AOPDF does not have to be positioned in the Fourier plane of a 4-F dispersive line, and the technique involves a long material interaction length, either a chirped pulse input or a compressor after the AOPDF is necessary for dispersion compensation. Digital micro-mirror device (DMD) based pulse shaper is also used for ultrafast pulse shaping [26,27]. In this technology, the frequency spectrum of the input pulsed laser is first spread horizontally, and then mapped to a thin strip on the DMD programmed with phase modulation patterns.
Among these approaches, AOM has advantages over LCM in the form of continuous modulation of intensity and phase with the ability of rapid waveform update rate. The AOM pulse shaper acts as diffracting modulator, which generates background free shaped optical pulses [28]. Above mentioned abilities of AOM gives it an edge over LCM as a modulator, when modulation of shot to shot pulse is required (i.e., molecular dynamics measurements). Driving AOM with tailored radiofrequency (RF) generates the acoustic waveform in the AOM, which acts as traveling wave grating. This traveling wave grating acts as a spatial filter and transforms the shape and phase of the input laser pulse [29]. Due to this traveling grating nature of the spatial filter, the refresh rate of a typical AOM is ∼100 kHz (depends on the length of the crystal). However, to function as a time-invariant filter (traveling acoustic waveform in the AOM), the repetition rate of the input laser has to be lower than the refresh-rate mentioned above. The calibration of the AOM is also necessary to design the tailored RF input. Previously, calibration was carried out by reducing the repetition rate of the laser, below the acoustic update rate of the AOM, using pulse picker [28–31]. This process ensures the concept of time-invariance of the traveling acoustic filter, and for this reason, pulse shaping at MHz repetition rate input pulse using AOM has been incomprehensible. In last few years, it has been shown, in pump-probe experiments and also in 2D electronic spectroscopy that, the technique involving pulses with high repetition rate (100 kHz) yields much faster data acquisition and enhancement in signal to noise (S/N) ratio [32,33] than the ones using low repetition rate laser pulses. Keeping that trend of innovative advancement in mind, we demonstrate here, femtosecond pulse shaping of MHz repetition rate pulses using AOM. In recent years, DMD-enabled ultrafast pulse shaper has been used to achieve continuous tuning of group velocity dispersion modulation at a rate of 2 MHz [26,27] but the efficiency of the shaper is relatively low (2%) because of the binary hologram setup of the DMD. Here we demonstrate a different technique to calibrate MHz AOM pulse-shaper using principles of Shift Theorem of Fourier Transform. We have characterized the shaped output pulses using this calibrated result. We have successfully generated and reproduced shaped pulses with ∼10 MHz repetition rate, keeping all the advantages provided by AOM pulse shaper (effective efficiency ∼12%). These kinds of pulses are quite advantageous for those experiments where the need of shaped pulse repetition rate is much higher (MHz).
2. Experimental setup
Schematic of the experimental setup is shown in Fig. 1, which utilizes a linear spectral filtering technique on the frequency components of the input pulse to generate a desired shaped output pulse. This AOM based pulse shaper consists of a holographic grating pair (1200 grooves/mm), a pair of concave mirrors (175 mm focal length), arranged in 4-F configuration, to create zero dispersion line on the focal plane (Fourier plane). The AOM (∼4.1 cm TeO2 crystal) is then placed in the Fourier plane as a linear time-invariant Fourier filter, which can modulate both amplitude and phase simultaneously. The first grating disperses incoming laser pulse to its frequency components. The first-order diffracted beam is thereafter focused to diffraction-limited spots on the AOM by the first curved mirror (this forward spatial Fourier transform converts angular dispersion (ω) into spatial dispersion on the Fourier plane). The second mirror converts spatial dispersion into angular dispersion (inverse Fourier transform), which is then re-collimated by the second grating. In the absence of AOM, the pulse shaper operates as a unit magnification device.
Fig. 1. Schematic layout of the optical set-up contains zero dispersion 4-F geometry with AOM in the Fourier plane. The generated RF wave is converted to an acoustic wave by a piezoelectric transducer to the AOM. This traveling acoustic wave inside AOM acts as a diffractive grating mask. In our particular method, only a fraction of pulses is diffracted. The need and usefulness of this fractionalized diffraction are discussed later.
We have used an oscillator (MIRA 900F, Coherent Inc.) as our laser pulse source, which produces bandwidth limited ∼150fs full width at half maxima (FWHM) pulses, centered at 780 nm, with a repetition rate of 76 MHz. The tailored RF pulses are generated by an arbitrary function generator (LeCroy, LW420A) with a resolution of 400 Msamples/s. We have used the quadrature circuit [34] for mixing shaped RF with a carrier RF (200 MHz). This synthesized RF has been used as input to the transducer of the AOM. The output optical pulse is then cross-correlated with the unmodulated zero-order pulse, which is the reflected beam from the first grating. A thin BBO (1 mm) crystal is used to generate a cross-correlated second harmonic signal, which is detected by a slow photomultiplier tube (1P28, Hamamatsu). The signal is then fed to a lock-in amplifier (SRS 830 DSP Lock-in Amplifier) to further amplify and increase the signal to noise by order of 200.
3. Calibration of the pulse shaper
In AOM pulse shaping, first laser pulse (temporal profile) Ein(tl) is diffracted by a grating into individual frequency components (ξin(ωl)). The ξin(ωl) is then modulated in the Fourier plane with a transfer function M(ωl) [2], which is applied through the AOM. The output pulse ξout(ωl)can be written as (ωl is the optical frequency, here we have used subscript ‘l’ for components of light and subscript ‘RF’ for components of applied radio signal):
The transfer function M(ωl) is directly related to the applied RF pulse (f(tRF)). To predict and/or routinely generate desired pulse shapes, one needs to know this one to one relationship. Previously [29,35], this relationship has been measured via propagating a small radio pulse through AOM with a series of triggered delays, and by measuring the corresponding diffracted spectrum. This also ensures the fact that ‘appropriate synchronization with the laser pulse to RF pulse’ has been achieved. The technique, however, required that the laser repetition rate must be lower than the acoustic update rate (100 kHz) of AOM. In this framework, the traveling acoustic wave acts as a linear time-invariant filter. In our case, the repetition rate of the laser pulse is much higher (76 MHz). This prevents precise selection of the spectrum from a single pulse, as several light pulses will be deflected even for the smallest RF pulse window. The concept of ‘triggering and synchronization’ between RF pulse and laser pulse, shot to shot, is pointless in case of repetition rate higher than 100 kHz.3.1 Optimized selection of RF window
Though selection and shaping of individual laser pulse is impossible, MHz repetition rate pulse shaping via AOM is achievable. Here, we call this MHz pulse shaping via AOM, an ‘averaged method of pulse shaping’. This is due to the fact that there is no possible way to deliver a single transfer function to individual laser pulses which are appearing at 76 MHz repetition rate to the AOM. The applied RF pulse creates a travelling grating in the AOM crystal and the neighboring optical pulses interact with different parts of the travelling grating. This can be summarized in the cartoon as shown in Fig. 2.
Fig. 2. Cartoon representation of the selection process involved in the MHz pulse shaping using AOM. The applied RF window creates traveling grating, which deflects several pulses (1-9) in sequence. The fact is that all deflected pulses do not contain the entire spectrum, as they are selected via a traveling grating inside the AOM.
Here, the acoustic update rate is much slower than the repetition rate of the laser pulse, leading to the point that a traveling grating in the AOM interacts and deflects several pulses, (Fig. 2) when laser repetition rate is in MHz range. Not only that, whenever an RF pulse arrives at the AOM crystal, it will always encounter the presence of the laser spectrum. Thus, whenever we try to modulate MHz repetition rate pulse using AOM, the need of previously mentioned method, “appropriate synchronization with the laser pulse to RF pulse”, becomes meaningless. When an RF pulse travels through AOM, it not only deflects several light pulses, but it slices different portions of the spectrum of the selected pulses (Fig. 2). Depending on the width of the RF pulse window, the number of deflected pulses will vary. The pulses 1-3 and 7-9 (Fig. 2) are partially diffracted. But pulses 4-6 are the copy of the original pulse (here ‘copy’ means pulses without any spectral slices, though, of course, in the strictest sense these copies will not be exactly identical to the original input as the dispersion from AOM and grating would have some influence). The deflected pulses are then cross-correlated with the pulses from the original source. Cross-correlation signal is generated as an average effect of all the pulses in correlation, which in this case, is an average measurement. This leads to deformation from the original pulse (here original pulse means output pulse from the shaper when the pulse is only diffracted by 200 MHz carrier radio frequency and not modulated at all by conventional intensity or phase modulation). If the numbers of these copies are increased, the averaged signal will also resemble the original pulse.
An RF pulse segment is created with 10µs duration and with a variable window (window duration of = 5µs, 4µs, 3µs, etc.).Here, a train of RF square pulse of different pulse width (variable RF window) has been generated and each pulse is separated by 10-ΔRF µs (see Fig. 2). It is then applied as an input to the AOM transducer, and the deflected pulses are then cross-correlated with the original pulse and cross-correlation signal is then measured (Fig. 3). This shows us, for our case, a 150fs FWHM Gaussian pulses, an RF segment of 10µs with the 2µs window is good enough to produce an optimized ratio between completely diffracted pulses to those of partially diffracted pulses (which are generating the average signal from the original pulses). Putting it simply, since the applied spectral phase is constant and equals to zero, it does not matter that the RF pulse is sliding through the spectrum for each laser shot. This is because, “constant zero is zero”, no matter at which frequency the input laser pulse is applied. This is the heart of this novel technique, generation of the optimized number of copied pulses.
Fig. 3. Cross-correlation signal from pulses deflected using 10 µs RF pulse segment with variable duration of radio pulse window: (a) When radio pulse window is too narrow (i.e., 0.1 µs and 0.05 µs), deformation in cross-correlation signal is significant as the number of completely diffracted pulses are rather low. (b) The enhanced cross-correlation features show the fact that pulse width decreases as the duration of radio window increases. After reaching 2 µs and beyond, the pulse width does not change and satisfactorily overlaps with the cross-correlation trace of the original pulse (output from the pulse shaper when no modulation is present).
We have captured the deflected pulse-train signals, generated by applying different RF windows (Fig. 4) in AOM. From these figures, we can easily comprehend the facts described in cartoon representation in Fig. 2. As the moving grating passes through the AOM, it deflects 76 MHz pulse train to a top-hat like waveform structure. The waveform has a rise time, an equilibrium time and a fall time. In this selection processes, inside the rising and falling part of the waveform, spectrally sliced pulses (Fig. 2— slices 1-3 and 7-9) reside, whereas the pulses within the equilibrium portion are completely diffracted pulses (slices 4-6). The total waveform created by a 2µs windowed RF pulse is 2.614µs, and the equilibrium portion is ∼1.69µs. This means that the percent of the completely diffracted pulses are 65% in this average selection and measurement. The copied pulses for the 1µs case are only 35% of the total deflected pulses and are ∼1% for the 0.5µs case. However, it is to be noted that, in this kind of selection procedure also, there is a reduction in the effective repetition rate of the shaped laser pulses. The use of, a 2 µs RF pulse window with a 10 µs RF pulse segment duration, means that ∼20% of the pulse from the 76 MHz pulse train are diffracted. Furthermore, only 65% of the diffracted pulses are “completely diffracted” pulses, which are correctly modulated. Combining these reduction factors, the effective repetition rate of the diffracted pulse is ∼ 10 MHz. This is several folds higher than the repetition rate achievable with other previous pulse shaping approaches using AOM. One more point to be noted that the shaped pulses are still separated ∼13 ns apart from each other.
Fig. 4. Measured Photodiode signals of the deflected pulses are presented here. (a) The deflected signal due to different RF window function. (b) 2µs windowed Photodiode signal and some of its zoomed in parts. (c) The envelope of the waveform signals in (a).
In our case, it is prominent from the pictorial representation that the spectrum of the input pulse is not spread all over the available aperture of the AOM. This has been done deliberately to minimize the amount of spectrally sliced pulses. The available resolution of the shaper thus has been reduced, but it is the necessary requirement of this technique to work. This method of pulse selection is also the main obstacle to previously mentioned triggering and synchronization procedure to calibrate pulse shaper. In regard to the points discussed above, we can find the available modulation pixel of the AOM. AOM is 4.1 cm in length, and the acoustic velocity in TeO2 crystal is 4.2 mm/µs, which translate into a transit time Δτ∼9.76 µs. In a TeO2 crystal, the minimum spatial feature size is of 21 µm [2,36]. The RF bandwidth of the AOM is 100 MHz, which translates ∼976 available AOM pixels. The effective modulation window of 2µs creates 8.4 mm spatial feature, and the spectral spread of the laser pulse on the AOM crystal is about 5.5 mm. Thus,
In our system, the diffraction efficiency of the grating pair is 85%. The efficiency of the curved mirrors is 95%, and the first order deflection efficiency of the AOM is ∼20%. The selection scheme further reduces the average output of the shaper by 13% (∼10 MHz output from 76 MHz input). The overall efficiency of the average power output from the shaper is thus 1.3% of the average input power. However, the peak output does not get reduced by the pulse selection routine, and therefore the effective efficiency of the peak power output is 12.35%.
3.2 Standardization between transfer function and RF pulse
In order to generate an identical output pulse as compared to that of the original pulse, a well-defined RF pulse window (the width of the window depends on the pulse-width of the laser pulse) has to be applied. In this framework, if a modulation (intensity and/or phase) is applied in the windowed part, it will modify the optical pulse at the Fourier plane of the 4-F. The question is, how much the output pulse shape resembles the expected modulation created by the RF, and if it is so, how a desired shaped pulse can be created where the optical pulse is in MHz repetition rate? To formulate the exact transformation between transfer function (M(ωl) and RF signal (f(tRF)), calibration of the pulse shaper is an absolute necessity. As mentioned previously, calibration via ‘triggering and synchronization’ is impossible in this case. Here, this technical barrier has been overcome by using Fourier Shift Theorem principle to calibrate our MHz AOM pulse shaper.
The Fourier shift theorem says that a delay in the time domain corresponds to a linear phase sweep term in the frequency domain. This can be represented as:
Applying linear phase sweep as a transfer function, we have generated several instances of the 2µs window, in which we have applied sinusoidal radio modulation of frequency 1 MHz, 5 MHz, and 10 MHz, etc. Then cross-correlation signals from the linear phase swept (spectral domain) output pulses were measured (Fig. 5). To our satisfaction, it was found that the peak shift occurs in the measured cross-correlation signal (temporal domain). The broadening of the output pulses has also been observed in the cross-correlation measurements, which is coming from higher order effects. In our 4-F alignment (see Fig. 1), the input optical pulse is non-normal to that of AOM, which invokes an inherent optical path length difference for each frequency component. This optical path-length difference for different frequency components induces dispersion and results in pulse broadening [36]. The interesting fact is that, when we prepared the sinusoidal modulation within in 1µs RF window, it shows the same peak shift effect. Not only that, both 2µs and 1µs window results in the exact same amount of peak shift in corresponding sinusoidal modulation frequency (Fig. 6). This proves another crucial point that, though the modulation experienced by first pulse and the last pulse (selected by the windowed RF) is significantly different, the average effect (cross-correlation measurement) merge to give a single output peak shift. A linear slope shifted in space (experienced by first and last pulses) is still a linear slope throughout, and thus the temporal shift is not affected (only an offset result, i.e., a zero-order spectral phase affecting the carrier-envelope phase that is not detected here). This experimental finding proves another curtail point, which is, the modulation function M(ωl) is a slowly and smoothly varying function in comparison to the spectrally dispersed optical pulse (ξin(ωl)), which requires the condition, that the smallest feature of M(ωl) is larger than the effective Gaussian beam waist (which, for our case 6.3 µm) at the central Fourier plane. This is because each of these pulses is temporarily very close to the other (only ∼13 ns apart), and the modulation window (2µs) is much larger than this pulse to pulse separation.
Fig. 5. Time segmented RF pulses applied to AOM, which creates the delayed optical pulses in the temporal domain. Applied RF pulse is a 2µs window within a 10µs time segment with 1 MHz to 40 MHz oscillation frequencies. In (a), the real part of the RF signal generated using 1 MHz oscillation frequency has been pictorially represented and (b) is the corresponding imaginary part of the sinusoidal modulation, and (c) shows the cross-correlation traces of delayed optical pulses created by different oscillation frequencies.
Fig. 6. Peak shift due to different linear phase sweep for both the case of 1 µs and 2 µs window is represented here. Though 2 µs window creates pulses whose average effect resembles that of original pulse and 1 µs window does not do so, the linear sweep results in the same shift for both the cases.
This explains why one can time-delay the pulses using a linear spectral phase even when the modulation RF wave has preceded through the crystal from laser shot to laser shot (Fig. 6). In other words, a straight line is translationally invariant. Thus, one can indeed obtain a calibration of the AOM pulse shaper, even with 76 MHz pulses as input. This linear relationship depends on several factors, such as response time of the AOM (depends on the material of AOM), the bandwidth of the light pulse, the repetition rate of the femtosecond laser pulse. To do MHz pulse shaping using AOM, one has to find out the RF window, as well as has to establish a linear relationship between the laser spectra to the RF pulse within the RF window (acoustic waveform in the AOM).
The above-described results are the facts that lead us to correlate the transfer function (M(ωl)) to the applied RF modulation frequency using the mathematical methods, described in Fourier Shift theorem. Using Matlab script, we have found out the exact relationship by back calculating the experimental data. First, we transform spectral and temporal representations of the pulse to the corresponding pixel representations (henceforth pixelation, which we denote as the dimensionless discretization of the relative variables for the numerical procedure). The smallest measured temporal resolution is dtl ( = 6.666 fs). Temporal domain pulse Ein(tl) is transformed to discretized version, Ein(pt) via pixel transform (PT), by replacing tl to tl/dtl. This generates Nt number of temporal pixel points, with pt (=tl/dtl) being the temporal pixel variable. Similarly, spectral domain pulse ξin(ωl) is also remodeled to ξin(pω), with Nω spectral points and pω (=ωl/dωl) being spectral pixel variable (Fig. 7).
Fig. 7. Pixelated representation of spectral domain pulse and radio pulse: The goal is to establish one to one correlation between the two-pixel forms.
The input RF function, f(tRF) is also transformed into f(pRF) via PT, and it contains NRF radio pixel points. In the relative pixelated representation, the number of pixels is set to equal, that is, Nt=Nω=NRF=N. We know that the transfer function, M(ωl), is directly related to f(tRF). The goal is to establish a relation between pixelated transfer function (M(pω)) and pixelated radio pulse f(pRF). Thus, pω is a scaled version of pRF (${p_\omega } = c \times {p_{RF}}$, where c is constant). We would like to extract this constant scaling value.
In our experiments, we have measured the amount of peak shift in temporal domain caused by application of smoothly varying phase sweep in the frequency domain. Fourier Shift Theorem says that linear phase sweep in frequency corresponds to peak shift in time and vice versa. In this scenario, if the spectral pixel imposes 2π linear phase sweep, it will generate a peak shift of 1 in a temporal pixel. In the experiment, we have measured the shift in peak position to that of the zero-delay position. A resultant peak shift of ‘α’ in the temporal profile will generate a peak shift of ‘a’ (=α/dtl) in pixel representation (Fig. 8). This means an applied radio-pulse, as a whole, has delivered a 2πa linear phase sweep to the spectral pixel. As the radio pulse is generated by sinusoidal radio frequency modulation, we can easily characterize it and can draw a conclusive correlation between radio pulse and transfer function in pixelated representation (Fig. 8).
Fig. 8. Flow chart to find out the exact correspondence between pixelated transfer function (M(pω)) and pixelated radio pulse f(pRF) from experimental measurements of peak shift via application of different sinusoidal modulation. Bold lines are main flow chart, and dotted lines are for comparison and checking.
4. Comparison among desired and measured shaped output pulses
4.1 Intensity modulation
Taking advantage of the mathematical relationship mentioned in the earlier section, we have designed some arbitrary RF pulses for intensity modulation and predicted cross-correlation of the shaped output. RF pulses are generated through a simple combination of ‘sin’ functions with 10µs time-segment and 1µs square window. A series of these linearly combined RF pulses are used to intensity modulate a Gaussian pulse. Two of them are represented here in Fig. 9(a) and 9(b) (using equations 2a and 2b) with appropriately tailored RF pulses:
Fig. 9. Intensity modulation through a combination of ‘sin’ functions and cross-correlation traces of shaped pulse is represented. Input RF pulse corresponding to equations (2a, 2b) are depicted in (a) and (b). Experimental and expected values corresponding to RF (a) and (b) is presented in (c) and (d). Using the previously mentioned method, we get the expected cross-correlation, which is in good agreement with experiment.
4.2 Intensity and phase modulation: square pulse
Due to a balancing problem in our RF-circuity, there is a constant DC leakage of the RF carrier with the tailored RF pulse, which in combination with the MHz repetition rate laser pulse creates a constant background. Though contribution from this is negligible, in the ‘sin’ amplitude modulation case, the effect is paramount when we tried to create temporal square pulses by using RF ‘sinc’ function with 10µs time segment and 2µs window.
The background signal from unmodulated pulse is comparable to the signal from the original square pulse. Still, by including a fixed background, the cross-correlation trace of the square pulse again matches the expected shape as calculated by using the pixel transform technique (Fig. 10(a)).
Fig. 10. Intensity modulation through ‘sinc’ RF function generates a square pulse in the temporal domain. Experimental cross-correlation of the generated square pulse is represented: (a) Cross-correlation of the output from only ‘sinc’ RF modulation; (b) same ‘sinc’ RF with a linear sweep and pre-compensated ‘sinc’ RF modulation with a linear sweep.
Interestingly, however, both the deformed time-shifted square pulses, generated from positive and negative linear phase swept ‘sinc’ modulation, creates cross-correlation traces that are mirror images of each other with respect to the zero position. This leads to the key information about this deformation, which is, that the linear sweep adds a higher order phase in the traveling acoustic wave in the AOM. This inherent higher order phase will invoke chirp to the input pulse. We can compensate for this chirp by using a pre-compensated RF pulse, which adds a positive or negative first order chirp based on the frequency sweep to the input pulse. The imaginary and real part of the Eq. (3) is presented in Fig. 11(a). Surprisingly, only a first-order correction in the input pulse compensates the deformation effectively, and the cross-correlation trace matches with the expected trace (Fig. 11(b)). This is done by adding only a cubic phase in the tailored RF pulse as shown in Eq. (3) above.
Fig. 11. (a) The plot of the imaginary and real part of the pre-compensated RF signal that is generated by adding a third-order phase with the linear frequency sweep. Experimental cross-correlation signal and expected square pulse (using αRF) are presented in (b) that are quite good agreement with each other.
5. Conclusion
In conclusion, we have used AOM pulse-shaping setup as our experimental device to successfully alter the temporal shape of the pulse and get the desired outcome by using high repetition rate laser (76 MHz) as input. We have achieved much higher repetition rate pulse shaping (∼10 MHz) via AOM pulse shaper technique. The essential part of this pulse shaping technique is the establishment of the square windows RF signal by ensuring the linear peak shift (linear relationship between the RF pulse and spectrum of the light pulse). We have taken full advantage of the principle of Shift-Theorem of Fourier transform to calibrate our high repetition rate laser pulse-shaping setup in a new way (by transforming into a pixelated form). Using this calibration, we have successfully demonstrated several shaped pulses applying intensity modulation, which are in good agreement with predicted shapes. Finally, we have successfully prepared a shifted square pulse that is free of deformation using a pre-compensation scheme of simultaneous intensity and phase modulation. Using our pixelated calibration scheme, we have characterized all the output shaped pulses. In this present first demonstration experiments for high-repetition experiments with AOM pulse shaping, a reduced effective resolution AOM pulse shaper was used that somewhat limits the possible freedom and capabilities of the AOM pulse-shaping. The limitation comes in the RF modulation, where the modulation function has to be smooth and slowly varying, compared to the time gap between laser shot to laser shot.
This kind of pulse shaping is beneficial for measuring optical free induction decay, where it is advantageous to have the full MHz shaped pulse train produced. The other point is that the shaped pulse sequences are created using an acousto-optic pulse shaper, which means that the experimental setup is the same regardless of whether a sequence of two or of hundreds of pulses is created. This technique is especially useful in samples where phase-matching is not possible such as single-molecule spectroscopy or in samples where it is impossible to detect the transmission, for example in highly scattering samples. To our success, we have shown the high fidelity of these shaped pulses in terms of reproducibility, i.e., in one case of not generating the shaped pulse, we recover a bunch of deformed shapes while in the other case when we do shape the pulse, we detect the shape faithfully.
Funding
Science and Engineering Research Board (SERB) (Individual Intramural Funds); Indian Space Research Organisation (ISRO) (STC).
Acknowledgment
SD and SNB are thankful for Senior Research Fellowship of the Council of Scientific and Industrial Research, India. All the authors acknowledge S. Goswami for extensive language editing of this article.
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