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Abstract
We describe a non-interferometric ultrashort-pulse measurement technique based on frequency-resolved optical gating (FROG) with which pulses can be reconstructed directly, i.e. non-iteratively. Two different FROG spectrograms are measured, which represent the only information required to reconstruct the amplitudes and phases of two independent input pulses. The direct reconstruction method is demonstrated with a single-shot FROG setup used to obtain the spectrograms generated from two synchronized input pulses. To demonstrate and determine the reconstruction quality for complex pulses, a programmable pulse shaper is used to modify the pulses sourced from a Kerr-lens mode-locked Ti:sapphire oscillator.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
While interferometric pulse measurement techniques usually employ direct, i.e. analytical reconstruction algorithms [1,2], this is rarely the case with non-interferometric techniques [3]. Classical non-interferometric pulse characterization techniques such as FROG have been using iterative reconstruction algorithms for decades [4]. Convergence tests have been conducted, memetic algorithms have been used, and even deep neural network techniques have been applied [5–7]. More recent non-interferometric pulse measurement techniques, such as dispersion-scan (d-scan), also use iterative reconstruction algorithms [8,9]. Hence, direct pulse reconstruction algorithms for non-interferometric pulse measurement techniques are currently absent from the literature. Even the most recent algorithms for special FROG schemes, such as SHG FROG, with very high reliability are still iterative [10].
In order to approach the problems associated with convergence in non-interferometric techniques, this work presents a fully non-interferometric pulse characterization technique that reconstructs amplitudes and phases of two input pulses, from two different FROG spectrograms using a direct reconstruction algorithm. This direct and analytical algorithm is evidently very fast and completely bypasses the convergence issue, thus presenting a significant difference to conventional FROG, d-scan and other non-interferometric techniques [11]. In addition, a pulse shaper is used for the purpose of experimental reconstruction of not only linear or quadratic phase functions. This allows the performance of the measuring method to be demonstrated more convincingly.
2. Theory
Stable, synchronizable and reproducible pulse trains consisting of identical pulses are assumed here so that, for example, the coherent artifact does not need to be considered [12–14]. Although some equations from tomography are used in the following, no tomographic technique is involved here. We report a purely spectrographic method.
It has been shown that the chronocyclic Wigner distribution for an ensemble of identical pulses $E\left ( t\right )$, expressed in the spectral domain as $\tilde {E}\left ( \omega \right )$, can be used to derive the partial differential equation
In the case of perfect phase matching in a sum-frequency generation (SFG) process, and assuming low conversion efficiency, i.e. no pump depletion, the FROG spectrogram generated by the pulses $\tilde {E}_{1}\left (\omega \right )$ and $\tilde {E}_{2}\left (\omega \right )$ can be expressed by formula (4).
In measurements there is always a noise level which restricts the measurement of small intensities in the spectrograms. Hence, the noise can localize the spectrograms spectrally and temporally. In this sense the spectrograms can be considered as functions with finite support. Because of the small difference between the spectrograms (4) and (5), the finite support of these spectrograms is almost identical for a given noise level. The integral in the numerator in formula (9) also has this same finite support. Note that this integral is zero if $\tau$ tends to infinity. So in the fraction in formula (9), the numerator and denominator have the same finite support in good approximation. In the areas where the numerator and denominator contain only noise, the result is only strong fluctuations. However, these fluctuations are not significant, because in these areas the amplitude is practically zero and therefore the phase is undetermined. To perform the Fourier transform (11), we first subtract the background from the spectrograms. Previously, the spectrograms were smoothed with a Fourier filter, taking advantage of the high resolution of the camera. In our experiment the background intensity is $4\%$ of the maximum intensity of the spectrograms. The noise level of this background is less than $1\%$ of the maximum of $I\left ( \Omega ,\tau \right )$. After the Fourier transform is performed, the background of the function $M\left ( \omega ,\Omega \right )$ is also subtracted to improve the result of the factorization of the functions $N\left ( \omega ,\Omega ^{\prime }\right )$ and $Q\left ( \omega ,\Omega \right )$.
Our method factors out the functions $N\left ( \omega ,\Omega ^{\prime }\right )$ and $Q\left ( \omega ,\Omega \right )$ and assumes perfect tensor products. However, due to noise and systematic errors, they are not perfect tensor products. The robustness of our reconstruction algorithm can probably be improved if additional outer products are taken into account. This has already been shown for FROG [5,17]. Because the direct algorithm presented here is novel, it is not yet on the level of other reconstruction algorithms that have been improved and optimized for decades. For example, we obviously do not reach the level of robustness that was achieved a few years ago with an iterative ptychographic reconstruction algorithm [18]. However, the important point here is that our algorithm is direct and can certainly be further improved.
There is an upper bound for $\Psi$, which is condition (8), but of course there is also a lower bound, because if $\Psi$ is chosen too small, the difference between the spectrograms (4) and (5) disappears. This means that the signal-to-noise ratio (SNR) is too small to allow a pulse reconstruction.
3. Experiment
The schematic of the experiment carried out in this work is illustrated in Fig. 1. The pulse source is a homemade Kerr lens mode-locked Ti:Sapphire oscillator which produces pulses with a duration of approximately $70$ fs, with a pulse energy of $3$ nJ, and a repetition rate of $78.8$ MHz. The carrier-envelope phase is not stabilized. The pump laser is an Opus $532$ nm $6$ W (Laser Quantum) that ensures stable pulse operation. The most stable pulses, which were used for the experimental work throughout, were obtained at a center wavelength of $883$ nm.
Fig. 1. Scheme of the optical setup: SLM, spatial light modulator; CL, cylindrical lens; GR, reflection grating; NPBS, non-polarizing 50:50 beam splitter; $E_{1}$ and $E_{2}$ are synchronized input pulses; BaF$_{2}$, barium fluoride.
To modify the pulse shape, a pulse shaper with the typical zero-dispersion compressor design is used, as shown in Fig. 2 [19].The pulses are first decomposed into their spectral components by a reflection grating. Each spectral component is then focused into the Fourier plane of the cylindrical lens which is placed after the grating. A reflective phase only spatial light modulator (SLM) (Holoeye, Pluto-2-NIR-015) is located exactly in the Fourier plane of the cylindrical lens, which allows the spectral phases of the pulses to be changed. By slightly tilting the SLM, the beam emerges a little above the input beam and is then sent with a mirror to the pulse measurement system. Chromatic dispersion caused by the cylindrical lens and SLM is partially compensated by a pre-chirp generated by a cavity external prism sequence in the Ti:sapphire oscillator. Since we are not dealing with few-cycle pulses, the influence of the chromatic dispersion is almost negligible. This also applies to the chromatic aberration generated by the cylindrical lens. Nevertheless, we observe weak diffraction effects in the measured spectra, probably because the spectral resolving power in the Fourier plane is not high enough. The active area of $15.36$ mm $\times$ $8.64$ mm of the SLM is used efficiently without producing diffraction effects at the borders. The SLM is a liquid crystal on silicon (LCOS) with a resolution of $1920\times 1080$ and a pixel size of $8$ $\mu m$ $\times$ $8$ $\mu m$. It should also be mentioned that the generation of complicated spatio-temporal effects due to chromatic aberrations is another systematic error that reduces the quality of the FROG spectrograms. The large maximum phase shift of $4.4\pi$ at $850$ nm allows the SLM to be used in the low phase flicker mode, because such large phase changes are not required. Also the longer response time of the SLM is not an issue here. Low phase flicker is more important for the resultant quality of the measurement. The phase flicker effect in LCOS devices is discussed in detail in [20]. Since the SLM is just an $8$-bit LCOS, the pulse shaping is strongly sampling limited. The effective efficiency of the pulse shaper is $33\%$. The input pulses coming from the Ti:sapphire oscillator have a pulse energy of $3$ nJ and thus leave the pulse shaper with $1$ nJ.
Fig. 2. Programmable pulse shaper: GR, reflection grating with 300 grooves/mm; CL, cylindrical lens with $f=200$ mm; SLM, spatial light modulator.
The pulse measurement system receives the synchronized pulses $E_{1}$ and $E_{2}$. The measurement technique presented herein generally works with different pulses, i.e. $E_{1}$ and $E_{2}$ need not be identical. Nevertheless, we test our method experimentally with almost identical pulses $E_{1}$ and $E_{2}$, in order to check if the pulse reconstruction confirms the similarity of the pulses and hence if the method is self-consistent in this sense. The pulses $E_{1}$ and $E_{2}$ are not exactly equal because of the chromatic dispersion of the non-polarizing beam splitter shown in Fig. 1. While $E_{2}$ passes directly to the FROG setup, $E_{1}$ is split previously into two sub-pulses. An interferometer is convenient for this purpose, since it allows the two subpulses to be brought together again collinearly into one beam. This avoids systematic error differences between the FROG spectrograms (4) and (5) caused by different beam directions. It is important that the sub-pulses leave the interferometer at different times and do not overlap. Of course, the sub-pulses can also be generated in other ways, as desired. In our experiment, a Michelson interferometer is used, with a chromatically dispersive element in one arm, as shown in Fig. 3(a). This means that one sub-pulse passes through air and the other sub-pulse passes through the dispersive element. The dispersion caused by the $2$ mm thick NPBS is low and both sub-pulses are equally affected. The pulses that would return to the laser are blocked in the setup so as not to destabilize the laser. From Eq. (2), it is clear that the dispersive element should ideally produce a perfect quadratic spectral phase. Therefore, a transparent medium with a minimized ratio of third-order dispersion (TOD) to group delay dispersion (GDD) must be used. The most suitable transparent material of those listed in Table 1 is barium fluoride (BaF$_{2}$), and is therefore used in the experiment. The $9$ mm thick BaF$_{2}$ glass (Eksma Optics) is passed through twice by the pulses, which is equivalent to $18$ mm of BaF$_{2}$ glass. At a wavelength of $883$ nm this corresponds to a GDD of $\Psi =604~$fs$^{2}$. The ideal amount of GDD introduced depends on the spectral bandwidth of the pulses. It is only a matter of keeping the condition (8). This is by no means a shortcoming as the optical components used in all pulse measurement techniques must always be adapted to the pulses used. A typical example is the thickness of the nonlinear crystals that are adapted to the spectral bandwidth of the pulses in order to achieve phase matching. Indeed, by similarly adapting components FROG has been demonstrated for the measurement of even single-cycle pulses [21]. However, the TOD to GDD ratio of BaF$_{2}$ is not small enough to characterize few-cycle pulses. The spectral bandwidth of the pulses $E_{1}$ can be limited to a few THz, e.g. with a bandpass filter. This ensures that the higher terms of the spectral phase function of the dispersive element can be neglected. Hence the pulses marked as $E_{2}$ can have a spectral bandwidth larger than $8$ THz, because these pulses do not pass the Michelson interferometer with the arm containing the dispersive element. Limiting the spectral bandwidth of the pulses $E_{1}$ also allows to keep condition (8) without having to change the dispersive element. In order to characterize e.g. $30$ fs pulses in this way, the cylindrical lenses in the setup should be replaced by concave mirrors to minimize the chromatic dispersion. A characterization of few-cycle pulses would be a challenge with our method because of the systematic errors. It must also be noted that if the pulses $E_{1}$ are spectrally filtered, and thus modified, then of course only the pulses $E_{2}$ can be considered independent.
Fig. 3. (a) Michelson interferometer: NPBS, non-polarizing 50:50 beam splitter; BaF$_{2}$, barium fluoride; M, mirror; (b) Single-shot FROG (without the beam splitting element): KDP, $0.5$ mm thick KDP crystal; CL1,CL2,CL3, cylindrical lenses with $f=100$ mm, $f=75$ mm, and $f=150$ mm respectively; A, aperture transmitting only the SFG beam; GR, holographic diffraction grating with 2400 grooves/mm; The input beams both lie in the xy-plane. The camera is located below the KDP crystal, outside the xy-plane. The spectrally resolved direction is indicated by $\lambda$, and the delay $\tau$ corresponds to the time axis of the spectrograms. The stray light filters are not shown.
Table 1. TOD to GDD ratios of several materials which are applicable dispersive elements.
In our experiment, the FROG spectrograms (4) and (5) are measured with one and the same FROG setup. There are two typical setups, the scanning FROG and the single-shot FROG, which have been in use for decades [22]. Our method can work for both, however throughout this study we have chosen the single-shot FROG setup shown in Fig. 3(b). A single-shot setup can be used for single-shot, but also for multi-shot measurements, as in our case. An advantage of a single-shot geometry is that it does not require a moving stage. In the nonlinear crystal the wavefronts of the incident pulses overlap each other. The beam diameters and the angle between the incident pulse fronts inside the crystal define the range of delay which can be obtained in a given setup. The delay ($\tau$) is a function of a specific transverse coordinate of the SFG beam, for example, for the SFG beam emanating from the crystal, it is the transverse coordinate axis, which lies in the plane of the incident beams. Time resolution is achieved using the delay information contained in the transverse coordinate of the SFG beam. This well-established method for the generation of a delay axis is described in detail in [22]. The FROG spectrograms of the pulses are obtained via SFG in a potassium dihydrogen phosphate (KDP) crystal, which is $0.5$ mm thick. The KDP crystal (Newlight Photonics) is designed for second-harmonic generation (SHG) (in: $800$ nm; out: $400$ nm) with type-I critical phase matching. The spectral acceptance bandwidth for $800$ nm input pulses is $24.6$ nm. Because the central wavelength of the pulses used in the experiment is $883$ nm instead of $800$ nm, the KDP crystal is slightly tilted, thus ensuring phase matching. The external crossing angle of the beams is $8^{\circ }$ and the beam diameter is $3$ mm. The pulses generated from the SFG process are then spectrally resolved by reflection from a holographic diffraction grating with 2400 grooves/mm. The FROG spectrograms are recorded with a monochromatic $8$-bit CMOS camera with a resolution of $1280\times 1024$. The high resolution of the camera facilitates the detection and reduction of noise. Therefore, small differences between the spectrograms can be measured despite the relatively low bit depth of the camera. By using a Fourier filter, the SNR can be increased considerably. Two BG40 colored glass bandpass filters ($335$ nm - $610$ nm) are installed directly in front of the camera to block infrared stray light. Additionally, a tube is installed in front of the camera to minimize the incidence of ultraviolet stray light.
4. Programmable pulse shaping and spectrogram simulations
To test the reliability of pulse measurement techniques, it is advisable to use complex structured pulses. This is often only simulated [23]. In order to test not only simulated pulses, reconstructions of octave-spanning ultrashort light pulses or supercontinuum pulses have been published [24,25]. It follows however, that the spectral phase is simply predetermined, and in the latter case it is extremely complex. To sufficiently control the spectral phase, femtosecond pulse shaping is required. This allows one to program the spectral phase and moreover provides a convincing test, in particular if later on the programmed phase matches the reconstructed phase. However, the controlled generation of highly complex pulses by means of pulse shaping is difficult [26]. Our pulse shaper almost reaches the level of professionally engineered pulse shapers, and is sufficient for the first tests of our pulse characterization method. To simulate the spectrograms, it is simply assumed that the pulses generated by the Ti:sapphire oscillator have a quadratic spectral phase and their spectral intensity is approximated with a Gaussian function with a full width at half-maximum (FWHM) of $6$ THz. For simplicity, for the simulation it is assumed that the pulse spectra are not modulated by diffraction effects generated by the SLM. To make the spectral phases of the pulses interesting for pulse characterization, a Gaussian function with negative sign is programmed with the SLM and added to the quadratic spectral phase of the pulses, as shown in Fig. 4. Later it can be verified whether this Gaussian phase function is reconstructed correctly. Furthermore, to break the symmetry, this Gaussian function is spectrally shifted by $0.31$ THz relative to the maximum of the spectral intensity function of the pulses. The Gaussian phase function we have chosen is sufficiently smooth so as to minimize the diffraction effects generated by the sampling limited $8$-bit LCOS. With the pulse shape shown in Fig. 4 the spectrograms $I\left ( \Omega ,\tau \right )$ and $I_{\Psi }\left ( \Omega ,\tau \right )$ can be simulated. Because the difference between both spectrograms is very small, Fig. 5 shows the spectrogram (4) and the difference between both spectrograms.
Fig. 4. Simulated pulses after leaving the pulse shaper. The black line is a Gaussian function with $\mathit {FWHM}=6$ THz that approximates the spectral intensity of the pulses. The green dashed line is a quadratic function that approximates the spectral phase of the pulses, before pulse shaping. The red dashed line is a Gaussian function, programmed with the SLM, with negative sign, a maximum phase change of $1.1$ rad, and $\mathit {FWHM}=1.3$ THz. The blue line represents the resulting spectral phase of the pulses leaving the pulse shaper. To break the symmetry, a small spectral offset of $0.31$ THz is inserted.
Fig. 5. Simulations: using the pulse from Fig. 4; (a) The spectrogram $I\left ( \Omega ,\tau \right )$ as defined in (4); (b) The modulus of the difference of the spectrograms (4) and (5); The maximum of the modulus of the difference is $5\%$ of the maximum of the $I\left ( \Omega ,\tau \right )$ spectrogram.
5. Spectrogram measurements and pulse reconstructions
The measured dual spectrogram of pulses produced by our experimental apparatus is presented in Fig. 6(a)-(b). Overall, the direct reconstruction algorithm successfully reconstructed the pulses emergent from the pulse shaper with error parameters discussed below.As in the simulations shown in Fig. 5, the maximum of the modulus of the difference in Fig. 6(c) is $5\%$ of the maximum of the measured spectrogram $I\left ( \Omega ,\tau \right )$ in Fig. 6(a). The SNR in Fig. 6(c) is low and the data quality is not as perfect as the simulation in Fig. 5(b). Despite the noise, both Fig. 5(b) and Fig. 6(c) show quite similar patterns. Of course, an exact match cannot be expected because the simulation in Fig. 5 is based on a relatively rough approximation of the pulses. Furthermore, the simulation completely neglects the systematic errors caused by all optical elements involved. If the time interval of the delay axis is sufficiently long, both spectrograms can be measured simultaneously. In our case the camera chip is not large enough to prevent interferences between both spectrograms. This means that the sub-pulses arrive the KDP crystal too close together temporally. A smaller imaging of the spectrograms would reduce the resolution. Therefore, the spectrograms in Fig. 6(a) and Fig. 6(b) were measured successively. The integration time was $1$ s each. The arms of the Michelson interferometer were adjusted so that both sub-pulses reached the nonlinear crystal simultaneously. To measure one of the spectrograms, the corresponding interferometer arm was then closed. When measuring the spectrograms one after the other, the temporal stability of the pulses is important and limits the quality of the measured difference in Fig. 6(c). In principle, two FROG setups could be used to measure both spectrograms simultaneously. Despite the low SNR in Fig. 6(c), our algorithm produces acceptable reconstruction results. This is an advantage of the FROG technique, because it uses the entire spectrogram information and not only a one-dimensional data set. In our algorithm this corresponds to Eqs. (13), (14) and (19). Applying the direct reconstruction algorithm, i.e. formulas (9) to (23), results in the reconstructed spectrogram in Fig. 6(d). It deviates only slightly from the measured spectrogram $I\left ( \Omega ,\tau \right )$ in Fig. 6(a). For a grid size of $512\times 512$, and following the defined rules for calculating a FROG error, we obtain the value $0.0021$. This is less than $1\%$ and is, therefore, considered acceptable [22]. Although in our experiment the pulses $E_{1}$ and $E_{2}$ are not exactly identical, as explained above, the measured spectrogram $I\left ( \Omega ,\tau \right )$ in Fig. 6(a) can be regarded approximately as an SHG FROG spectrogram. So we applied a standard SHG FROG algorithm to this spectrogram, and obtained a FROG error of $0.0018$. The largest part of the calculated error values comes from spatio-temporal effects, which are generated by the pulse shaper and beam splitters [27]. These effects can be avoided with a greater experimental effort. Imprecise spectral correction data for the grating and camera efficiencies also constitute systematic errors. The spectrogram in Fig. 6(a) is not perfectly symmetrical with respect to the delay. This asymmetry is caused by the NPBS shown in Fig. 1, and the NPBS shown in Fig. 3(a). If the symmetry of this spectrogram were better, the SHG FROG error would be much smaller. However, the FROG error of our method would not become smaller because the data quality in Fig. 6(c) would not increase. Since our direct method is a finite-difference method, it does not achieve the robustness of the standard SHG FROG due to the lower SNR.
Fig. 6. (a)-(b) Measured dual spectrogram. (a) $I\left ( \Omega ,\tau \right )$ spectrogram (4); (b) FROG spectrogram (5) which is modified by the dispersive element (BaF$_{2}$). (c) The modulus of the difference of the spectrograms shown in (a) and (b). (d) FROG spectrogram reconstructed with the direct, non-iterative, algorithm. The FROG error is $0.0021$.
The reconstructed pulses, $E_{1}$ and $E_{2}$, and the measured spectrum of the pulses before they are split by the NPBS are shown in Fig. 7.All have weak fringes caused by diffraction and spatio-temporal effects generated by the pulse shaper, which do not match exactly. This is because there is a distance of about $2$ m between the position where the spectrum is measured and the position of the KDP crystal. The moderate match of measured and reconstructed spectra is, therefore, no shortcoming of our pulse measurement method. The fact that the reconstructed pulses $E_{1}$ and $E_{2}$ are almost identical shows that the pulse characterization works well despite the systematic errors. The spectral phases in Fig. 7(a) show the Gaussian phase function programmed with the SLM. The maximum phase change of $\left ( 1.2\pm 0.2\right )$ rad and the $\mathit {FWHM}=\left ( 1.5\pm 0.3\right )$ THz confirm the programmed phase function in Fig. 4. In Fig. 7(c), it can be seen that the reconstructed pulses in the time domain are slightly asymmetrical. A simulation shows that this asymmetry is caused by the small spectral offset indicated in Fig. 4. Furthermore it shows that the programmed phase function changes the pulse duration, i.e. the FWHM, only very slightly. The actual changes of the temporal pulse structure are the appearing pulse wings in Fig. 7(c).
Fig. 7. Reconstructions of both pulses, $E_{1}$ and $E_{2}$; (a) in the spectral domain; The black dashed line in (a) shows the measured spectrum of the pulses before they are split by the NPBS in Fig. 1. (b) in the spectral domain; The black dashed lines in (b) show the simulated pulses from Fig. 4. Because these simulated pulses are only an approximation, we shifted these lines spectrally by $1$ THz. (c) in the time domain; Both pulses in (c) are slightly asymmetrical and have a duration of approximately $70$ fs. (d) in the spectral domain; The black dashed lines in (d) show the reconstructed pulse obtained from a standard SHG FROG algorithm applied to the spectrogram in Fig. 6(a). The FROG error is $0.0018$.
The uniqueness property of a pulse characterization method must also be validated for direct reconstruction algorithms and, above all, must be clearly defined. This has already been discussed in previous studies of this approach to pulse measurement, which did not use a fully direct algorithm [11]. Proper signal processing is essential for a direct reconstruction algorithm, because there is no iterative error correction process. For example, we have replaced the Fourier transform in formula (11) by a chirp z-transform [28]. This increases the resolution in the spectral domain, which is important for the subsequent numerical steps. Formula (11) is the only Fourier transform in the algorithm. In an iterative Fourier transform reconstruction algorithm, however, each iteration step corresponds to two Fourier transforms. In this sense, our direct algorithm is equivalent to a half iteration, and resultantly is a much faster algorithm. Whether this benefit can always be put to experimental use depends on many factors. Experimentally it is advantageous to record both spectrograms simultaneously with one camera. This is efficient and avoids errors due to pulse fluctuations. With a scanning FROG, this may be difficult to achieve because the scanning speed is usually not very fast. If the pulses are sufficiently intense, then the integration time of the camera can be reduced. The reconstruction speed then ultimately still depends on the properties of the camera used.
6. Conclusion
We have experimentally shown that one of the most widely used pulse measurement techniques, namely the FROG technique, can be employed for a novel pulse measurement scheme with a direct reconstruction algorithm. To our knowledge, this work reports the first methodology for which all information contained in FROG spectrograms is effciently used for direct pulse reconstruction. For its successful application only a special chromatically dispersive element is needed, which can be easily implemented experimentally. The issue of robustness is most strongly linked to the systematic errors incurred, therefore further reduction of the systematic errors described should yield improved robustness of the method in terms of its calculated FROG error. Also by using more advanced numerical methods the robustness can probably be further increased. Nevertheless, the signal strength in our method is quite good compared to pulse measurement techniques based on third-order nonlinear processes. It is also conceivable that the robustness of the reconstruction algorithm can be increased by modifying the algorithm in a way that only one pulse is reconstructed, as in the conventional SHG FROG technique. However, the independent reconstruction of both input pulses has many advantages. Differences of the input pulses no longer represent non-correctable systemic errors and can easily be corrected mathematically. Furthermore, the spectral domains of the two input pulses need not necessarily coincide in non-interferometric measurement techniques, which gives the dual pulse retrieval a further advantage. There are also techniques, such as the time-resolved pump-probe technique, where two pulses have to be reconstructed. These topics are current and are, therefore, also addressed by other modern pulse measurement techniques [29].
Funding
ANID – Millennium Science Initiative Program (ICN17_012); Comisión Nacional de Investigación Científica y Tecnológica (PIA/Anillo ACT192023).
Disclosures
The authors declare no conflicts of interest.
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