We demonstrate the generation of temporally clean few-cycle pulses at 2.1 μm by shortening of 6-optical-cycle pulses via cross-polarized wave (XPW) generation in BaF2, CaF2 and CVD-Diamond crystals. By combining spectra and single-shot third-order intensity cross-correlation traces in a novel Bayesian pulse retrieval technique, we measured pulse durations of 20 fs, corresponding to 2.8 optical cycles. Our results show that XPW generation in the infrared could provide a high-fidelity source of few-cycle pulses for strong-field physics applications. It could also serve as an injector for high-peak power ultrafast mid-IR wavelength parametric amplifiers.
© 2013 Optical Society of America
Over the past two decades, the generation of intense few-cycle pulses has been a major research goal driven by applications in coherent high-energy density physics. Such pulses are particularly attractive in the infrared (IR) spectral region because of favorable wavelength scaling of strong-field processes  such as the generation of attosecond soft x-ray bursts. Ultrafast sources in this wavelength range are based on optical parametric amplification (OPA) or optical parametric chirped pulse amplification (OPCPA). While OPCPA can nowadays deliver intense ultrashort and carrier-to-envelope phase (CEP) stable pulses in the true mid-IR [2, 3], OPA, driven by an existing Ti:Sapphire system, provides a convenient solution at near-infrared (NIR) wavelengths . Using OPA, CEP-stable pulses with few optical cycles at 2.1 μm and energies approaching one millijoule can be straightforwardly achieved . Reaching the true few-cycle regime however comes at some energy expense and requires either three-wave mixing , nonlinear propagation and post-compression [7,8], or multi-spectrum waveform synthesis and electronic feedback loops . These methods do not necessarily produce pulses of high temporal quality.
Here, we achieve two-fold compression of six-optical-cycle pulses at 2.1 μm using cross-polarized wave (XPW) generation. This technique, which exploits the anisotropy of the third-order tensor in a nonlinear medium, is a well-proven method for obtaining temporally clean few-cycle pulses at 800 nm [10–12]. Both the incoherent temporal contrast and the coherent pedestal due to spectral amplitude and phase distortions are enhanced by this method. The generated pulse consequently presents a steeper rising edge and smoother spectral properties. Because XPW is an achromatic third-order nonlinear process occurring in materials with a near-instantaneous electronic response, it is applicable over a large spectral range: from the UV  to the near-IR , and here at 2.1 μm. As a proof-of-feasibility, by trying different nonlinear materials, we obtain compressed pulses of 2.8 cycle duration, with good spectral quality and with an XPW efficiency of 15%, corresponding to a peak power transmission of 30%. Efficiency could be further improved by implementing a more sophisticated XPW filtering stage as will be discussed later in the article. Therefore, this process can provide clean seed pulses for high-peak power infrared pulse amplification systems as well as a high-dynamic range temporal characterization device for few-cycle infrared pulses . In addition, as temporal characterization of 2 μm-wavelength few-cycle pulses still remains a challenge, we introduce in this article a new method combining spectral measurements and third-order intensity cross-correlation traces with a numerical retrieving algorithm based on Bayesian inference and a Monte-Carlo code.
2. Theoretical overview of the XPW process
The XPW generation process is driven by the third-order nonlinearity of the crystal, , and the anisotropy of the χ(3) tensor . Furthermore, the XPW pulse energy and spectrum also depend on the input spectral phase of the driving laser pulse . In addition, for ultrashort input pulses, the dispersion of the nonlinear material itself must be taken into account. The process, including self-steepening, all four-wave mixing terms and dispersion, is described by the following coupled wave equations where A and B represent the E-field of input and crossed-polarized waves respectively (Eq. 1):16]. M is the matrix containing the six four-wave mixing terms between A and B.
In the experiment, we used the following three crystals exhibiting different dispersive properties over the OPA spectrum: 2 mm -cut BaF2, 2 mm -cut CaF2 and 1.2 mm -cut synthetic diamond grown using the chemical vapor deposition (CVD) technique (denoted CVD-C in the following). The relevant characteristics of these crystals for XPW generation are summarized in Table 1. At 2.1 μm, CaF2 is in the anomalous dispersion regime while CVD-C is in the normal regime. The dispersion of BaF2 is one order of magnitude lower, i.e. closer to the zero dispersion regime over the considered spectral range.
We solve equation 1 numerically with a 1-D split-step propagation code in order to illustrate the XPW generation with the three crystals. Figure 1(a) shows the XPW conversion efficiency as a function of the input intensity on each crystal in conditions similar to the experiment described below (42 fs pulse at 2.1 μm). The input spectral phase is chosen to be zero and reflection losses are not taken into account in the simulation. CVD-C efficiency curve is steeper than BaF2 in spite of the fact that they are 1.2 mm and 2 mm thick respectively. This is due to the very high nonlinearity of CVD-C. On the other hand, at a given intensity, CaF2 exhibits a much lower efficiency because of its lower nonlinearity.
In the temporal domain, the nonlinear process induces temporal shortening and cleaning of the pedestal or satellite pulses. The latter property is characterized by a cleaner XPW spectrum. In first approximation, the cubic response of the process generates a pulse with a duration reduced by a factor of . For short pulses, this translates to an increased spectral bandwidth by the same factor if the input pulse is transform-limited. For few-cycle pulses, the crystal dispersion limits the spectral broadening . However it was observed at 800 nm that highly efficient XPW produces a spectral broadening factor of greater than due to self-phase modulation (SPM) of the incident pulse. This may produce a broader XPW spectrum. Additionally, because of the simultaneous temporal filtering, the XPW pulse does not acquire the spectral amplitude and phase modulations usually associated with SPM . This feature is confirmed by the calculations (Fig. 1(b)) where spectral broadening in excess of a factor of is predicted for the three involved crystals (the simulated intensities were chosen to match the experimentally observed efficiencies). Figure 1(b) demonstrates the high sensitivity of spectral broadening to the global strength of nonlinearity and crystal dispersion. The major role of dispersion is highlighted by the case of CaF2, which gives rise to a spectrum much larger than BaF2 and CVD-C despite its lower nonlinearity and conversion efficiency. This is a consequence of its anomalous dispersion that allows the preservation of the input pulse duration inside the crystal by compensating the phase due to SPM. Conversely, in the case of CVD-C, phase due to normal dispersion adds to SPM and quickly broadens temporally the compressed input pulse. As a confirmation, running the code with zero dispersion, all other parameters being unchanged, produces a spectrum 40 nm narrower in the case of CaF2, but 20 nm wider for CVD-C and unchanged for BaF2. Experimentally, the issue of crystal dispersion can be partially tackled by pre-compensating the second-order phase so that the input pulse is best compressed half-way through the crystal.
3. Experimental setup
The experimental layout is depicted on Fig. 2. The laser source was a multipass Ti:Sa amplifier followed by a home built two-stage OPA which delivered six-optical-cycle pulses at 2.1 μm with a 3 kHz repetition rate and energies up to 250 μJ . After the OPA source, the XPW crystal was placed at the focus of a 500 mm CaF2 lens. A 280 μm diameter diamond pinhole positioned before the crystal served as a spatial filter to optimize the XPW conversion efficiency. The energy after the pinhole is 150 μJ. The XPW signal was then isolated using a linear polarizer (Thorlabs LPMIR050). For each crystal, we inserted in the OPA beam various plates of ZnSe and SiO2 and a wedge pair in order to achieve pulse compression in the XPW crystal and thus optimize both the efficiency and spectral broadening of the XPW process. We followed the same procedure for the optimization of the post-compression of the generated XPW pulse.
As expected, XPW generation is more efficient in BaF2 and diamond where internal conversion of 20% and 15% respectively can be reached whereas it is limited to 10% in CaF2. Since the diamond crystal is thinner than BaF2 and CaF2, its internal efficiency per unit length is comparatively higher. However its overall performances in terms of energy are mitigated by the high Fresnel losses at the crystal faces (≈ 30%). Conversely, BaF2 and CaF2 have much lower reflection losses (≈ 5%). Hence, the XPW output energy is 28.5 μJ, 16 μJ and 14 μJ in BaF2, diamond and CaF2 respectively.
Despite the lower energy after XPW, it was demonstrated in a previous study  that amplification from tens of μJ to hundreds of mJ in an OPCPA scheme could preserve the incoherent contrast. In fact, final contrast in OPCPAs is ultimately limited by the parametric fluorescence of the pump pulse. Using short ps-duration pump pulses precisely synchronized with the injector reduces contrast degradation (during amplification) to sub-5 ps feature . In addition, global throughput of the XPW set-up could be improved by either using a two-crystal scheme [22,23] or working in the saturation regime of the XPW generation with a waveguided XPW setup  for example. Moreover, both methods would limit the degradation of the energy stability to a factor lower than 2 instead of approximately 3 here. The latter technique has also proven to be ideal for optimizing the spatial quality of the output pulses thanks to the guiding inside the hollow-core fiber, and is applicable to pulse energies up to several millijoules. The source could then seed an OPCPA stage with near-mJ energy.27] using the homemade single-shot device  depicted on Fig. 3. The peak signal-to-noise ratio of our cross-correlation traces varied from 50–150, whilst that of the spectra was 800–1200.
Using the spectrum and the third-order intensity cross-correlation trace as input, we inferred the pulse temporal profile and its associated uncertainty. Owing to its novelty, we now describe the procedure in detail. We expect that the precision of this approach — and the occurrence of ambiguities — to be between that obtained when combining the spectrum with the second-order intensity autocorrelation  and combining the spectrum with the temporal intensity . The former possesses many ambiguities , whilst the latter  possesses a two-fold ambiguity for symmetric profiles and is sensitive to noise. These facts suggest that the spectrum and third-order intensity cross-correlation do not, in general, provide enough information for complete pulse characterization. However, here we are concerned with the ambiguities present in our specific measurements — that is, the set of pulses consistent with the measured data. We therefore used a Bayesian algorithm which yields a distribution of reconstructed pulses, sampled according to their consistency with the measured data. From this ensemble, distributions of any desired pulse property, such as the FWHM duration or the spectral phase coefficients, may be computed. Ambiguities, if present, are automatically reflected as multi-modality in the distributions, whilst excessive noise sensitivity produces broad distributions. We emphasize that we are not claiming that the third-order intensity cross-correlation together with the spectrum provide a complete and ambiguity-free characterization; we do claim that our algorithm will detect when different pulses are consistent with a given input.
We use Differential Evolution Markov Chain , which combines a genetic algorithm, differential evolution , with the Metropolis-Hastings algorithm (MH) [35, 36], a standard procedure for generating samples from a distribution. We used simulated annealing to assist convergence. Since the spectrum is known, the unknown pulse is parameterized by spectral phase Taylor coefficients ϕ = ϕ2, ϕ3,...,ϕN. The absolute phase and arrival time of each pulse is adjusted such that the temporal peak occurs at t = 0 and the instantaneous temporal phase at t = 0 is zero. Formally, our target distribution is defined by Bayesian inference: P(ϕ|M) ∝ P(M|ϕ). In other words, the probability (in the Bayesian sense) that the spectral phase coefficients were ϕ given the measured data M is proportional to the probability of obtaining the measured data M given that the coefficients were indeed ϕ. We calculate P(M|ϕ) by computing the theoretical third-order intensity cross-correlation trace corresponding to the measured spectrum and ϕ, and assuming additive white Gaussian noise is present on each pixel of the detector. The noise amplitude is inferred from signal-free pixels. We checked the convergence by ensuring that the distributions 1) were identical between multiple runs, 2) did not change significantly as the maximum order N of the phase coefficients is increased, 3) were not sensitive to all algorithmic parameters. Typically, we found N = 6, a population in the genetic algorithm of 100 members, and 10 000 generations, to be sufficient for the experimental and numerical results presented here. Finally, we note that since single-shot autocorrelators are sensitive to the spatial profile, for each crystal we obtained several cross-correlation traces from different positions in the beam by adjusting the delay stage (Fig. 3). We performed the reconstruction algorithm on each cross-correlation trace independently, and then merged the resulting pulse distributions. In this way, the uncertainty described by the final pulse distribution included systematic errors caused by the spatial profile in the single-shot autocorrelator.
As a verification of our method, we numerically illustrate the reconstruction of a pulse in the presence of an ambiguity. The unknown pulse has centre wavelength 1.8 μm, a Gaussian spectrum with a bandwidth corresponding to a transform limit of 30 fs, and a second-order spectral phase of 500 fs2. We computed the third-order cross-correlation and, along with the spectrum, used it as input to the algorithm described above. We assumed a peak signal-to-noise ratio (SNR) of 30 for the cross-correlation, whilst noise on the spectrum was taken as negligible. Figure 4 shows various aspects of the results. Figure 4(a) compares the cross-correlation trace of the unknown pulse (blue) with the distribution of reconstructed cross-correlation traces (greyscale). Because of the finite SNR, the reconstructed traces lie in a distribution about the “measured” value. Figure 4(b) compares the temporal intensity of the unknown pulse (blue) with the distribution of reconstructed pulses (greyscale). Again, the reconstructed distribution lies around the true value, showing that the algorithm has correctly inferred the temporal profile within the precision allowed by the SNR. Figure 4(c) compares the true spectral phase (blue) with the distribution of reconstructed spectral phase curves (greyscale). The latter is concentrated around two parabolas whose curvatures are of similar magnitude but opposite sign. The upper parabola agrees with the spectral phase of the unknown pulse. This illustrates that the algorithm has detected a two-fold ambiguity — in this case, the ambiguity is caused by symmetric pulse profiles. The bimodality becomes clear in a histogram of the second-order spectral phase coefficients, shown in Fig. 4(d).
By way of contrast, Figure 5 shows the same information for a reconstruction which is free from trivial ambiguities. The unknown pulse has the same spectral intensity, but has a cubic spectral phase of 27 × 103 fs3. The cross-correlation trace (Fig. 5(a)) is asymmetric, and the retrieved temporal intensity (Fig. 5(b)) and spectral phase (Fig. 5(c)) distributions are unimodal and centered on their correct values (blue curves) in regions of significant intensity. The histogram of cubic spectral phase coefficients (Fig. 5(d)) is also unimodal and centred on the correct value.
4. Spectral and temporal characterization of the XPW pulse
The experimental results for the three XPW crystals, as well as the OPA output, are shown in Fig. 6. The measured cross-correlation traces are shown by the blue lines in the second column. The different lines correspond to different time delays on the single-shot autocorrelator as discussed in Section 3, and represent the systematic errors resulting from variation in the spatial profile of the measured pulses. For comparison, the theoretical traces corresponding to transform-limited pulses are also shown, by the magenta lines. The measured cross-correlation traces were combined with the spectral intensity profiles shown by the blue lines in the first column of Fig. 6 using the algorithm of Section 3. The resulting spectral phase distributions are shown in greyscale in the first column of Fig. 6. In general, the spectral phase distributions are multimodal, showing that ambiguities are present. Some of these ambiguities result from the finite precision of the measured cross-correlation traces. Others result from slight systematic differences between the cross-correlation traces obtained at different time delays. However, these uncertainties correspond to only a minor degree of uncertainty in the temporal intensity profiles, which are shown in greyscale in the third column of Fig. 6. Therefore, whilst the reconstructions are not complete, they are sufficient to quantify the temporal compression resulting from XPW.
The OPA pulse exhibits a temporal pedestal and residual spectral phase leading to a best compressed duration of about 40 fs. As expected, the temporal profile is filtered during the process and significant spectral broadening and smoothing are observed for the three involved crystals. In good agreement with the calculations, despite its lower nonlinear characteristics, CaF2 anomalous dispersion regime at 2.1 μm gives rise to the broadest spectrum. As already shown in the NIR, the XPW pulse has a cleaner spectral phase, enabling good temporal compression [17, 37]. This is re-assessed here in the infrared spectral range with our cross-correlation measurement. A transient-grating FROG  could have been used to confirm this feature of XPW filtering. Unfortunately we were not able to implement such technique at the time of the experiment. The cross-correlations and inferred temporal profiles, represented as probability distributions, show that best results in terms of pulse compression were obtained with CVD-C and CaF2 crystals. Pulses were compressed by a factor of 2, leading to minimum pulse duration of 20 fs, close to Fourier-transform limit, corresponding to 2.8 optical cycles at 2.1 μm. In the case of BaF2, the spectral shape is not as Gaussian as in the case of the two other crystals. Residual modulations on the XPW spectrum may arise from uncompensated higher order phase in the input or from the fact that we are close to the white-light generation regime. Therefore, we obtained slightly longer pulses, with 26 fs duration.
In conclusion, we generated 20 fs pulses at 2.1 μm (corresponding to 2.8 optical cycles) via temporal shortening of six-optical-cycle pulses using XPW. So far, the best trade off between internal efficiency (excluding Fresnel losses), temporal shortening and spectral quality was achieved with CVD-Diamond. The use of a longer crystal, or the implementation of a double crystal setup , or a waveguide-XPW filter  along with index matching surface coatings could improve the conversion efficiency of CVD-C, which has a high optical damage threshold suitable for high-power laser systems. Moreover, our laser source now delivers > 600 μJ, 40 fs CEP-stable pulses , raising the possibility of high-contrast CEP-stable > 150 μJ, < 20 fs pulses at 2.1 μm wavelength and 3 kHz repetition rate.
The authors acknowledge financial support from the Agence Nationale pour la Recherche through program ANR-09-JC-JC-0063 (UBICUIL) and the RTRA-Triangle de la Physique through program 2011-024T - NewXPW; from the Spanish Ministerio De Economia Y Competitividad (MINECO) through its Consolider Program (SAUUL-CSD 2007-00013), “Plan Nacional” (FIS2011-30465-C02-01) and the Catalan Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) with SGR 2009–2013; from LASERLAB-EUROPE (228334) and COST Action MP1203, and from the Fundação para a Cie̊ncia e a Tecnologia (grant SFRH/BD/69913/2010). D.R.A acknowledges support from a Marie Curie Intra-European Fellowship (project 276556-BAXHHG). The authors also wish to thank N. Forget for suggesting the use of CVD-Diamond.
References and links
1. P. Colosimo, G. Doumy, C.I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A.M. March, G.G. Paulus, H.G. Muller, P. Agostini, and L.F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386–389 (2008) [CrossRef] .
3. J. Biegert, P.K. Bates, and O. Chalus, “New Mid-Infrared Light Sources,” J. Sel. Top. Quantum Electron. 18, 531 (2012) [CrossRef] .
4. G. Cerullo, A. Baltuska, O.D. Mücke, and C. Vozzi, “Few-optical-cycle light pulses with passive carrier-envelope phase stabilization,” Las. Phot. Rev. 5, 323 (2011) [CrossRef] .
5. F. Silva, P. K. Bates, A. Esteban-Martin, M. Ebrahim-Zadeh, and J. Biegert, “High-average-power, carrier-envelope phase-stable, few-cycle pulses at 2.1 μm from a collinear BiB3O6 optical parametric amplifier,” Opt. Lett. 37, 933–935 (2012) [CrossRef] [PubMed] .
6. J. Biegert and J.-C. Diels, “Compression of pulses of a few optical cycles through harmonic generation,” J. Opt. Soc. Am. B 18, 1218 (2001) [CrossRef] .
7. B.E. Schmidt, P. Bejot, M. Giguere, A.D. Shiner, C. Trallero-Herrero, E. Bisson, J. Kasparian, J.-P. Wolf, D.M. Villeneuve, J.-C. Kieffer, P.B. Corkum, and F. Legare, “Compression of 1.8 μm laser pulses to sub-two optical cycles with bulk material,” Appl. Phys. Lett. 96, 121109–1211035 (2010) [CrossRef] .
8. C.P. Hauri, R.B. Lopez-Martens, C.I. Blaga, K.D. Schultz, J. Cryan, R. Chirla, P. Colosimo, G. Doumy, A.M. March, C. Roedig, E. Sistrunk, J. Tate, J. Wheeler, L.F. DiMauro, and E.P. Power, “Intense self-compressed, self-phase-stabilized few-cycle pulses at 2 μm from an optical filament,” Opt. Let. 32, 868–870 (2007) [CrossRef] .
9. C. Manzoni, S.W. Huang, G. Cirmi, P. Farinello, J. Moses, F.X. Kärtner, and G. Cerullo, “Coherent synthesis of ultra-broadband optical parametric amplifiers,” Opt. Lett. 37, 1880–1882 (2012) [CrossRef] [PubMed] .
10. N. Minkovski, G.I. Petrov, S.M. Saltiel, O. Albert, and J. Etchepare, “Nonlinear polarization rotation and orthogonal polarization generation experienced in a single-beam configuration,” J. Opt. Soc. Am. B 21, 1659–1664 (2004) [CrossRef] .
11. A. Jullien, X. Chen, A. Ricci, J.P. Rousseau, R. Lopez-Martens, L.P. Ramirez, D.N. Papadopoulos, A. Pellegrina, F. Druon, and P. Georges, “High-fidelity front-end for high-power, high temporal quality few-cycle lasers,” App. Phys. B 102, 769 (2011) [CrossRef] .
12. J.M. Mikhailova, A. Buck, A. Borot, K. Schmid, C. Sears, G.D. Tsakiris, F. Krausz, and L. Veisz, “Ultra-high-contrast few-cycle pulses for multipetawatt-class laser technology,” Opt. Lett. 36, 3145–3147 (2011) [CrossRef] [PubMed] .
14. A. Cotel, A. Jullien, N. Forget, O. Albert, G. Chériaux, and C. Leblanc, “Nonlinear temporal pulse cleaning of a 1-μm optical parametric chirped-pulse amplification system,” App. Phys. B 83, 7–10 (2006) [CrossRef] .
15. A. Trisorio, S. Grabielle, M. Divall, N. Forget, and C.P. Hauri, “Self-referenced spectral interferometry for ultra-short infrared pulse characterization,” Opt. Lett. 37, 2892–2894 (2012) [CrossRef] [PubMed] .
16. L. Canova, S. Kourtev, N. Minkovski, A. Jullien, R. Lopez-Martens, O. Albert, and S. M. Saltiel, “Efficient generation of cross-polarized femtosecond pulses in cubic crystals with holographic cut orientation,” Appl. Phys. Lett., 92, 231102–231104 (2008) [CrossRef] .
17. L. Canova, O. Albert, N. Forget, B. Mercier, S. Kourtev, N. Minkovski, S.M. Saltiel, and R. Lopez-Martens, “Influence of spectral phase on cross-polarized wave generation with short femtosecond pulses,” App. Phys. B 93, 443 (2008) [CrossRef] .
18. D. E. Adams, T. A. Planchon, J. A. Squier, and C. G. Durfee, “Spatiotemporal dynamics of cross-polarized wave generation,” Opt. Lett., 35, 1115–1117 (2010) [CrossRef] .
19. F. Silva, P.K. Bates, A. Esteban-Martin, M. Ebrahim-Zadeh, A. Grun, S.L. Cousin, and J. Biegert, in Conference on Lasers and Electro-Optics Europe (CLEO EUROPE/EQEC), Munich - Germany, 22–26 May 2011.
20. A. Buck, K. Schmid, R. Tautz, J. Mikhailova, X. Gu, C. Sears, D. Herrmann, F. Krausz, and L. Veisz, in Frontiers in Optics, OSA Technical Digest (CD) , FMN2 (2010).
21. I. Ahmad, S. A. Trushin, Z. Major, C. Wandt, S. Klingebiel, T. -J. Wang, V. Pervak, A. Popp, M. Siebold, F. Krausz, and S. Karsch, “Frontend light source for short-pulse pumped OPCPA system,” Appl. Phys. B 97, 529–536 (2009) [CrossRef] .
22. A. Jullien, S. Kourtev, O. Albert, G. Chériaux, J. Etchepare, N. Minkovski, and S. Saltiel, “Highly efficient temporal cleaner for femtosecond pulses based on cross-polarized wave generation in a dual crystal scheme,” App. Phys. B , 84, 409–414 (2006) [CrossRef] .
23. A. Jullien, O. Albert, G. Chériaux, J. Etchepare, S. Kourtev, N. Minkovski, and S.M. Saltiel, “A two crystal arrangement to fight efficiency saturation in cross-polarized wave generation,” Opt. Exp. 14, 2760–2769 (2006) [CrossRef] .
24. L.P. Ramirez, D.N. Papadopoulos, A. Pellegrina, P. Georges, F. Druon, P. Monot, A. Ricci, A. Jullien, X. Chen, J.P. Rousseau, and R. Lopez-Martens, “Efficient cross polarized wave generation for compact, energy-scalable, ultrashort laser sources,” Opt. Exp. 19, 93–98 (2011) [CrossRef] .
25. I. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009) [CrossRef] .
26. L. Dahlstroem and B. Kallberg, “Third-order correlation measurement of ultrashort light pulses,” Opt. Commun. 4, 285–288 (1971) [CrossRef] .
28. J. Janszky, G. Corradi, G. Gyuzalian, and R. N. Gyuzalian, “On a possibility of analysing the temporal characteristics of short light pulses,” Opt. Commun. 23, 293–298 (1977) [CrossRef] .
29. J. Peatross and A. Rundquist, “Temporal decorrelation of short laser pulses,” J. Opt. Soc. Am. B, 15, 216 (1998) [CrossRef] .
30. R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).
31. J.-H. Chung and A.M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” J. Sel. Top. Quant. 7, 656–666 (2001) [CrossRef] .
32. A.M.J. Huiser, A.J.J. Drenth, and H.A. Ferweda, “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik 45, 303–316 (1976).
33. C. Braak, “A Markov Chain Monte Carlo version of the genetic algorithm Differential Evolution: easy Bayesian computing for real parameter spaces,” Stat. and Comp. 16, 239–249 (2006) [CrossRef] .
34. R. Storn and K. Price, “Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341 (1997) [CrossRef] .
35. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087 (1953) [CrossRef] .
36. W.K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57, 97 (1970) [CrossRef] .
37. T. Oksenhendler, S. Coudreau, N. Forget, V. Crozatier, S. Grabielle, R. Herzog, O. Gobert, and D. Kaplan, “Self-referenced spectral interferometry,” Appl. Phys. B 99, 7–12 (2010) [CrossRef] .