Open Access
Add to BibSonomyAbstract
We present the design of a novel single-cycle infrared source tunable from 2 to 10 μm. We simulate the optical parametric amplification (OPA) in BBO and the difference frequency generation (DFG) in AGS based on coupled second-order three-wave nonlinear propagation equations. We combine this with the unidirectional pulse propagation equation, which models the generation of the initial supercontinuum seed in sapphire and the final self-compression in YAG, ZnS, and GaAs, respectively. The obtained results indicate that single-cycle pulses can be produced in a tunable range of 2 to 10 μm.
© 2017 Optical Society of America
1. Introduction
In the reaction of an ultrashort laser and material, the ponderomotive potential plays the dominant role [1–3], especially in the generation of the attosecond pulse in a high harmonic generation (HHG) process [4–7]. The ponderomotive potential in an oscillating electric field takes [8, 9], when the driving wavelength is longer, the ponderomotive potential is larger, and the generated order of the HHG is higher. Thus, an intensive infrared pulse is preferred in an HHG experiment with higher photon energy output. In addition, in an experiment for the HHG, the electric field of the laser pulse is high enough to extract an electron by tunnel ionization. The electron set free into the continuum is subsequently accelerated in the laser field and brought to recollide with the parent ion, leading to the emission of an XUV continuum by recombination into the ground state [10]. The sequence of emission recollison events during a driving pulse leads to a train of XUV pulses [11, 12]. The use of a single-cycle pulse as a driving pulse for HHG would enable to naturally limit emission/recollision to a single event, greatly increasing the efficiency of isolated attosecond pulse generation [13–15]. On the other hand, many vibrational “footprint” spectrums of the materials are located in the mid-infrared range [16, 17]. Thus, a tunable ultrashort mid-infrared source is the critical part in the ultrafast dynamic observation of a chemical process [18, 19].
Many efforts have been made to research and improve upon the near-cycle mid-infrared source in HHG experiments for the attosecond pulse [20–22]. There are two types of sources: a near-cycle pulse generated by a spectral coherent combination [10, 23], and a near-cycle pulse generated by the compression of a broader pulse in gas or solid material [24, 25]. In these methods, generation by the filamentation in solid material has the simplest structure. This method does not need a complex beam coherent synthesis arrangement or a vacuum system. On the other hand, Sugih Arto illustrated a difference frequency generation (DFG) source that is tunable from 2.6 to 20 μm [26]. However, no research has been reported on a source that is single cycle and tunable from 2 to 10 μm.
In this paper, we designed a single-cycle source that is tunable from 2 to 10 μm by combining supercontinuum generation, optical parametric amplification, difference frequency generation, and self-compression in bulk material. In our design, several-cycle infrared pulses are produced by the difference frequency of the pulses exported from the OPA. Then, the generated infrared pulse is compressed to a single cycle by optimizing the spectral broadening and dispersion compensation in the proper solid.
2. Concept
In order to generate a widely tunable infrared laser, we adopt an OPA-DFG structure, as shown in Fig. 1. The device is driven by an 800-nm/110-fs/3.2-mJ/1-KHz pulse from a Ti:sapphire CPA system. The OPA part consists of two amplification stages and two BaB2O4 (BBO) nonlinear optical crystals. First, the incident pump laser is split with a beam splitter (BS1). Then, the smaller transmitted beam is further split using BS2, and a small portion of the pump beam is focused onto a 2-mm sapphire plate by lens1 to generate a while-light seed beam. This beam is focused onto the first BBO crystal (BBO1) by lens2. A variable neutral density filter fine-tunes the input energy in the white-light generator. An iris diaphragm is used to adjust the self-focusing condition in the sapphire plate. The larger reflected beam of BS2 focused by lens3 is made collinear with the white-light seed beam and is directed onto BBO1 by dichroic mirror DM1. The temporal overlap is controlled by a delay line (DL1). The generated IR beams (signal and idler) are collimated with lens4, and are then directed onto BBO2.
Fig. 1 Tunable single-cycle source design concept. BS, beam splitters; VNDF, variable neutral density filter; λ/2, half-wave plate; DM, dichroic mirror; DL, delay line. White light is generated by focusing the laser onto a 2-mm sapphire plate; BBO1 and BBO2, BBO nonlinear optical crystals for first and second amplification stages, respectively; AGS, nonlinear optical crystal for DFG stage. PM, parabolic mirror. YAG, ZnS, and GaAs, bulk material for self-compression. Other abbreviations defined in text.
A YVO4 birefringent crystal is inserted to separate the signal and idler pulses from BBO1. The residual pump beam is then removed by dichroic mirror DM2. The larger portion of the first pump beam is combined with the seed beams by dichroic mirror DM2. Similarly, the temporal overlap is controlled by another delay line (DL2). The residual pump beam is then removed by dichroic mirror DM3. This is the two-stage OPA part. The two BBO crystals are cut for type-I phase matching. The crystal thicknesses of the BBO crystals are 3 and 1.5 mm, respectively. The crystal surfaces are antireflection coated at 800 nm to minimize the refraction loss of the pump beam. Then, the signal and idler pulses from BBO2 are separated by DM4, and the separated pulses are combined by DM5 and are directed onto the AGS crystal (different frequency generation).
The temporal overlap between the signal and idler pulses is controlled by another delay line (DL3), and the DM6 is used to separate the shorter wavelength pulses and the longer wavelength pulse. This is the DFG part, and the AGS crystals are cut for type-I phase matching at θ = 40° and ϕ = 45°. The crystal thickness is 1 mm, and the crystal surfaces are antireflection coated at 1000–13,000 nm to minimize the refraction loss of the incident beam. After the DFG process, the output pulse is tunable from 2 to 10 μm. The produced tunable (2–10 μm) pulse is then focused by a parabolic mirror (PM1) into the YAG, ZnS, and GaAs, and the output beam from the crystal is collimated by another parabolic mirror (PM2). The focal lengths of the parabolic mirrors are 200 mm. The self-compression in the bulk material can produce a corresponding single-cycle pulse.
3. Generation of the supercontinuum seed in sapphire
We begin our simulations using an incident 800-nm pump laser. The parameters of this laser are set according to an experimental test of our similar OPA [27]. The bandwidth (fwhm) of the incident 800-nm laser is set to 33 nm, and the corresponding pulse width (fwhm) is set to 110 fs. As shown in Fig. 1, a small portion of the pump laser is focused into a sapphire crystal to generate the seed white light. We generate the initial seed via a supercontinuum process in the sapphire. We simulate pulse propagation in the sapphire by considering the dispersion, diffraction, self-phase modulation (SPM), self-steepening, photon-ionization, and plasma effects. We model the nonlinear process based on equations from [28, 29]:
Here, is the complex field amplitude, is its Fourier transform, is its Hankel transform, is the field intensity, η is the time in the retarded frame of reference, z is the coordinate along the propagation path, r is the transverse coordinate, is the space-to-frequency Hankel transform operator, is the time-to-frequency Fourier transform operator, ω0 is the central frequency of the laser field, , is the refractive index at frequency ω, n2 is the nonlinear coefficient of the Kerr-effect-induced change in the refractive index, , ρ is the electron density and obtained by jointly solving. , U0 is the field-free ionization potential, , W(I) is the ionization rate, σ(ω) is the cross section of avalanche ionization calculated with the Drude formula , and τc is the collision time. is the effective electron mass, e is the electron charge, c is the speed of light in a vacuum, and ε0 is the dielectric permittivity of a vacuum.To model the field evolution in the sapphire, we use for the band gap of the sapphire, , , and [30–32]. The dispersion of the sapphire is included in the model, with all orders through its spectral profile calculated using the Sellmeier equation with coefficients specified in [33]. The initial condition in our simulation is defined in terms of the GDD (−1100 fs2) and the spectrum of the input field, which was chosen (shown by the blue line in Fig. 2) to model the short pulse of the 800-nm pump laser in our experiment. We assume the energy of the pulse injected into the sapphire is 1 μJ, the input diameter of the spot after the iris is approximately 2 mm, and the focal length is 100 mm. The supercontinuum spectrum after the 2-mm sapphire is shown in Fig. 2 (red line). The simulated result infers that the spectrum of the supercontinuum can cover a range of 1 to 1.6 μm, which is necessary for our following parametric amplification.
4. Optical parametric amplification in BBO crystals
As an efficient way to amplify the infrared pulse [34, 35], we choose parametric amplification to amplify the initial seed. In order to amplify a signal range of 1.2–1.6 μm pumped by 800-nm laser, the type-I phase matching angle should vary from 19.8° to 20.2°. Hence, we need to cut BBO crystals at 20°. After some test calculations, we choose the thicknesses of the first and second crystals as 3 and 1.5 mm, respectively. The initial conditions of our OPA include: a pump duration of approximately 110 fs, the diameters of the pump spots of the first and second stages are 2.4 and 12 mm, the diameters of the signal spots of the first and second stages are 2.2 and 10 mm, the pump energies of the first and second stages are 100 μJ and 3 mJ, and the injected energy of the signal in the first OPA stage is 0.5 μJ.
Based on these parameters, the output of the supercontinuum stage is then numerically propagated through two OPA stages in order to amplify their energy and produce the corresponding idler pulse for the next DGF process. In order to simulate the OPA process in the BBO crystals considering the dispersion and optical parametric amplification, we model the parametric process based on equations of [27, 36]:
Here, the physical meanings of some parameters and operators are the same as in Eq. (1),, μ0 is the permeability of a vacuum, and χeff is the effective second-order susceptibility. After amplification by the two-stage OPA, the results are shown in Fig. 3.Fig. 3 Simulated results of OPA. (a) Output signal spectrums, (b) idler spectrums, (c) signal energies, and (d) idler energies for distinct tuned angle.
In order to produce an output of the wavelength from 2.4 to 10 μm in the next DFG stage, we tune the angles of the BBO crystals. The distinct output spectrums are shown in Fig. 3 (a, b). The output spectrum covers a range of 1.2–2.4 μm. The output energies are calculated as well, and are shown in Figs. 3(c) and 3(d). Because the signal and idler pulses have the same polarizations in the type-I phase-matching OPA, the output beams should be separated by the dichroic mirror to adjust their relative delays (as shown in Fig. 1). Considering the loss of optics, we assumed that 80 percentages energies of the output pulse of the OPA stage were transmitted into the DFG stage, which can produce a longer wavelength.
5. Different frequency generation in AGS crystal
The output of the OPA stage is then numerically propagated through a DFG part with an AGS crystal, as shown in Fig. 1. The relative delay of the two incident pulses has been optimized. We simulate the DFG process based on the coupled three-wave Eq. (2). We assumed the thickness of AGS was 1 mm. The phase-matching angle should be changed from 33.7° to 43.6° when the output wavelength of the DFG is tuned from 3 to 10 μm. Hence, we need to cut the crystal at approximately 40°. We adopted the output of the OPA stage as the input of the DFG stage, and the simulated results are shown in Fig. 4.
Fig. 4 Simulated results of DFG: (a) output spectrums for distinct tuned angle, and (b) output energies (blue) and durations (green) for distinct tuned angle.
In the DFG process, incident pulses may temporally walk off, which will decrease the gain bandwidth. The longer wavelength of the output in the DFG process corresponds to less walk-off of the incident pulses, which means the corresponding output spectrum is wider, as shown in Fig. 4(a). On the other hand, the bandwidth of the angle frequency (instead of the bandwidth of the wavelength) has a direct relationship with the pulse width according the Fourier transform. The bandwidths of wavelengths of 3 and 10 μm pulses are 315 and 4250 nm, respectively, but the bandwidths of the angle frequencies of 3 and 10 μm are 63 and 87 THz. Therefore, the bandwidth of a 10-μm pulse is just a little larger than that of a 2-μm pulse.
In the DFG process, the output energy is inversely proportional to the wavelength, so the output energy will decrease when the output wavelength is tuned to be longer, as indicated by the blue line in Fig. 4(b). The output energy corresponding to a 10-μm pulse is 12.13 μJ. The green line in Fig. 4(b) illustrates the output durations of the DFG part. The durations are almost maintained in a range of 50–60 fs, we think this is the result of the interaction of the temporal walk-off and the durations of the incident pulses. In the DFG process, the shorter wavelength output corresponds to shorter duration of the incident pulse, but the amount of temporal walk-off is larger for the shorter wavelength output. Hence, after the DFG process in 1-mm AGS, the shorter duration pulse will be stretched more due to the larger amount of the walk-off. Therefore, the final duration of the DFG is nearly constant throughout the spectrum as shown in Fig. 4(b). The optical cycle of 10 μm is approximately 33 fs, and the optical cycle of 3 μm is approximately 10 fs. Hence, we need to compress the duration of a 10-μm pulse approximately twice to reduce it to a single cycle, but we need to compress the duration of a 3-μm pulse approximately six times to do the same. In order to compress the duration of the pulse from DFG, we focus the output pulse onto the proper bulk material, as shown in Fig. 1. Considering the loss of optics, we assumed 80 percentages energies of the output pulse of the DFG stage must be transmitted into the compression stage.
6. Self-compression with YAG, ZnS, and GaAs
6.1 Dispersion of crystals
We compared several bulk materials for our pulse self-compression. Considering the range of the wavelength needed for compression, and the transmittance of the materials, we chose YAG, ZnS, and GaAs as our material. According to the calculation of the dispersion, we find that the zero-GVD points of YAG, ZnS, and GaAs are 1.6, 3.6, and 6.6 μm. The best situation for pulse self-compression is to use a negative GVD range because the dispersion effect can be balanced with the SPM effect. This balance will guarantee the generated frequency converges in the center of the pulse, and self-compresses the pulse during the propagation of the laser beam. Based on the calculation of the GVD (shown in Fig. 5), we plan to compress the 2/3/4-μm pulses with the YAG crystal, the 5/6/7-μm pulses with ZnS, and the 8/9/10-μm pulses with GaAs. We should point out that the 2-μm pulse is directly from the OPA part. The specific parameters of the compressor are listed in Table 1.
Table 1. Characteristics of materials for self-compression.
6.2 Self-compression
Based on the parameters listed in Table 1, we simulated the self-compression process by solving Eq. (1). We considered the dispersion, diffraction, SPM, self-steepening, photon-ionization, and plasma effects. When designing, we attempted to adopt the same thickness for one material for the convenience of future experiments. However, the 4-mm YAG crystal is better for the compression of a 2-μm pulse, and the 2-mm YAG crystal is better for the compression of a 4-μm pulse. In the simulations, we optimized the compression results by varying the location relative to the focal point (Ldis in Fig. 1) of the bulk materials. The values of the corresponding Ldis are listed in Table 1. The simulated results are shown in Figs. 6–8.
When the pulse propagates in the material, if the anomalous dispersion is balanced with the SPM effect, then the spectrum of the incident pulse will be broadened and the duration of the pulse can be compressed. We can achieve this balance by optimizing the incident intensity, which can be adjusted by Ldis in Fig. 1. On the other hand, an incident intensity that is too high may lead to pulse splitting. In addition, the plasma affects the duration of the infrared laser. The plasma will absorb the rear of the pulse, as shown in Figs. 6(a) and 6(d). The simulated results in Fig. 7 demonstrate that the spectrum of pulses from the DFG part have broadened, and the pulse can be compressed to a single cycle by adopting the proper negative GVD bulk materials (shown in Fig. 8). In some cases, the compression retains some residual pedestal, as in Fig. 6(a) and 6(f), and the compression process may result in pulse splitting, as in Figs. 6(d) and 6(g). There are some spectral modulation in Figs. 7(h) and 7(i). After the comparative analysis, we find these spectral modulations are invoked by the interaction of the spatio-temporal coupling and dispersion in the recompression process (the incident pulse is stretched first and then compressed in time domain). These spectral modulations also lead to the pedestal in their corresponding pulse shapes. The structure of the compressor is shown in Fig. 1, and the self-compressing materials should be switched for a distinct input pulse.
Figure 4(b) implies that the incident energy of the compressor decreases for longer wavelengths. Meanwhile, the incident pulse of 2 μm is directly from the OPA part. Therefore, the corresponding output energy of the compressor is lower for the longer wavelength, as shown by the blue line in Fig. 8, and the final energy of 10 μm is 8.5 μJ. The durations were optimized to approximate one optical cycle, as indicated by the green line in Fig. 8. In some cases, the duration is a little larger than one optical cycle, such as 8 μm. This is because further compression may enlarge the amplitude of the split pulse. In addition, considering the limit of the optical damage, we also calculated the corresponding critical plasma densities and plasma frequencies based on the equations and . The calculated critical plasma densities (ρc), the maximal plasma densities (ρmax), the center angular frequencies (ω0) of the incident pulses and the maximal plasma frequencies (ωpmax) corresponding to the maximal plasma densities are listed in Table 2.
Table 2. Characteristics of plasma in the self-compression.
The results demonstrate that the maximal plasma densities are smaller than the corresponding critical plasma densities, in addition, the center angular frequencies are obviously larger than the corresponding maximal plasma frequencies. Therefore, the plasma haven’t limited the compression in our simulations.
7. Conclusion
We designed an all-solid-state single-cycle tunable source in the infrared spectrum by using a cascade of carefully optimized supercontinuum-generation, parametric-amplification, difference-frequency-generation, and self-compression stages. We simulated the optical parametric amplification in BBO and the difference frequency generation in AGS based on coupled second-order three-wave nonlinear propagation equations. We combined this with a unidirectional pulse propagation equation that models the generation of the initial supercontinuum seed in sapphire and the final self-compression in YAG, ZnS, and GaAs. The obtained simulated results indicate that the anomalous dispersion of the corresponding solid in the compressor plays a significant role under the conditions of our simulations. In addition, single-cycle tunable pulses of 2 to 10 μm can be produced by optimizing the thickness and location of the bulk material in the compressor.
Funding
This work was financially supported in part by the 100 Talents Program of CAS, the National Natural Science Foundation of China (Grant Nos. 61475169, 61521093, and 11127901), and the Youth Innovation Promotion Association of CAS. The Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB16. International S&T Cooperation Program of China, Grant No. 2016YFE0119300.
References and links
1. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). [CrossRef]
2. D. Popmintchev, C. Hernández-García, F. Dollar, C. Mancuso, J. A. Pérez-Hernández, M.-C. Chen, A. Hankla, X. Gao, B. Shim, A. L. Gaeta, M. Tarazkar, D. A. Romanov, R. J. Levis, J. A. Gaffney, M. Foord, S. B. Libby, A. Jaron-Becker, A. Becker, L. Plaja, M. M. Murnane, H. C. Kapteyn, and T. Popmintchev, “Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionized plasmas,” Science 350(6265), 1225–1231 (2015). [CrossRef] [PubMed]
3. T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef] [PubMed]
4. M.-C. Chen, C. Mancuso, C. Hernandez-Garcia, F. Dollar, B. Galloway, D. Popmintchev, P.-C. Huang, B. Walker, L. Plaja, A. A. Jaro -Becker, A. Becker, M. M. Murnane, H. C. Kapteyn, and T. Popmintchev, “Generation of bright isolated attosecond soft X-ray pulses driven by multicycle midinfrared lasers,” Proc. Natl. Acad. Sci. U.S.A. 111(23), E2361–E2367 (2014). [CrossRef] [PubMed]
5. A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9(7), 242 (2007). [CrossRef]
6. C. Yu, H. He, Y. Wang, Q. Shi, Y. Zhang, and R. Lu, “Intense attosecond pulse generated from a molecular harmonic plateau of H 2 + in mid-infrared laser fields,” J. Phys. B 47(5), 055601 (2014). [CrossRef]
7. X. Yuan, C. Liu, P. Wei, Z. Zeng, and R. Li, “Contribution of multiple electron rescatterings on high-order harmonic generation in the mid-infrared wavelength regime,” Chin. Opt. Lett. 14(3), 030201 (2016). [CrossRef]
8. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
9. E. Toma, H. Muller, P. Paul, P. Breger, M. Cheret, P. Agostini, C. Le Blanc, G. Mullot, and G. Cheriaux, “Ponderomotive streaking of the ionization potential as a method for measuring pulse durations in the XUV domain with fs resolution,” Phys. Rev. A 62(6), 061801 (2000). [CrossRef]
10. O. D. Mücke, S. Fang, G. Cirmi, G. M. Rossi, S.-H. Chia, H. Ye, Y. Yang, R. Mainz, C. Manzoni, P. Farinello, G. Cerullo, and F. X. Kartner, “Toward waveform nonlinear optics using multimillijoule sub-cycle waveform synthesizers,” IEEE J. Sel. Top. Quantum Electron. 21(5), 1–12 (2015). [CrossRef]
11. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292(5522), 1689–1692 (2001). [CrossRef] [PubMed]
12. V. V. Strelkov, E. Mével, and E. Constant, “Generation of isolated attosecond pulses by spatial shaping of a femtosecond laser beam,” New J. Phys. 10(8), 083040 (2008). [CrossRef]
13. A. Couairon, H. S. Chakraborty, and M. B. Gaarde, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77(5), 053814 (2008). [CrossRef]
14. H. S. Chakraborty, M. B. Gaarde, and A. Couairon, “Single attosecond pulses from high harmonics driven by self-compressed filaments,” Opt. Lett. 31(24), 3662–3664 (2006). [CrossRef] [PubMed]
15. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-harmonic generation of attosecond pulses in the “single-cycle” regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997). [CrossRef]
16. H. Arnolds and M. Bonn, “Ultrafast surface vibrational dynamics,” Surf. Sci. Rep. 65(2), 45–66 (2010). [CrossRef]
17. M. Lim, T. A. Jackson, and P. A. Anfinrud, “Mid‐infrared vibrational spectrum of CO after photodissociation from heme: Evidence for a ligand docking site in the heme pocket of hemoglobin and myoglobin,” J. Chem. Phys. 102(11), 4355–4366 (1995). [CrossRef]
18. H.-S. Tan, I. R. Piletic, R. E. Riter, N. E. Levinger, and M. D. Fayer, “Dynamics of water confined on a nanometer length scale in reverse micelles: ultrafast infrared vibrational echo spectroscopy,” Phys. Rev. Lett. 94(5), 057405 (2005). [CrossRef] [PubMed]
19. J. J. van Thor, K. L. Ronayne, and M. Towrie, “Formation of the early photoproduct lumi-R of cyanobacterial phytochrome cph1 observed by ultrafast mid-infrared spectroscopy,” J. Am. Chem. Soc. 129(1), 126–132 (2007). [CrossRef] [PubMed]
20. O. Chalus, A. Thai, P. K. Bates, and J. Biegert, “Six-cycle mid-infrared source with 3.8 μJ at 100 kHz,” Opt. Lett. 35(19), 3204–3206 (2010). [CrossRef] [PubMed]
21. I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, T. Paasch-Colberg, I. Znakovskaya, M. Pescher, W. Schweinberger, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015). [CrossRef]
22. Z. Huang, D. Wang, Y. Dai, Y. Li, X. Guo, W. Li, Y. Chen, J. Lu, Z. Liu, R. Zhao, and Y. Leng, “Design of intense 1.5-cycle pulses generation at 3.6 µm through a pressure gradient hollow-core fiber,” Opt. Express 24(9), 9280–9287 (2016). [CrossRef] [PubMed]
23. C. Manzoni, O. D. Mücke, G. Cirmi, S. Fang, J. Moses, S. W. Huang, K. H. Hong, G. Cerullo, and F. X. Kärtner, “Coherent pulse synthesis: towards sub‐cycle optical waveforms,” Laser Photonics Rev. 9(2), 129–171 (2015). [CrossRef]
24. T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32(22), 3330–3332 (2007). [CrossRef] [PubMed]
25. V. Shumakova, P. Malevich, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016). [CrossRef] [PubMed]
26. A. Sugiharto, C. M. Johnson, H. De Aguiar, L. Alloatti, and S. Roke, “Generation and application of high power femtosecond pulses in the vibrational fingerprint region,” Appl. Phys. B 91(2), 315–318 (2008). [CrossRef]
27. W. Li, Y. Chen, Y. Li, Y. Xu, X. Guo, J. Lu, and Y. Leng, “Astigmatism transfer phenomena in the optical parametric amplification process,” Appl. Phys. B 123(1), 37 (2017). [CrossRef]
28. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. R Góngora, and M Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011). [CrossRef]
29. E. A. Stepanov, A. A. Lanin, A. A. Voronin, A. B. Fedotov, and A. M. Zheltikov, “Solid-state source of subcycle pulses in the midinfrared,” Phys. Rev. Lett. 117(4), 043901 (2016). [CrossRef] [PubMed]
30. E. R. Dobrovinskaya, L. A. Lytvynov, and V. Pishchik, Sapphire: Material, Manufacturing, Applications (Springer Publishing Company, Incorporated, 2009).
31. A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29(6), 602–604 (2004). [CrossRef] [PubMed]
32. A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71(12), 125435 (2005). [CrossRef]
33. G. Li, M. Bass, and E. V. Stryland, Handbook of Optics, Volume IV: Optical Properties of Materials, Nonlinear Optics, Quantum Optics (McGraw-Hill, 2010).
34. G. Andriukaitis, T. Balčiūnas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M.-C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36(15), 2755–2757 (2011). [CrossRef] [PubMed]
35. S. Li, P. Ju, Y. Liu, X. Jiang, R. Ni, G. Zhao, X. Lv, and S. Zhu, “Efficiency-enhanced picosecond mid-infrared optical parametric downconversion based on a cascaded optical superlattice,” Chin. Opt. Lett. 14(4), 041402 (2016). [CrossRef]
36. G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14(10), 2543–2549 (1997). [CrossRef]
