Generation of few-cycle infrared pulses from a degenerate dual-pump OPCPA

A degenerate dual-pump optical parametric chirped-pulse amplifier (OPCPA) for generation of few-cycle intense pulses centered at 1.6 μm is theoretically investigated. By adding the optimized linear chirp to the two pump pulses from Ti:sapphire source and carefully adjusting the delays between the two pumps and seed, the longand short-wavelength components of the seed pulse are efficiently amplified during the parametric process. Our simulations show that a broadband spectrum spanning from 1.3 μm to 2.1 μm is attained with a conversion efficiency of 22.6%. Signal pulse with a near transform-limited (TL) duration of 10.1 fs can be achieved by simply removing the linear chirp from the output signal. Besides, the compressed signal beam manifests good quality both spectrally and temporally, which allows tightly focusing the signal beam for further use. © 2014 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.4970) Parametric oscillators and amplifiers; (140.7090) Ultrafast lasers. References and links 1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). 2. S. Backus, C. G. Durfee III, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998). 3. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003). 4. D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Fewoptical-cycle pulses tunable from the visible to the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010). 5. O. D. Mücke, S. Ališauskas, A. J. 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Introduction
Laser sources with femtosecond pulse duration as well as high wavelength tunability are of utmost importance for a wide range of scientific applications in time-resolved optical spectroscopy, quantum optics, nanoscience, nonlinear optical microscopy and high-field physics [1,2].Optical parametric amplification (OPA), owing to its unique properties including broad gain bandwidth, broadband tunability and large gain per pass, has become an ideal source for such types of laser pulses [3][4][5][6].Nevertheless, the potential of producing more energetic pulses in OPA is severely restricted by the available pump energy.To increase the maximum applicable pulse energy limited by the damage threshold and size of the nonlinear crystal, the well-known chirped-pulse amplification (CPA) concept is combined with OPA, and the OPCPA technique is proposed [7][8][9][10][11][12].
As a more energy-scalable method than OPA, the advent of OPCPA has prominently stimulated the development in the study of strong-field physics [13].On the other hand, recent progresses on high intensity isolated attosecond pulses (IAPs) generation [14][15][16][17][18], controlling and tracking electron dynamics in atoms and molecules within attosecond timescale [19][20][21][22][23][24] require not only enough pulse energy but also even shorter durations down to few-cycle region.However, the possibility of achieving millijoule, few-cycle pulses in OPCPA still faces several challenges such as the chirp-induced phase-mismatch between the pump and seed, compensation of the output pulse chirp and the gain-narrowing effect resulted from the temporal dependence of the pump intensity.
In order to obtain pulses with few-cycle durations in OPCPA, ultrabroadband phase-matching is essentially required [25][26][27].An effective method of broadening the OPCPA gain bandwidth is the two-beam-pumping scheme.By using two pump pulses with different wavelengths [27,28] or input angles [29,30], different regions of the signal spectrum are amplified, respectively, thus a combined broadband gain can be achieved.For example, Harth et al. [27] employed the second (SH; 515 nm) and third harmonic (TH; 343 nm) from a Yb:YAG regenerative amplifier as two-color pump sources, and two neighboring regions (700-1300 nm and 430-700 nm) of the broadband seed from a Ti:sapphire oscillator are amplified in two successive stages, resulting in an output spectrum ranging from 430 nm to 1300 nm, with a TL duration of sub-3 fs and pulse energy of 1 µJ.
The chirp-compensation scheme [31][32][33] provides another approach for broadband gain, in which the broadband pump and seed are both carefully chirped to ensure that their phasematched spectral components are overlapped at all times.In Ref. [34], Limpert et al. utilized a photonic crystal fiber to produce a degenerate supercontinuum seed with a quadratic chirp.By linearly chirping the pump pulse to optimize the temporal overlap of the phase-matched frequency components, the gain bandwidth was enhanced.As a result, an output spectrum covering 630 nm to 1030 nm with an energy of ∼5 µJ was generated.A chirp-compensation OPCPA scheme centered at non-degenerate wavelength was proposed by Tang et al. [35].The nonlinearly chirped seed was amplified by a linearly chirped pump within a broad phase-matching bandwidth, and the obtained idler pulses with >165-nm bandwidth and ∼7-µJ energy could be a suitable source for a large OPCPA system.Despite the gain bandwidth broadened by the two-beam-pumping or chirp-compensation scheme, the output pulses encounter compressing difficulties due to the nonlinear temporal chirp.In order to achieve near-TL pulse durations, the complex spectral phase has to be compensated.Besides, the resulting microjoule energy is insufficient for many strong-field experiments.
In this article, we report a novel degenerate dual-pump OPCPA scheme producing 1.6-µm pulses with few-cycle duration and millijoule energy.By optimizing the linear chirp and delay of the two pumps and seed, the instantaneous wavelengths of the interacting pulses are phasematched at all times.The long-and short-wavelength regions of the seed are amplified by two inversely chirped pumps, respectively.The results of the numerical simulations show that the scheme is capable of efficiently amplifying a broadband spectrum around degeneracy, which supports a sub-two-cycle TL duration.Near-TL pulses are attainable by simply compensating the linear temporal chirp.Moreover, energy scalability of the system is remarkably improved by temporally stretching the interacting pulses, meanwhile the possible detrimental effects originated from high peak intensity can be avoided.Note that only one commercially available Ti:sapphire laser is utilized in this scheme as the source for both the pump and seed pulses, allowing all-optical synchronization without the need of complicated electric synchronizing.The rest of this paper is organized as follows.In Sec. 2, we illustrate the concept of the dual-pump OPCPA scheme.In Sec. 3, the numerical model is introduced.In Sec. 4, the simulation results and discussion are presented.Finally in Sec. 5, the conclusions are drawn and the prospect is discussed.

Concept of dual-pump OPCPA
In a conventional OPCPA, a temporally stretched seed is amplified by a long-duration TL pump (e.g. a nanosecond [43,44] or picosecond [42,45] pulse) through parametric amplification in a nonlinear crystal.During this parametric process, the phase-mismatch (∆k = k p − k s − k i ) is a critical parameter for obtaining a broadband gain.However, the phase-matching condition in OPCPA is normally optimized only for a particular seed wavelength.In this case, perfect phasematching no longer exists (∆k ̸ = 0) as the instantaneous wavelength of the chirped seed varies in time (λ s (t)) while the wavelength of the TL pump (λ p ) is fixed.This chirp-induced phasemismatch between pump and seed leads to the preferential amplification of phase-matched frequency components and the consequent narrowing of the gain bandwidth.1.The variation of pump wavelengths with the phase-matched seed wavelengths: calculated data (solid curves) and linear fitting (dashed lines).
Figure 1 shows the phase-matching curves in type-I BBO crystal pumped by a 0.8-µm Ti:sapphire femtosecond laser in a collinear geometry.Three crystal angles are calculated for comparison.For example, when the crystal is cut at 19.89 • , signal wavelengths of 1.47 µm and 1.76 µm are phase-matched to 0.8-µm pump.As the signal wavelength changes, the phasematched pump wavelength changes correspondingly.Intuitively, a broad phase-matching bandwidth can be realized when the corresponded instantaneous wavelengths between the pump and seed pulses fit the curve.Since this wavelength relation is difficult to be achieved in experiment, the broadband phase-matching can be expected with an alternative way of fitting the curve with two straight lines.As shown in Fig. 1, the long-and short-wavelength regions of the blue curve (θ = 19.89• ) are linearly fitted by two dashed lines, respectively.Normally, the pulse duration of a Ti:sapphire laser is 30-40 fs, with a bandwidth of approximately 20 nm (790-810 nm).When the crystal angle is smaller (19.87 • , red curve in Fig. 1), the long-wavelength region of the pump can not be phase-matched to any signal spectral components, thus the 20-nm bandwidth are not fully utilized.While with a larger crystal angle (19.91 • , green curve in Fig. 1), two phase-matched spectral regions are separated from each other, leading to an undesired gap at ∼1.6 µm.Therefore the angle of 19.89 • is selected in our scheme.The linear fittings are realized by introducing linear temporal chirp to both pump and seed pulses, and the pump-seed chirp ratios are adjusted according to the slopes of two lines.In this way, two neighboring regions of the seed spectrum (1.4-1.6 µm and 1.6-1.8µm) can be amplified by two pump pulses, respectively, resulting in a broad overall gain bandwidth.
The dual-pump OPCPA scheme is introduced based on the above illustration.The conceptual layout is depicted in Fig. 2. The initial TL seed is negatively chirped by a pulse stretcher.Two TL pump pulses, divided by a beam splitter, are negatively and positively chirped, respectively.The overlap of the phase-matched components between pump and seed pulses are achieved by adjusting the pump delay with two delay lines.The two pump beams can be combined using a dichroic mirror in a collinear geometry.It is also possible to employ the two pump beams by using slightly different input angles.Thereafter, the stretched and synchronized pulses enter the nonlinear crystal where the parametric amplification takes place.After separating and recompressing the output signal and idler pulses, femtosecond pulses at degeneracy are obtained.

Numerical model
To quantitatively determine the performance of the dual-pump OPCPA, we numerically solve the coupled-wave equations [31,46,47], in which the nonlinear interaction and the 1st-to 4thorder dispersions are included.The nonlinear partial differential equations are calculated with the split-step Fourier-transform algorithm [48][49][50][51].The crystal is divided into a number of small segments, the linear and nonlinear steps are integrated separately.During each step, it is necessary to Fourier transform the pulses back and forth because the linear steps are handled in frequency domain and the nonlinear steps are handled in time domain.All related parameters must be prepared before the calculation.Since the essential physics of the system can be revealed by a one-dimensional (1D) model [31,[50][51][52], we perform most of the following simulations with a simplified 1D model except that the spatial profiles of the amplified signal pulse are studied with a two-dimensional (2D) model (simplified from the 3D model in [51]).
Before we continue, the relation between the linear temporal chirp and the group delay dis-persion (GDD) needs to be clarified.The GDD, also known as second-order dispersion in the frequency domain, is the derivative of the group delay with respect to the angular frequency, The GDD is specified in the frequency domain, whereas the linear chirp is a concept in the time domain.The Fourier transformation is therefore employed to connect these two parameters.The laser field with a Gaussian envelope and GDD can be described in the frequency domain as where η m = 2ln2/∆ω m 2 , ∆ω m is the spectral bandwidth of the pulse, D m is the GDD, and m = p, s and i correspond to the pump, signal and idler pulses, respectively.Transforming A m (ω) inversely into the time domain, we will obtain The linear chirp rate α m in the time domain can be inferred from Eq. ( 2) Eq. ( 3) shows the relation between the linear temporal chirp (α) and the GDD (D m ).Hereafter, the GDD value is employed to characterize the linear chirp rate of the interacting pulses.
We assume the original Gaussian-shaped 0.8-µm pump and 1.6-µm seed pulses with TLdurations of 30 fs and 10 fs, respectively.The ultrabroadband seed can be generated from the white-light generation in a piece of sapphire or YAG crystal.By focusing a small fraction of the 0.8-µm pump into the WLG crystal, a supercontinuum extending up to 2.1 µm can be produced [6].The seed is chirped with a GDD of 1800 fs 2 , stretching the pulse to a duration of 500 fs.To ensure that the instantaneous wavelengths of input pulses are phase-matched at all times, the chirp and delay of the pump pulses are carefully adjusted based on the linear fitting of the phase-matching curve.In this case, the GDDs of initial pumps are -5600 fs 2 and 5000 fs 2 and the delays are 270 fs and -300 fs, respectively.Gaussian beams with a diameter of 10 mm are used, the peak intensities of pump and seed are 20 GW/cm 2 and 0.1 MW/cm 2 , corresponding to pulse energy of 17.6 mJ and 40 nJ, respectively.The BBO crystal is cut at θ = 19.89• .We employ the energy-bandwidth product (EBP), i.e. the ratio of signal energy to the TL pulse duration, to optimize the spectral-temporal performance of the dual-pump system [53,54].In this case, BBO crystal with a thickness of 3.2 mm is selected, in which the highest EBP is achieved.The calculated spatial walk-off angle ρ of the extraordinary pump is 2.8 • , resulting in a spatial walk-off of 0.06 mm.Comparing to the 10-mm beam diameter, the spatial walk-off is neglectable in the simulations.In order to separate the degenerate signal and idler beams in experiment, a very small non-collinear angle between the input pump and seed beams (α = 0.4 • ) can be introduced by projecting the wave vector of the pump beam onto the wave vector of the seed.The phase-matching curve in this near-collinear geometry is almost the same as the blue curve shown in Fig. 1.The near-collinear angle is taken into account in the following simulations.

Results and discussion
First, in order to determine the validity of the dual-pump OPCPA, we compare this scheme with several single-pump cases: (a) the pump pulse is linearly chirped, (b) the pump pulse is quadratically chirped to fit the phase-matching curve, and (c) the pump pulse is nonlinearly chirped precisely based on the phase-matching curve for perfect phase-matching at all times.The results are plotted in Fig. 3. Figure 3(a) shows the relationship between the instantaneous wavelengths of the pump and seed pulses.The black line, calculated from the Sellmeier equation, is the phase-matching curve around degeneracy, which can be well fitted by the quadratically chirped pump (the red line).To simplify this chirp-compensation concept in experiment, we replace the quadratic pump chirp by synthesizing two linearly chirped pulses, depicted by the blue lines in Fig. 3(a).The GDDs of the pump and seed pulses in the linear chirp scheme are determined by maximizing the EBP.The ∆k dependence on signal instantaneous wavelength is shown in Fig. 3(b).The phase-mismatch in quadratic chirp scheme is negligible throughout the 1.4-1.8µm range, implying broadband phase-matching around degeneracy.In the linear chirp scheme, as we stated above, perfect phase-matching can only be realized for two signal wavelengths, thus ∆k increases rapidly as the signal instantaneous wavelength shifts away from degeneracy.As for the dual-pump scheme, when two pumps are synthesized, the trailing edge of the leading pulse temporally overlaps with the leading edge of the trailing pulse.The superposed wave vector k p changes with the instantaneous wavelengths of both pump pulses in the overlapped region, and ∆k changes correspondingly.As blue line in Fig. 3(b) shows, for near-degenerate signal components, ∆k oscillates near 0 and the influence on gain efficiency is minor.Consequently, the dual-pump scheme is a reasonable approximation and experimental simplification of the quadratic-chirp scheme.In the numerical simulations of the four schemes, seed pulses are identical while the pump pulses are chirped in different ways.Therefore, the peak intensities of the input pumps are adjusted to ensure the equal pump energy, which are 29.1 GW/cm 2 for quadratically chirped pump, 30.5 GW/cm 2 for nonlinearly chirped pump, and 18.9 GW/cm 2 for linearly chirped pump, respectively.Here, the crystal lengths are also selected to optimize EBP of the schemes, as can be seen in Fig. 3(c).As the figure shows, the largest EBP is achieved in the perfectly phase-matched OPCPA owing to the highest gain efficiency.The quadratically chirped scheme can realize an EBP slightly lower than the perfect phase-matching case, proving the validity of the chirp-compensation concept.The highest EBP in dual-pump scheme is smaller than the two cases.Because of both the pump intensity modulation and the phase-mismatch oscillation, different signal spectral components approach saturation at different crystal lengths, therefore the energy conversion efficiency is lower.This phenomenon is more obvious in the linearly chirped pump case, in which the ∆k is quite large for non-degenerate spectral components.In this case, different components are amplified at different gain efficiency, thus the saturation energy in the three previous cases is unreachable in the linearly chirped OPCPA.Besides, the crystal length needed for maximal EBP in dual-pump scheme is shorter than that in the linearly chirped single-pump scheme, which is more beneficial for suppressing the super-fluorescence and the B-integral accumulation during the parametric process.Figure 3(d) shows the output signal spectra of the four schemes where the maximized EBPs are achieved.Despite the strong intensity modulation, the output spectrum of the dual-pump scheme covers the broadest range among the four schemes.The output TL pulse durations are 9.0 fs (dual-pump OPCPA), 9.6 fs (quadratically chirped OPCPA), 9.2 fs (perfectly phase-matched OPCPA) and 10.1 fs (linearly chirped OPCPA), respectively.
Furthermore, a single-pump chirp-free OPA system is compared in Fig. 3(d), the spectrum in this scheme is broader than the single-pump linear-chirp scheme, which implies that an OPA with TL input pulses is still more favorable for a broadband gain.On the other hand, for the dual-pump scheme, the resulting spectrum is slightly broader than the chirp-free OPA system, showing the effectiveness of the chirp-compensation method.In the meantime, the energy scalability in the OPCPAs can be remarkably improved comparing to the chirp-free OPA because of the dual-chirp configuration.
After the comparison between the dual-pump scheme and three single-pump schemes, we investigate the spectral-temporal characteristics of the system and plot the results in Fig. 4. As shown in Fig. 4(a), the system is capable of amplifying a spectrum ranging from 1.3 µm to 2.1 µm, which supports a sub-two-cycle TL duration of 9.0 fs (red dashed line in Fig. 4(b)).The output signal energy is 3.97 mJ, corresponding to a conversion efficiency of 22.6%.However, a strong modulation is observed in the spectrum.To determine the origin of the modulation, the spectral phase of the output signal is investigated (green solid line in Fig. 4(a)).Even though ∆k oscillates throughout the whole spectrum, the effect on the spectral phase is inconspicuous.The resulting phase curve fits well with the quadratic initial seed phase, implying that the signal spectral phase has no evident variation during the amplification.Moreover, we simulated the dual-pump scheme without the phase-mismatch, and the resulting signal spectrum is nearly the same as the spectrum in Fig. 4(a), which also indicates that the small ∆k is of minor influence on the spectral modulation.Except for the ∆k oscillation, intensity modulation also occurs in the overlapped region between two pump pulses.For the nearly phase-matched signal components around degeneracy, the output spectrum inherits the shape of the pump intensity profile, resulting in the spectral modulation.Despite the strongly modulated spectrum, the near-quadratic spectral phase is beneficial for pulse compression.By compensating the GDD of the spectrum, i.e. linear chirp in time domain, near-TL 10.1-fs pulses are produced (Fig. 4(b)).The temporal phase is quite flat in the main pulse, which proves the effective compensation of the linear temporal chirp.Therefore, prism or grating pairs can be utilized in this scheme to recompress the output pulses.
We also simulated two similar OPCPA schemes pumped by two single pulses (one by the leading pump pulse and the other by the trailing pump pulse) for comparison.The resulting spectra are plotted as the red dashed curves in Fig. 4(a).With the same chirp and delay of the pump pulses in previous simulations, broadband spectra at non-degenerate regions are attained.
Nevertheless, when only one pump pulse is employed, the pump duration is shorter than the seed duration, and the non-degenerate range of the signal gets amplified faster than the degenerate components because of the higher corresponded pump intensity, thus the gain bandwidth decreases during the amplification.This gain-narrowing effect leads to a spectral gap near 1.6 µm.On the other hand, the degenerate gap can be filled in dual-pump scheme owing to the intensity superposition in the overlapped region, resulting in ∼800-nm overall gain.Generally, there is a trade-off between the bandwidth and energy of the output signal pulse in OPA.We study this energy-bandwidth trade-off by varying the GDD of the input seed, and the result is shown in Fig. 5.With a small GDD, the chirped seed is quite short comparing to the chirped pump, only a small fraction of the pump energy is transferred to the signal.
In the meantime, most of the signal spectral components are amplified because of the higher pump intensity in the center temporal coordinates.In contrast, a larger GDD corresponds to a longer chirped seed and more pump energy is transferred to the signal, thus a higher conversion efficiency is attained.Under this condition, the edges of the signal can only be amplified by the low-intensity pump edges, resulting in the gain-narrowing of the output spectrum.Since the pulse duration and conversion efficiency manifest the same trend in the figure, the EBP is also determined for clarification.As depicted by the red dots in Fig. 5, with the seed GDD of 1800 fs 2 , the maximal EBP can be achieved.Next, to determine the possibility of generating even shorter pulses in this scheme, bandwidth limit of the input pulses is discussed.We employ seed pulses with different TL durations varying from 16 fs to 6 fs, and compare the TL signal durations and pulse energy after calculating the amplification processes.In the simulations, the peak intensities of different seeds are adjusted to maintain the 40-nJ input pulse energy, and all cases are compared at EBP-maximized crystal lengths.As the blue circles in Fig. 6(a) show, for >10-fs seed, the TL duration of output signal decreases by shortening the initial seed, while for <10-fs seed, the bandwidth enhancement is barely observed.As depicted by the phase-matching curve, the gain bandwidth is affected by both the pump and seed bandwidths, thus we simulate the processes using shorter initial pump pulses (with optimized chirp and delay) and compare the results with the 30-fs case.As the blue circles in Fig. 6(b) show, using 25-fs input pumps, the output TL signal are clearly shorter.Besides, the signal energy in both conditions (green triangles in Fig. 6) increases as shorter seed pulses are employed, resulting in a remarkable enhancement in the attained pulse EBP.The bandwidth of 25-fs pump is larger than that of 30-fs pump, according to the phasematching curve, more spectral components of the seed can be phase-matched by shorter pump pulse, thus a broader gain bandwidth is obtained.Besides, with equal GDD, a shorter seed pulse is stretched to a longer duration.In this case, the durations of the chirped seed and pump pulses are closer and the temporal overlap between the input pulses is improved, which is more beneficial for sufficient energy transfer.With optimized parameters, the output signal with a TL duration of 7.8 fs and an energy of 4.62 mJ can be generated, corresponding to a conversion efficiency of 26.3% and a peak power of ∼ 6 × 10 11 W.
Apart from the bandwidth limit, we also studied the influence of the seed shape (Gaussian and super-Gaussian envelope) and center wavelengths (near-degenerate and non-degenerate).For a 10th-order super-Gaussian seed pulse with a TL-duration of 10 fs, the output spectrum is almost the same as the Gaussian seed, and the conversion efficiency is nearly equal.The bandwidth of input super-Gaussian seed is well preserved, corresponding to a TL duration of 8.9 fs, even shorter than that in the Gaussian-seed case.It can be inferred that the dualpump scheme is suitable for seed pulses with various kinds of shapes.For non-degenerate seed pulses, the symmetry of the phase-mismatch and the wavelength no longer exists.When the center wavelength is close to degeneracy (e.g.1500 nm-1700 nm), the pulses can be amplified in a broad spectrum by adjusting the pulse delay.Although the ideal phase-matching conditions are deteriorated due to the wavelength shift, the ∆k near degeneracy is quite low.Despite the slightly lower conversion efficiency, the theoretical TL duration of the near-degenerate pulses are still less than 10 fs.When the center wavelengths further shifts into the non-degenerate region, it is difficult to optimize the phase-matching condition by simply adjusting the pulse delay, thus the gain bandwidth dramatically decreases.As the phase-matching curves in Ref. [34] show, the symmetry center of the curve in a non-degenerate geometry has a small shift away from degeneracy, which is also observed in 800-nm pump cases.It is therefore deduced that the dual-pump scheme may be available for non-degenerate pulses in a configuration with a larger non-collinear angle.Considering the strong modulation of the signal spectrum, the obtained beam quality is also of concern.The spatial properties are studied by solving the 2D coupled-wave equations with the previous parameters.Figure 7 depicts the transversal intensity distribution with respect to the spectrum (a) and temporal profile (b).The spectrum (the upper cross-section in Fig. 7(a)) in 2D model is identical to Fig. 4(a).For the transversal profile at 1.6 µm, the beam center has reached saturation whereas the two sides are unsaturated, leading to the flattop cross-section in Fig. 7(a).After compensating the linear temporal chirp, the temporal-spatial profile of the recompressed signal is shown in Fig. 7(b).Both the temporal and spatial intensity profiles of the signal are close to Gaussian envelope, revealing a good beam quality in the near field.The uncompressed signal energy in 2D model is 2.90 mJ, corresponding to a conversion efficiency of 16.5%, lower than that in the 1D model.The efficiency decrease is resulted from the Gaussian envelope of the beam transversal profile.At the beam edges with lower pump intensities, the gain efficiency is smaller than in the beam center, thus the conversion efficiency degrades comparing to the ideal flattop beams in 1D model.The potential of attaining higher peak intensity can be achieved by tightly focusing the output beam.We calculated the temporal-spatial property of the beam at the focus of a f=100 mm concave mirror using the transfer-matrix method [55].A good focusability is observed in Fig. 8, with a peak intensity of ∼10 17 W/cm 2 .
Note that when the pulse duration is down to few-cycle regime, the carrier-envelope phase (CEP) of the pulse becomes important [1,26,[56][57][58][59].It has been proposed and demonstrated by Baltuška et al. that self-CEP-stabilized idler pulses can be generated during the OPA process [60].In direct analogy, the dual-pump OPCPA is also capable of producing CEP-stabilized idler pulses.In our simulation, the resulting idler spectrum (not shown in the article) covers a broad spectral range similar to the signal, supporting a TL duration of 11.5 fs and a compressed duration of 13.1 fs.Owing to the very small near-collinear angle we introduced between the pump and seed beams, the output signal and idler can be easily separated.Even though the idler beam contains angular dispersion because of the near-collinear angle, the dispersion is quite small.Considering the large beam diameter, the slightly angularly dispersed beam can be directly used in the following stages.Besides, it is possible to compensate the angular dispersion using a hollow fiber [61].Furthermore, because of the high single-pass gain and bandwidthpreserving character, the dual-pump scheme is capable of efficiently amplifying low-energy, CEP-stable pulses [41,62], which makes it an ideal secondary amplification stage in a large OPCPA system.

Conclusion
In summary, we proposed a novel OPCPA scheme, the degenerate dual-pump OPCPA, for the generation of few-cycle millijoule pulses at 1.6 µm.In this scheme, the broadband 0.8µm pump and 1.6-µm seed are linearly chirped and delayed to ensure that the instantaneous wavelengths of input pulses are phase-matched at all times.As our simulations show, the longand short-wavelength regions of the negatively chirped seed are amplified by two pump pulses with positive and negative chirps, respectively.The combined gain spectrum ranges from 1.3 µm to 2.1 µm, supporting a TL duration of 9.0 fs.The uncompressed energy of the output signal is 3.97 mJ (2.90 mJ in 2D model), corresponding to a conversion efficiency of 22.6%.A near-TL 10.1-fs pulse centered at 1.6 µm is attained by compensating the linear temporal chirp, which clearly simplifies the compressing stage.If we use even shorter initial pump/seed pulses, the resulting pulse durations can be further shortened down to 7.8 fs.Besides, the dual-pump scheme is also suitable for super-Gaussian or near-degenerate seed pulses.Due to the high single-pass energy gain (over 10 5 ) and high conversion efficiency, the scheme can be an ideal secondary stage for a low-energy OPA/OPCPA system.Moreover, the 2D simulations reveal a good beam quality and good focusability of the compressed pulse, a peak intensity of ∼10 17 W/cm 2 is readily available by tightly focusing the output signal beam.
The dual-pump OPCPA, employing only one commonly used Ti:sapphire femtosecond laser, allows low-timing-jitter all-optical synchronization instead of costly electric synchronizing.Furthermore, high energy scalability can be expected in this scheme.On one hand, the damage threshold of BBO crystal is over 200 GW/cm 2 , and the commercially available BBO aperture has reached 20×20 cm 2 .On the other hand, the initial GDDs suitable for the dual-pump scheme is not restricted to our selected parameters.As long as the pump-seed chirp ratio is fixed (pulse delays also adjusted), the relationship between the instantaneous wavelength of the pump and seed pulse is valid, allowing further stretching of the input pulses to tens of picoseconds.In this case, the dual-pump scheme is still capable of amplifying broadband spectrum at degeneracy, meanwhile the available pulse energy is remarkably increased.Nowadays, 200-TW class (e.g. 25 fs, 5 J) Ti:sapphire laser systems are commercially available.With the conversion efficiency of dual-pump OPCPA, an uncompressed signal energy of hundreds of millijoule or even joule-level can be expected.These attributes make it possible to apply the dual-pump scheme to a much more energetic OPCPA system, thus achieving TW-level peak intensity of few-cycle IR pulses.We believe that the versatile dual-pump OPCPA scheme with such simplicity and energy-scalability will contribute to the development of not only the attosecond physics but also the strong-field laser researches.

Fig. 3 .
Fig.3.Comparison between the dual-pump scheme (blue lines) and several single-pump schemes with different chirp conditions: linear pump chirp (green lines), quadratic pump chirp (red lines), and ideal nonlinear pump chirp (black lines).(a) The variation of pump wavelengths with the phase-matched signal wavelengths, (b) the phase-mismatch variation with respect to the signal instantaneous wavelength, (c) the EBP evolution during the parametric amplification in the crystal, and (d) the output signal spectra at EBP-maximized positions.The green dashed line in (d) shows the output spectrum of a single-pump chirpfree OPA system.

Fig. 4 .
Fig. 4. Spectral-temporal characteristics of the output signal pulse.(a).Normalized intensity (blue line) and uncompensated phase (green line) of the amplified spectrum.The output spectra of two separated single-pump OPCPA systems (red dashed lines, all parameters are the same as the dual-pump system) are plotted for comparison.(b) Normalized intensity (blue line) and phase (green line) of the compressed pulse, the intensity profile of the TL pulse (red dashed line) is plotted for comparison.

Fig. 5 .
Fig. 5. Dependence of output signal duration (blue circles) and energy (green triangles) on the GDD of input seed.The red dots are the EBP evolution.

Fig. 6 .
Fig. 6.Dependence of output signal duration (blue circles) and energy (green triangles) on the TL duration of original seed.The TL durations of the pump are (a) 30 fs and (b) 25 fs, respectively.