Dual-chirped optical parametric ampliﬁcation for ...ufolab.phys.hust.edu.cn/__local/F/AC/EC/CC249319315A35CE5196020… · Dual-chirped optical parametric ampliﬁcation for generating

Dual-chirped optical parametricamplification for generating few

hundred mJ infrared pulses

Qingbin Zhang1,2, Eiji J. Takahashi1,∗, Oliver D. Mucke3,Peixiang Lu2, and Katsumi Midorikawa1,2

1 Extreme Photonics Research Group, RIKEN Advanced Science Institute, 2-1 Hirosawa,Wako, Saitama 351-0198, Japan

2 Wuhan National Laboratory for Optoelectronics, Huazhong University of Science andTechnology, Wuhan 430074, China

3 Photonics Institute, Vienna University of Technology, Gusshausstrasse 27-387, A-1040Vienna, Austria∗[email protected]

Abstract: An ultrafast high-power infrared pulse source employing adual-chirped optical parametric amplification (DC-OPA) scheme based ona Ti:sapphire pump laser system is theoretically investigated. By chirpingboth pump and seed pulses in an optimized way, high-energy pump pulsescan be utilized for a DC-OPA process without exceeding the damagethreshold of BBO crystals, and broadband signal and idler pulses at 1.4μm and 1.87 μm can be generated with a total conversion efficiencyapproaching 40%. Furthermore, few-cycle idler pulses with a passivelystabilized carrier-envelope phase (CEP) can be generated by the differencefrequency generation process in a collinear configuration. DC-OPA, aBBO-OPA scheme pumped by a Ti:sapphire laser, is efficient and scalablein output energy of the infrared pulses, which provides us with the designparameters of an ultrafast infrared laser system with an energy up to a fewhundred mJ.

© 2011 Optical Society of America

OCIS codes: (320.7110) Ultrafast nonlinear optics; (190.4970) Parametric oscillators and am-plifiers; (190.2620) Harmonic generation and mixing.

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#140098 - $15.00 USD Received 22 Dec 2010; revised 4 Mar 2011; accepted 9 Mar 2011; published 30 Mar 2011(C) 2011 OSA 11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7190

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1. Introduction

High-order harmonic generation (HHG) [1] represents one of the best methods of producingultrashort fully coherent light covering a wavelength range from the extreme ultraviolet (XUV)region to the soft x-ray region. Since most experiments have employed femtosecond Ti:sapphirelaser (800 nm) technology, this symbiosis has fostered rapid progress in both research fields.Thus, HHG has succeeded in opening the door to research in attosecond science [1] and nonlin-ear optics in the XUV region. The recent years have witnessed the birth of attosecond science, inwhich the shortest pulse duration of isolated attosecond pulses attained is 80 as [2]. Moreover,high-power HHG sources have been utilized for several applications, such as the observationof a two-XUV-photon absorption process [3], holographic diffractive imaging [4], and so forth[5, 6]. HHG contributes not only to the ultrafast community, but also to the accelerator commu-nity when using an HHG source for seeding a free-electron laser (FEL) to improve its temporalcoherence [10].

To further develop other applications of HHG, one of most important issues is the extensionof the wavelength range into the soft- and hard- x-ray region. The maximal harmonic photonenergy Ec is given by the cutoff law Ec = Ip +3.17Up [1] , where Ip is the ionization potentialof the target atom and Up[eV] = 9.38×10−14 I [W/cm2] (λ [μm])2 the ponderomotive energy,with I and λ being the intensity and wavelength of the driving field, respectively. Since Up

scales as λ 2, the laser wavelength is an effective control knob for the ponderomotive energy


and cutoff, and a promising route for generating harmonics of higher photon energy is to usea driving laser of longer wavelength. This has motivated HHG experiments with near-infrared(NIR) to mid-IR pulse sources. Using a 1.55-μm driving field from an optical parametric am-plifier (OPA), for example, Takahashi and coworkers have recently succeeded in generatingharmonics with photon energies of 300 eV from Ne and 450 eV from He gas, which lies wellin the water-window region [11, 12].

Recently, the intriguing prospects of using HHG for the realization of table-top x-ray sourcesin the sub-keV to keV region have stimulated increased research efforts in the development offew-cycle NIR OPA systems [13, 14, 15, 16, 17, 18, 19] and optical parametric chirped pulseamplification (OPCPA) systems [20, 21, 22]. The broad gain bandwidths of OPA and OPCPAhave been widely exploited for few-cycle pulse generation in the NIR region. For CEP-stableIR sources, although pulse durations in the few-cycle regime can be achieved, the output energyof the IR pulses barely reaches the mJ level. To generate a harmonic beam of not only higherphoton energy but also higher photon flux, NIR pulses having a sufficient output energy andultrashort pulse duration are required to examine the energy scaling [23] of HHG under phase-matching conditions. Although few-cycle OPCPA at 0.8 μm with energy over 100 mJ has al-ready been demonstrated [24], such 0.8-μm systems are not in our interested wavelength rangefor long-wavelength HHG. Therefore, the scalability of the pulse energy at long-wavelength IRis of paramount importance for the development of intense high-order harmonic sources.

Generally, the power-scaling potential of OPA/OPCPA is limited by the pump laser energyand the size of the nonlinear crystals. Although periodically poled nonlinear crystals such asLiNbO3 (PPLN) and stoichiometric LiTaO3 (PPSLT) are attractive media for obtaining a broad-band IR pulse with high conversion efficiency, the acceptable pump intensity in the OPA is quitelow owing to the damage threshold of the crystal and its AR coatings. Therefore, OPCPA withPPLN and PPSLT crystals might be suitable for generating ultrashort pulse durations at highrepetition rates (>1 kHz). On the other hand, β -BaB2O4 (BBO) is one of the most outstandingnonlinear optical crystals for obtaining broadband IR pulses, which has unique properties: widetransparency region (0.19 μm - 3.5 μm), wide phase-matching range (0.41 - 3.5 μm), large non-linear coefficient, and high damage threshold. In the OPA scheme with BBO, an output energyexceeding 7 mJ with a pulse width of 40 fs [25] was achieved at a signal wavelength near 1.4μm using a terawatt Ti:sapphire laser system (0.8 μm). Because the OPA scheme in generaldoes not require a pulse compressor, high-power NIR pulses can easily be obtained with highefficiency. By using BBO crystals with 0.8 μm pumping, Brida et al. [17] also demonstratedsub-two-cycle near-IR pulses (8.5 fs) at the degeneracy wavelength (1.6 μm). Further increasein the output power of OPA requires BBO crystals with a large aperture. However, the powerscalability of OPA is limited by the available aperture size of BBO crystals (typically ∼ 20 ×20 mm2) and the intensity of the pump laser owing to the damage threshold of nonlinear crys-tals. To increase the acceptable pump energy in a parametric amplifier, OPCPA has attractedmuch attention as a promising route for the scaling of output power from the visible to theNIR region. Rudd et al. [26] reported a high-power OPCPA system that utilized a 10 Hz, 300ps Nd:YAG pump laser system, a 1.575 μm fiber oscillator and amplifier as seed source, andrubidium titanyl phosphate RTiOPO4 (RTP) and KTiOAsO4 (KTA) crystals. Although the out-put energy was ∼ 30 mJ at a wavelength of 1.55 μm, the pulse duration was 260 fs due to thecollinear geometry in the OPCPA geometry, which is not sufficient for generating the HHG.To obtain a broadband IR pulse, Kraemer et al. [27] proposed and demonstrated a noncollinearOPCPA of 1.56-μm pulses using KTA, in which 100-fs pulses from an erbium fiber laser werestretched to 100 ps and the KTA crystals were synchronously pumped by high-power 100-pspulses from a Nd:YLF regenerative amplifier at 1.053 μm. This work disclosed a promisingroute for power scaling of OPCPA to simultaneously obtain a high peak power and an ultra-


short pulse duration less than 100 fs. Following this pioneering work, Mucke et al. [28, 29]have recently demonstrated the generation of CEP-stable multi-mJ 1.5-μm ∼ 70-fs pulses froma 4-stage KTP/KTA OPCPA based on the fusion of a diode-pumped solid-state (DPSS) fem-tosecond Yb:KGW MOPA system and picosecond Nd:YAG solid-state technology.

Even if the IR output energy reaches the multi-mJ level in this IR-OPCPA with KTP/KTAcrystals [28, 29], OPCPA in general encounters important technical challenges [30], such as therequirement of a specific pump laser, the problem of synchronization with an external pumplaser, and the unwanted generation of parasitic superfluorescence accompanying the primarypulse in broadband high-parametric-gain configurations [31, 32]. On the other hand, a BBOOPA pumped by 800-nm pulses can easily produce a few-cycle IR pulse with a multi-mJ outputenergy. Since the maximum output is limited by the damage threshold of BBO crystals, how-ever, the power scalability for the pump laser is inferior to that of OPCPA. Although Ti:sapphirelaser systems with >100 TW peak power [33] and 10 Hz repetition rate are already available,they can not be applied in an OPA as they stand owing to the damage threshold of the BBO crys-tals. Moreover, the concept of OPCPA does not work if we employ a high-energy Ti:sapphirelaser. If we only stretch the seed pulse in OPCPA, the intensity of the pump pulse from thehigh-energy Ti:sapphire laser is still far beyond the crystal damage threshold. If a Ti:sapphirelaser with sufficient energy can be applied to pump an OPA while preventing damage to BBOcrystals, we will conveniently obtain a high IR energy. In this paper, we propose and investigatein detail a novel OPA method, dual-chirped OPA (DC-OPA), which permits to simultaneouslyobtain high peak power and ultrashort pulse duration in the IR region. DC-OPA allows us tonot only apply the powerful scalability of the IR energy while maintaining ultrashort pulseduration (< 50 fs), but also employ high pump energy in the OPA. Moreover, DC-OPA canalso produce a self-CEP-stabilized IR idler pulse [34]. Previously, Isaienko and Borguet [35]applied a similar method to the analysis of a noncollinear KTP OPA. However, the large-anglenoncollinear interaction reduces the conversion efficiency down to 2%, and the idler pulses areproduced with an unavoidable angular dispersion. Besides, the amplified signal pulse is diffi-cult to compress to the transform limit (TL), even after sufficient compensation of higher-orderdispersion, owing to the divergence and pulse-front tilting of the seed pulse. In DC-OPA, weemploy chirped pump and seed pulses, which interact in BBO crystals in a collinear configu-ration. By controlling the chirping value of the pump and seed pulses around , we achieve arelatively high conversion efficiency and broadband signal and idler pulses without exceedingthe damage threshold of the BBO crystals.

The rest of this paper is organized as follows. In Sec. 2, we explain the concept of the DC-OPA. In Sec. 3, we describe the numerical model and clarify the pump and seed parameters.In Sec. 4, the third-order nonlinear effect and the amplification of quantum noise in the DC-OPA scheme are discussed. In Secs. 5 and 6, we discuss the characteristics of the amplifiedsignal and idler pulses in the DC-OPA scheme. We also show how to achieve a high conversionefficiency and a broad bandwidth for signal and idler pulses in DC-OPA. In Sec. 7, we presenta summary of our conclusions and discuss the prospects for high-power IR pulse sources basedon DC-OPA.

2. Concept of DC-OPA

First, to clarify the concept of DC-OPA, we discuss the conceptual differences between OPA,OPCPA and DC-OPA. In the OPA scheme shown in Fig. 1(a), transform-limited (TL) ultrashortpump and seed pulses are synchronized and interact in the nonlinear crystal, yielding amplifiedultrashort signal and idler pulses. As we have discussed in the introduction, however, the maxi-mum acceptable pump energy is limited by the aperture and damage threshold of the nonlinearcrystal. Naturally, the concept of chirped pulse amplification (CPA) is transferred to OPA in or-


der to solve this problem, forming a new scheme called OPCPA. Besides, OPCPA inherits theadvantages of OPA, such as broad gain bandwidth and low thermal loading [36]. Figure 1(b)shows the schematic of OPCPA. A temporally chirped broadband seed pulse with a markedlyreduced peak intensity makes it possible to use a much higher pump energy, and therefore toobtain a much higher amplified pulse energy than from an OPA. In this scheme, one no longerneeds an ultrashort pump pulse, instead one can resort to the use of a high-energy laser systemwith a comparatively long pulse duration (for example, a picosecond Nd:YAG system) as thepump source. If the seed pulse temporally overlaps well with the pump pulse, good energy ex-traction can be achieved, and it is possible for the amplified signal and idler pulses to reach ahigh power after subsequent recompression. Note that seed and pump pulses are derived fromdifferent laser sources in many OPCPA experiments [26, 37]. Actually, an additional timingcircuitry for synchronizing pump and seed pulses is necessary, because the synchronization be-tween the pump and seed pulses is critical for efficient and stable amplification in OPCPA sys-tems. Alternatively, low-jitter all-optical synchronization between pump and seed pulses can beachieved by injection seeding the pump laser with a part of the seed spectrum. Having in mindthe objective of producing high energy and ultrashort pulses, the laser source itself is prefer-ably powerful and broadband. The table-top Ti:sapphire laser system, which has developed intoa rather mature stage in recent years, might be an attractive candidate. For a Ti:sapphire lasersystem, the DC-OPA scheme as shown in Fig. 1(c) is proposed. The DC-OPA is seeded with achirped, broadband seed pulse and pumped by a stretched, broadband pump pulse. In our envis-aged DC-OPA system, automatically synchronized pump and seed pulses are obtained becausethey come from a common source. The pump pulse can be spatially separated into two pulseson a beam splitter, one strong and the other one weaker. The strong one will be used as pumppulse, and the weaker one will be used to generate the seed pulse via white-light generation,e.g., in a sapphire plate.

To achieve even shorter pulse durations down to a few optical cycles, the noncollinear con-figuration of OPA was proposed [38]. In the noncollinear geometry, the signal group velocityequals the projection of idler group velocity along the signal direction, resulting in the compen-sation of group-velocity mismatch (GVM) [36, 38, 39]. However, the idler pulse generated fromnoncollinear geometry is inherently accompanied by angular dispersion, which makes furtherapplication of the idler pulse difficult. Although we can obtain an ultrashort signal pulse du-ration using a noncollinear configuration, angular dispersion becomes a disadvantage for idlerpulse compression. Therefore, a collinear configuration is better suited for the utilization of theidler pulses.

The principle of optical parametric amplification is quite simple; in the difference frequencygeneration (DFG) process, the instantaneous angular frequency ωm(t), where m denotes thepump (p), signal (s), or idler (i), should satisfy the law of energy conservation, i.e.,

ωi(t) = ωp(t)−ωs(t). (1)

For linearly chirped pump and seed pulses, a fairly obvious way to describe ωp(t) and ωs(t) is

ωp(t) = ωp +βpt, (2)

ωs(t) = ωs +βst, (3)

where ωm is the central angular frequency, and βm = dωm(t)/dt is a well-known linear chirp.Inserting Eq. (2) and Eq. (3) into Eq. (1), one obtains

ωi(t) = (ωp −ωs)+(βp −βs)t. (4)

Equation (4) indicates that the chirp of the idler pulse is determined by the chirps of the pumpand seed pulses.


Fig. 1. Scheme of (a) OPA, (b) OPCPA and (c) DC-OPA.

Both OPCPA and DC-OPA allow us to choose different chirps for optimizing the amplifica-tion of the signal and idler pulses. Figure 2 shows the possible chirp conventions of the pumpand seed pulses in OPCPA, and the values of chirp are represented by the slopes of the lines.Note that the narrowband pump pulse has a long temporal duration, but shows no frequencychirp. The seed pulses with positive and negative chirps shown in Figs. 2(a) and (b) interactwith unchirped pump pulses, producing idler pulses with negative and positive chirps, respec-tively. Compared with the OPCPA scheme, the DC-OPA scheme has more degrees of freedomfor combining pump and seed pulses, since the chirp of the pump pulse is also variable. In Figs.3(a)-(d), we chirped the pump pulses with the same positive chirp, the seed pulses have fourdifferent chirps. The lines represent ωp(t) and ωs(t). In Fig. 3(a), ωp(t) and ωs(t) are parallelto each other. This indicates that the pump and seed pulses have the same chirp, therefore thegenerated ωi(t) would be constant in time, i.e., the idler pulse contains no chirp. In Fig. 3(b),the seed pulse has negative chirp; thus, ωi(t) increases with time, resulting in a positive chirp ofthe idler pulses. In Figs. 3(c) and (d), both seed pulses are positively chirped but with differentvalues. If the chirp of the seed is smaller than that of the pump, the produced idler pulse wouldbe positively chirped, as shown in Fig. 3(c); in the opposite situation [Fig. 3(d)], the idler pulsehas a negative chirp. In Figs. 3(e)-(h), the sign of the chirp for the pump pulse is changed frompositive to negative, and the sign of the chirp of the seed pulse is reversed accordingly; however,the physical properties are identical to Figs. 3(a)-(d). How to choose the chirping combination


shown in Fig. 3 to create broadband signal and idler pulses will be discussed in later sections.

Fig. 2. Time-dependent angular frequencies of pump and seed pulses for OPCPA.

Fig. 3. Time-dependent angular frequencies of pump and seed pulses for DC-OPA.

Apart from the output energy and bandwidth, the CEP of the output pulses is also an im-portant parameter. In the few-cycle regime, the CEP variation strongly affects the waveform,producing many CEP-dependent phenomena [40, 41, 42]. These phenomena can be isolatedon a sub-cycle time scale; thus, CEP control is very important in attosecond metrology. To in-vestigate these CEP-dependent phenomena, a CEP-stabilized laser system is strongly desired.Baltuska et al. [43, 34] have proposed and demonstrated self-stabilization of the CEP using theDFG process. In this scheme, a white light continuum (WLC) generated by the pump pulse isused as the seed pulse, thus the seed pulse (and consequently the signal pulse) inherits the phaseof the pump φ0. In this case, the phases of the pump (φp), signal (φs), and idler (φi) pulses aregiven by

φp = φ0, (5)

φs = φ0 −π/2, (6)

φi =−π/2+φp −φs, (7)

Since the idler pulse arises from the DFG between the pump and signal pulses, the phase φ0,which for a non-CEP-stabilized pump laser fluctuates from shot to shot in the pump and signalpulses, automatically cancels out in the idler pulse according to Eq. (7). In direct analogy, DC-OPA can also produce a self-CEP-stabilized idler pulse.


3. Numerical model

To quantitatively evaluate the conversion efficiency and the bandwidth of the DC-OPA scheme,we carried out numerical calculations based on coupled wave equations describing three-waveinteractions. We assume that the Gaussian-type electric fields of the pump, signal and idlerpulses in the DC-OPA processes are expressed as

Em(x,y,z, t) =12

Am(x,y,z, t)exp[ j(kmz+ωmt)]+ c.c, (8)

where m = p, s and i correspond to the pump, signal and idler pulses, respectively; z and (x,y)are the propagation and transverse coordinates; Am is the complex field amplitude including theGaussian spatial and temporal shapes. The spatial shape is assumed to be exp[−(x2 + y2)/w2

0],where w0 is the waist size. Because the laser beams are loosely focused and the Rayleigh rangeis much greater than the propagation length in the following calculations, we approximate thephase for the spatial component as that of the plane wave. For a chirped pulse, its linear chirpshould be taken into account; thus, the temporal shape is described by

Am(t) = exp(−αmt2)exp( jβmt2), (9)

where αm = 2ln2/D2t (αm > 0), and Dt represents the full width at half maximum (FWHM)

of the temporal intensity profile. For the collinear scheme we get the following coupled waveequations:

∂Ap(x,y,z, t)

∂ z+

3

∑n=1

(− j)n−1

n!k(n)p

∂ nAp(x,y,z, t)

∂ tn +

(∂∂x

+ρp∂∂y

)Ap(x,y,z, t) =

−12

κpAp(x,y,z, t)− jχ(2)ωp

2npcAsAie

−iΔk(t)·z− j3χ(3)ωp

2npc(γpp|Ap|2+γps|As|2+γpi|Ai|2)Ap,(10)

∂As(x,y,z, t)∂ z

+3

∑n=1

(− j)n−1

n!k(n)s

∂ nAs(x,y,z, t)∂ tn +

(∂∂x

+ρs∂∂y

)As(x,y,z, t) =

−12

κsAs(x,y,z, t)− jχ(2)ωs

2nscApA∗

i eiΔk(t)·z − j3χ(3)ωs

2nsc(γss|As|2+γps|Ap|2+γsi|Ai|2)As, (11)

∂Ai(x,y,z, t)∂ z

+3

∑n=1

(− j)n−1

n!k(n)i

∂ nAi(x,y,z, t)∂ tn +

(∂∂x

+ρi∂∂y

)Ai(x,y,z, t) =

−12

κiAi(x,y,z, t)− jχ(2)ωi

2nicApA∗

s eiΔk(t)·z − j3χ(3)ωi

2nic(γii|Ai|2 + γis|As|2 + γip|Ap|2)Ai, (12)

where c is the velocity of light in vacuum; nm (m = p, s and i) denote the refractive indicesof the pump, signal and idler pulses, which can be evaluated from the Sellmeier equations[44]; k(n) is the nth-order dispersion coefficient; the angles ρm account for the possibility ofspatial walk-off; κm are the absorption coefficients; χ(2) and χ(3) are the second- and third-order nonlinear coefficients, respectively; the time-dependent wave-vector mismatch is givenby Δk(t) = kp(t)− ks(t)− ki(t); and γm,n (m, n = p, s and i) are the correction factors for aKerr-effect nonlinear refractive index accounting for self-phase modulation (SPM, m = n) andcross-phase modulation (XPM, m �= n). The γm,n depend on the polarizations of the light; it isequal to 1 if the light beams have the same polarization, otherwise it is 1/3 [45].


To integrate the nonlinear partial differential equations (10)-(12), we employ the split-stepFourier-transform algorithm [46]. The BBO crystal is divided into a large number of small seg-ments, in each of which the propagation is calculated in the spectral domain. The propagationstep is calculated in the reference frame of the pump for convenience. Then the obtained so-lution is back-Fourier-transformed into the time domain, where the nonlinear source terms aretaken into account, and solved with a fourth-order Runge-Kutta method. After the treatments inthe time and space domains, the spatial walk-off is performed in the space-frequency domain.Before the simulation is carried out, all the parameters appearing in the coupled wave equa-tions should be well prepared. Since a full three-dimensional calculation is time-consuming,and a one-dimensional model already reveals the essential physics, we can simplify the cou-pled wave equations by neglecting the effects of the transverse distribution of each of the fields.In the present study, most of the calculations are performed using a simplified one-dimensionalmodel except those for the spatial profile of the signal pulse after propagation.

Before we introduce the laser parameters used in the present study, the relation between linearchirp and group delay dispersion (GDD) should be clarified in advance to avoid confusion. Thelinear chirp is a concept in the time domain, whereas the GDD is introduced in the frequencydomain. We employ the Fourier transform to relate these two quantities. Starting from thefrequency domain, it is often helpful to expand the spectral phase into a Taylor series [47];

φm(ω) = φ(ωm)+φ ′(ωm)(ω −ωm)+12

φ ′′(ωm)(ω −ωm)2 + · · · . (13)

The spectral phase of zero-order term is a phase constant in not only frequency domain but alsothe time domain, since this term has no influence on the Fourier transformation. The coefficientof first-order φ ′ is nothing but a time shift in the time domain according to the Fourier shifttheorem. We mainly concentrate on the second-order dispersion φ ′′ (also termed GDD), whichaccounts for the linear chirp of the laser pulse. Considering only GDD, the laser field with aGaussian envelope in the frequency domain can be described approximately as

Am(ω) = exp(−ηm(ω −ωm)2)exp( j

ζm

2(ω −ωm)

2), (14)

where ηm = 2ln2/D2ω (ηm>0), and Dω denotes the FWHM of the pulse in the frequency do-

main; ζm represents the GDD value. Transforming Am(ω) back into the time domain and drop-ping the term exp( jωmt), we obtain

Am(t) = exp

(− η2

m

4η2m +ζ 2

mt2)

exp

(j

−ζm

8η2m +2ζ 2

mt2). (15)

Comparing Eqs. (9) and (15), αm and βm are given by

αm =η2

m

4η2m +ζ 2

m, (16)

βm =−ζm

8η2m +2ζ 2

m. (17)

Eq. (17) indicates that a positive GDD value ζm corresponds to a negative chirp, and vice versa.It is also found that |βm| does not monotonically depend on |ζm|. The critical point is |ζc|= 2ηm,beyond which |βm| decreases with increasing |ζm|.

In Table 1, we summarize the parameters for the pump and seed pulses that will be used.The original pump and seed pulses are 35-fs Gaussian TL pulses at 0.8 μm and 1.4 μm, and


their corresponding bandwidths are 27 nm and 82.5 nm, respectively. For the DC-OPA scheme,various chirps are introduced to stretch the pump and seed pulses to appropriate pulse dura-tions. The chirp can be introduced by stretchers, such as bulk materials, prism pairs, gratingpairs, or an acousto-optic programmable dispersive filter (AOPDF). As shown in Table 1, westretch the pump and seed pulses with the absolute value of GDD varying from 0 to 10000fs2, producing pulses with durations from 35 to 792 fs. In this situation, the critical point |ζc|equals 442 fs2, which means that a larger |ζm| corresponds to a smaller |βm| when |ζm|> |ζc|.After stretching, the pump and seed pulses interact in a BBO crystal with an aperture diameterof 5 mm. Although the chirped pump pulses have different pulse durations, we keep the peakintensity of each pump pulse at 100 GW/cm2. Obviously, the available pump energy is higherfor the chirped pump pulse owing to the longer pulse duration, which benefits the productionof amplified signal and idler pulses with higher energies. Thus, we can easily scale the energyof the pump and seed pulses to the aperture of the BBO crystals. The parameters of the pumppulse marked by an asterisk∗ are employed to discuss the B-integral in Sec. 4. In the follow-ing calculations, both Type-I (pump extraordinary, signal ordinary, idler ordinary) and Type-II(pump extraordinary, signal extraordinary, idler ordinary) phase-matching configurations areconsidered. The BBO crystals are cut at θ = 20◦ and θ = 27◦ to satisfy the Type-I and Type-IIphase-matching conditions, respectively, in the 1 - 2 μm region.

Table 1. Parameters of pump and seed pulses.

wavelength |GDD| pulse duration intensity energy(μm) (fs2) (fs) (GW/cm2)

pump 0.8 0 - 10000 35 - 792 100 0.685 mJ - 15.5 mJpump∗ 0.8 10000 792 50 7.75 mJseed 1.4 0 - 10000 35 - 792 0.04 - 0.9 6.25 μJ

4. B-integral and superfluorescence buildup in the DC-OPA scheme

During the OPA process, several undesirable effects may limit the choice of parameters such aspump intensity and the length of a nonlinear crystal. We discuss these effects before exploringthe optimized generation conditions for DC-OPA. For high optical intensities, which often oc-cur when ultrashort pulses are amplified, inevitable parasitic nonlinear effects can accumulateduring pulse propagation. Here, a parameter called B-integral is introduced to evaluate theseunwanted nonlinear effects. In analogy to its definition in the OPA scheme [48], the B-integralin the DC-OPA scheme is defined as

B =2πλs

∫ L

0n2(γss|As|2 + γsp|Ap|2 + γsi|Ai|2), (18)

accounting for both SPM and XPM, where the same notations as in Eq. (11) are used. λs is thewavelength of the signal wave, L is the length of the BBO crystal, and n2 = 2.9×10−16 cm2/Wis the Kerr-effect nonlinear refractive index coefficient of BBO [49]. Note that |Ap|2, |As|2,and |Ai|2 should be normalized to yield intensity. For large B-integrals, the Kerr effect causesa time-dependent phase shift in accordance with the time-dependent pulse intensity, and thenonlinear lensing effect can become sufficiently strong to collapse the beam to a very smallradius at which the optical intensities are markedly increased and easily exceed the damagethreshold. Good temporal and spatial beam quality of the amplified pulses are of paramountimportance for the successful recompression of the amplified stretched pulses down to the TLpulse duration.


0 5 100.0

0.5

1.0

1.5

2.0

2.5

3.0

Crystal Length (mm)B

−in

tegr

al

100 GW/cm2

50 GW/cm2

Fig. 4. Operational limits for pump intensity and BBO crystal length in a DC-OPA asdetermined by the nonlinear B-integral limit (dashed line) for BBO Type-I phase matching.Pulse parameters as in Table 1.

For further detailed investigation, a numerical simulation is performed to estimate a con-crete value of the B-integral for Type-I phase matching. The B-integral is calculated from theposition-dependent values of the pump, signal and idler intensities, which are obtained froma solution of the coupled wave equations. In this calculation, pump pulses with intensities of50 GW/cm2 (indicated by pump∗ in Table 1) and 100 GW/cm2 (indicated by pump in Table 1)are used in the DC-OPA scheme, and the intensity of the input seed pulse is 0.1 GW/cm2. Thecalculated B-integral is plotted in Fig. 4. It is shown that the B-integral is larger for a higherpump intensity and a thicker BBO crystal. As pointed out by Ross et al. [50, 51], the B-integralis required to be less than 1; therefore, the thickness of the BBO crystal is limited to 8 mmand 4 mm for pump intensities of 50 GW/cm2 and 100 GW/cm2, respectively. Simulations forType-II phase matching yield a result similar to that for Type-I phase matching.

Besides the possible detrimental effects caused by the Kerr nonlinearity, we should also con-sider the superfluorescence background, which originates from the parametric amplification ofthe vacuum or quantum noise in OPA [52]. The amplification of vacuum noise, also known asoptical parametric generation (OPG), will occur at those wavelengths at which the parametricinteraction is phase-matched, i.e., where spontaneous conversion of pump photons into a signaland idler photon pair takes place [52, 53]. The OPG process is very useful for many applica-tions, such as in a high-repetition-rate femtosecond laser system [54, 55]; however, we musttake the possible drawbacks brought by the OPG for our DC-OPA scheme into account. First,the amplified noise may decrease the pulse contrast ratio [31]. Second, the CEP stabilization ofthe idler pulse may be destroyed by the OPG process, because the CEP relationship betweenthe pump and OPG pulses is completely random [56].

Table 2. Types of pump and seed pulses

pump seedC-OPG chirped pulse (noise)OPG TL pulse (noise)DC-OPA chirped pulse chirped pulseOPA TL pulse TL pulse

We evaluate the energy evolution of the signal pulse during the C-OPG, OPG, DC-OPAand OPA processes as a function of BBO. The newly emerging C-OPG is an abbreviation for


0 5 10pJ

nJ

mJ

J

Crystal Length (mm)

log 10

[Ene

rgy]

0 5 10pJ

nJ

mJ

J

Crystal Length (mm)

log 10

[Ene

rgy]

C−OPG

OPG

DC−OPA

OPA

(a) Type−I (b) Type−II

JμJμ

Fig. 5. Energy evolution during C-OPG, OPG, DC-OPA and OPA processes inside a BBOcrystal. TL pump: 35 fs, 685 μJ (intensity of 100 GW/cm2). Chirped pump: 792 fs (GDDvalue of -10000 fs2), 15.5 mJ (intensity of 100 GW/cm2). TL seed: 35 fs, 6.25 μJ (intensityof 0.9 GW/cm2). Chirped seed: 792 fs (GDD value of 10000 fs2), 6.25 μJ (intensity of 0.04GW/cm2). Initial noise: 1 pJ. (a) Type-I and (b) Type-II phase matching.

chirped optical parametric generation. We summarize the types of pump and seed pulses forthe four parametric processes in Table 2. The C-OPG/OPG process is initiated by a randomnoise field with an energy of approximately 1 pJ, while a seed pulse with an energy of 6.25μJ is used for the DC-OPA/OPA process. In Fig. 5, we observe that the saturation of signalenergies is slightly faster for Type-I phase matching compared to the Type-II case for all of thefour processes, since the nonlinear coefficient is higher for Type-I phase matching in the BBOcrystal. It is also found that the C-OPG process reaches saturation more rapidly than the OPGprocess, which means that a short pump pulse favors the suppression of the superfluorescence.This is because noise can only be amplified within the time window defined by the pump pulse.In order to maintain a high signal-to-noise contrast ratio and ensure CEP stabilization of theidler pulse, the output energy of the signal generated by C-OPG should be much lower thanthat of the DC-OPA. It is evident from Fig. 5 that the signal energy of C-OPG is rather lowwhen the DC-OPA process reaches saturation. The signal energy amplified by DC-OPA is stillmore than 3 orders of magnitude higher than that amplified by C-OPG for propagation lengthsof 3 mm under Type-I, and 5 mm under Type-II phase matching. Thus, we can suppress theamplification of noise by imposing proper restrictions on the propagation length. In a practicalOPA/DC-OPA process, the amplification of noise and seed is inseparably entwined, so that thesignal energies coming from C-OPG and DC-OPA are indistinguishable. Possible means ofestimating the energy amplified from noise is to measure the signal output energy in absence ofthe seed [31] or to employ a spectral-hole pulse-shaping technique using an AOPDF [21].

5. Analysis of the signal pulse

To produce a high-power IR source with a pulse duration of a few tens fs, it is necessaryto investigate the simultaneous optimization of conversion efficiency and signal bandwidth.Several previous works have addressed this issue for the OPCPA scheme, and it is found thatthere is a trade-off between the conversion efficiency and bandwidth of the amplified signalpulse determined by the chirp of the seed pulse [50, 57, 32].

Here we study the conversion efficiency and bandwidth for the DC-OPA scheme. In Fig. 6,we perform a time-frequency analysis of the seed pulse, shown in combination with a pump gainprofile to illustrate the gain-narrowing effect. The chirp situation corresponds to that shown inFig. 3(b). During the parametric process, only the central part of the pump strongly couples withthe signal and idler; therefore, the temporal gain profile (white dotted-dash line) is narrowerthan the profile of the pump pulse (yellow dashed line). Considering the temporally varying


wave vector mismatch induced by the chirp, there would be further gain reduction in the wingsof the stretched seed pulse [32]. As expected, the gain profile for the TL seed shown in Fig.6(a) is wider than that for the chirped seed pulse shown in Figs. 6(b) and (c). In Fig. 6(a),the TL seed pulse is sufficiently short that it is fully contained in the gain profile; thus, thewhole phase-matching bandwidth can be preserved. In contrast, the wings of the contour plotfor the chirped seed pulse in Fig. 6(c) immensely exceed the gain profile of the pump pulse,and a temporal narrowing occurs due to the fact that the signal can only be amplified efficientlywithin the gain profile. The gain-induced narrowing will shape the signal pulse to a form withsteeper leading and trailing edges. We keep in mind that the GDD of the shaped pulse is stillunchanged. If we consider two stretched pulses having different durations but the same GDDvalue, the Fourier principle tells us that the shorter (longer) stretched pulse corresponds to anunstretched TL pulse with longer (shorter) pulse duration and narrower (broader) bandwidth.Therefore, the temporal narrowing will reduce the output bandwidth.

Figure 7 shows a comparison of the conversion efficiencies and bandwidths for the signalpulse for Type-I and Type-II phase matching, respectively. Basically, the conversion efficiencyis improved as the added GDD value increases. The conversion efficiency is mainly determinedby the temporally overlapping area of the pump and seed pulses. For very short seed pulses,only a small fraction of the pump-pulse energy is depleted; in contrast, a large amount of energycan be transferred from the pump pulse to the seed pulse when the pulse duration of the seedpulse is sufficiently long. We now address the bandwidth of the amplified signal pulse. Theresulting bandwidth of the signal exhibits a different trend from the conversion efficiency. Thesignal bandwidth firstly increases, then saturates, and finally decreases with increasing GDDvalue of the seed pulse. This result seems to contradict our earlier findings: a short seed pulseis favorable for preserving the input bandwidth. However, this puzzle can be resolved by takingthe group-velocity mismatch (GVM) between the interacting pulses into account. The GVMfor parametric interactions is defined as

δsp = 1/vgs −1/vgp, (19)

δip = 1/vgi −1/vgp, (20)

δsi = 1/vgs −1/vgi, (21)

where vgp, vgs, and vgi are the group velocities of the pump, signal and idler pulses. As pointedout by Cerullo et al. [36], δsp and δip limit the interaction length over which parametric am-

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plification takes place. Because of the relative movement between the pump and signal pulses,the trailing edge of the moving signal pulse will be amplified leading to a modest temporalbroadening and asymmetry of the temporal profile coming along with a gradual reduction inbandwidth. With the parameters for Fig. 7, δsp is estimated to be about 7 fs/mm for Type-I phasematching. If we employ a 35-fs TL pulse as the seed pulse, the relative delay between pumpand signal pulses is about 20% of the input seed pulse duration after propagation through 1 mmof BBO. However, the relative shift is less than 3% of the pulse duration if we input a 318-fsseed pulse. Hence, GVM can be neglected for limited propagation length if we use a seed pulsewith sufficiently long pulse duration. Note that the output bandwidth of a signal pulse is nar-rower for Type-II phase matching, especially for a short seed pulse. E.g., employing a TL seedpulse, the bandwidth of the output signal is 76 nm for Type-I phase matching (Fig. 7(a)), whileit is only 48 nm for Type-II phase matching (Fig. 7(b)). We believe that this is because of thedifferent acceptable bandwidths for the two types of phase matching. As previously reportedin [36], the acceptable bandwidth increases as the absolute value of GVM between the signaland idler pulses decreases. The calculated |δsi| for Type-I phase matching is 8.5 fs, which ismuch smaller than that of 72 fs for Type-II phase matching. From the above discussion, we canconclude that the bandwidth of the output signal is mainly determined by the GVM for shortTL seed pulses, and the gain-narrowing effect plays a most important role for long chirped seedpulses. As can be seen from Fig. 7, it is suitable to choose the seed pulse GDD value withinthe range from 3000 fs2 to 5000 fs2 for both Type-I and Type-II phase matching, in order tosimultaneously achieve a high conversion efficiency and a broad bandwidth of the amplifiedsignal pulse. In other words, the seed pulse duration is preferentially 30% to 50% of the pumppulse duration.

Figure 8 shows the output signal conversion efficiency as a function of BBO thickness, wherethe pumping intensity is fixed to be 100 GW/cm2. The GDD values of the pump and seed pulsesare -10000 fs2 and 4000 fs2, respectively. These design parameters are chosen for the simulta-neous optimization of the conversion efficiency and bandwidth, according to the result shownin Fig. 7. The conversion efficiency increases with the increasing thickness of the BBO crys-tal until it reaches the gain saturation position, after which back-conversion occurs. Note thatthe conversion efficiency reaches its maximum faster for Type-I phase matching owing to thehigher nonlinear coefficient. The back-conversion to some extent may broaden the bandwidth ofthe signal pulse. This can be understood by the fact that the gain is different for different wave-lengths, resulting in spectral shaping. Similar results were found in previous works [58, 59].


However, the spectrum may suffer strong modulation if the propagation length is far beyondthe saturation position, leading to multiple satellite pulses in the temporal domain. Therefore,one should carefully choose the propagation length in the BBO crystal.

To recompress the amplified signal pulse, it is very important to determine the spectral in-tensity and phase. For Type-I phase matching as shown in Fig. 9(a), the amplified signal pulseat 1.4 μm exhibits a smooth distribution with a bandwidth of 79 nm, which is just slightlynarrower than the injected spectra of the seed pulse. The spectral phase shows a parabolic pro-file originating from the added GDD and accumulated phase in the BBO crystal. The spatialprofile of the produced signal pulse is also calculated with our three-dimensional model, in or-der to check the beam quality. Figure 9(b) shows the temporal profile of the amplified signalpulse after phase compensation. The compensation is realized by adding the inverse GDD ofthe seed pulse to that of the amplified signal pulse. A signal pulse with a pulse duration of 40fs is obtained, which approaches the TL of 36 fs supported by the 79 nm bandwidth. We alsoshow the temporal phase that was compensated for without considering the rapidly changingphase induced by the carrier frequency. The temporal phase profile of the signal pulse is veryflat, which provides evidence for the good compensation of the second-order spectral phase.For comparison, the spectral and temporal properties of the generated signal pulse for Type-IIphase matching are shown in Fig. 10. They are similar to the Type-I case in Fig. 9. One detailthat does differ between Figs. 9 and 10, however, is the bandwidth of the output signal pulse,which is narrower for Type-II phase matching. When compensating for the spectral phase ofthe produced signal pulse with a bandwidth of 65 nm, we obtain a 53-fs signal pulse.

We have already discussed in detail the situation in which the pump and signal pulses arechirped by GDD values of -10000 fs2 and 4000 fs2, which corresponds to the situation shownin Fig. 3(b). Note that according to Eq. (17) a negative GDD value induces a positive chirpfor a pulse with a standard Gaussian temporal envelope, and vice versa. Actually, several otherchoices for chirping the pump and seed pulses to achieve a relatively high conversion efficiencyand a broad bandwidth are available as well, as long as the ratio of the seed pulse duration tothe pump pulse duration are maintained at a proper value. We employ the schemes shown inFigs. 3 (d), (f), and (h) to perform an optimization of the bandwidth and conversion efficiency.As expected, we obtain a very similar result to the scheme shown in Fig. 3(b), only a very slightdifference in the maximum bandwidth of the output signal pulse is found. The situations shownin Figs. 3 (a), (c), (e) and (g) cannot be applied for DC-OPA, because the duration of the seedpulse is comparable to or longer than that of the pump pulse. As we have explained earlier, aserious gain narrowing would occur in these situations.

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Fig. 9. (a) Spectrum of the injected seed pulse (yellow solid line), and spectral intensityof the amplified signal pulse (blue solid line) together with its phase (green solid line)before phase compensation. Inset: Calculated spatial profile for the obtained signal pulse.(b) Temporal intensity (red solid line) and phase (green solid line) profiles of the amplifiedsignal pulse after phase compensation. The curves are calculated for the saturation pointshown in Fig. 8, for the case of Type-I phase matching. The GDD values for the pump andseed pulses are -10000 fs2 and 4000 fs2, respectively.

Before we continue, we want to mention a few pioneering works on chirp-compensationschemes [50, 60, 61, 62]. These OPA systems are designed to choose pump and seed chirpssuch that at each temporal coordinate the combination of pump and seed wavelengths are phasematched, resulting in a bandwidth enhancement of the output signal. This idea has been suc-cessfully demonstrated at pump wavelength of 0.4 μm and seed wavelength of 0.7 - 0.8 μm.Although our proposed system also chirped pump and seed pulses, it is operated at 0.8 μmpump and 1.4 μm seed. Around our interested wavelength, the phase-matching curve is flat-ter and we do not employ the ultra-broadband pump and seed (TL pulse duration above 30fs), therefore the chirp-compensation mechanism does not work in our system. In the previ-ous paragraph, we demonstrate that the signal output bandwidth almost keeps unchanged evenfor four different chirp combinations of pump and seed pulses, this result strongly supportsour judgement. However, if we consider ultrabroadband input pump and seed pulses (TL pulseduration below 10 fs), the bandwidth narrowing induced by mismatching of pump and seedgradually plays a role, in this case we should take the chirp-compensation scheme into ac-count for optimization of the output bandwidth. Moreover, the experimental works which usechirp-compensation scheme [60, 61] only report on few-μJ systems, and they do not show thepotential benefit of chirp-compensation for minimizing limitations due to B-integral. Since our

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Fig. 10. (a) Spectrum of the injected seed (yellow solid line), and spectral intensity (bluesolid line) and phase (green solid line) of the amplified signal before phase compensation,using the same parameters as in Fig. 9, but for Type-II phase matching. Inset: Calculatedspatial profile for the obtained signal pulse. (b) Temporal intensity (red solid line) and phase(green solid line) profiles of the amplified signal pulse after phase compensation.


target output energy is hundred mJ, the reduction of B-integral is the key aspect in our system.We also discuss the shortest signal pulse duration realizable in DC-OPA. Here, we apply the

DC-OPA scheme with a targeted gain of 103 to pump and seed pulses with 10-fs TL. Accord-ing to the above discussion, the pump and seed pulses are chirped to optimize the conversionefficiency and bandwidth. The bandwidth of the injected seed pulse is 288 nm; however, thebandwidth of the output signal pulse decreases to 210 nm. We attribute this narrowing of band-width to several factors: First, we can reduce the gain-narrowing effect, but not completelyavoid it. Second, the inherent existence of an acceptable bandwidth for the BBO crystal it-self prevents us from using a seed pulse with a broad bandwidth. Third, we do not optimizethe chirp ratio for the ultrabroadband pump and seed pulses according to chirp-compensationscheme. In principle, the 210 nm bandwidth corresponds to a ∼14-fs TL Gaussian-type pulse;however, we only obtain a 17.5-fs pulse even after compensation of the spectral phase to allorders. This is because the amplified temporal intensity profile of the pulse deviates from astandard Gaussian type, consequently resulting in a reduction of the compression factor. In thisevaluation, the collinear DC-OPA enables us to generate a few-cycle IR pulse with the shortestduration of ∼17 fs at the signal wavelength. Certainly, if the amplifier is operated at the degen-eracy wavelength (pump: 0.8 μm, signal/idler: 1.6 μm), it is helpful to improve the acceptablebandwidth to support a shorter pulse [17]. Furthermore, if we slightly detune the BBO crystalsimilar to [17], two different wavelengths would be simultaneously phase matched, generatingoutput pulses of broader bandwidth. In this case, the shortest achievable pulse centered at 1.6μm is down to 13 fs.

6. Analysis of the idler pulse

The possibility of producing pulses with passively stabilized CEP attracts us to also study theidler pulses. Here, we employ a collinear DC-OPA scheme to avoid angular dispersion of theidler pulses. For the utilization of the idler pulses, bandwidth and conversion efficiency shouldbe further investigated. During DC-OPA, idler pulses are generated as soon as pump and seedpulses are injected into the BBO crystal, and then they experience energy gain and bandwidthmodulation. Intuitively, we must pay more attention to the initial bandwidth of the idler pulses,because an initial idler pulse with a sufficiently broad bandwidth is the prerequisite for obtainingbroadband idler pulses after propagation in the BBO crystal.

Figure 11 shows the initially generated bandwidth of the idler pulse by chirping the pump andseed pulses with different GDD values. The idler pulse centered at a wavelength of 1.87 μm isgenerated by a DFG process from 0.8-μm pump and 1.4-μm seed pulses. For comparison, theideal case of unchirped pump and seed pulses is also shown. The GDD value of the pump pulseis fixed at -10000 fs2, and it is found that the bandwidth of the idler pulse gradually increases toa value close to that in the ideal case when the GDD value of the seed pulse varies from -10000fs2 to 4000 fs2. To explain this finding, we recall the second-order nonlinear interaction term,

Ape jωpt(Asejωst)∗ = e−(αp−αs)t2e[ j(ωp−ωs)t+ j(βp−βs)t2], (22)

where the first exponential term on the right-hand side of the equation stands for the envelope,and the second exponential term represents frequency conversion during DFG, (ωi(t) =ωp(t)−ωs(t)). Using Eq. (4), it is found that the linear chirp for the idler pulse vanishes, when the pumpand seed pulse are chirped with the same GDD value. However, the envelope of the idler pulseis a product of the stretched envelopes of the pump and seed pulses, which can be describedby e−(αp−αs)t2 according to Eq. (22). Therefore the initially generated idler pulse still has longpulse duration, which is independent of the chirp signs of the pump and seed pulses. As is well-known, long pulses without chirp induce a narrow bandwidth. So we expect to obtain an outputidler pulse with narrow bandwidth using pump and seed pulses with the same chirp value. On


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the other hand, if the chirps for the pump and seed pulses have opposite signs, the absolutevalue of linear chirp for the idler pulse will increase corresponding to a smaller absolute valueof GDD. It is known that a smaller absolute value of GDD is needed when we stretch a pulsewith broader original bandwidth to a certain pulse duration. Thus we expect to produce anidler pulse with relatively broad bandwidth using pump and seed pulses with opposite signs ofchirp. Then we return to the OPCPA scheme, considering pump with GDD of 0 fs2 interactingwith a chirped seed pulse which results in a narrower bandwidth than the output bandwidth atoptimized chirp combination for DC-OPA.

We choose seed GDD values of -10000 fs2, -4000 fs2 and 4000 fs2 for the conditions of DC-OPA, and then check the spectral properties of the output idler pulse. The spectral phase shownin Fig. 12(a) is attractive, since it is almost flat, which means that we can obtain a TL pulse with-out compensation. Unfortunately, the corresponding bandwidth is very narrow as expected. InFig. 12(b), the bandwidth of the idler pulse increases because the long idler pulse still containschirp coming from the inequality of the pump and seed chirps. The bandwidth shown in Fig.12(c) is further increased to 192 nm by changing the sign of the added GDD value of the seedpulse used in Fig. 12(b). In Figs. 12(a), (b) and (c), we obtain spectral phases of the idler pulseswith zero, negative, and positive chirps, which agrees well with our prediction using Eq. (4).By compensating for the spectral phase shown in Fig. 12(c), an idler pulse with a duration of 31fs is obtained (see Fig. 12(d)). Moreover, comparing the spectral phases shown as green solidand green dashed lines in Fig. 12(c), it is found that the GDD sign of the idler pulse changeswhen we invert the GDD sign of the input pump and seed pulses. This feature is useful for con-ventionally compensating for the GDD of the idler pulse. If we use transmission compressorssuch as bulk materials and prism pairs, the powerful idler pulse in the DC-OPA scheme maysuffer degradation in beam quality due to high-order nonlinear effects. An appropriate choiceis to use reflective compressors such as grating pairs. As is well-known, parallel grating pairscan only provide negative GDD. Therefore, a high-power idler pulse with a positive GDD (seegreen dashed line in Fig. 12(c)) can conveniently be recompressed by a grating pair.

Finally, we analyze the relation between the conversion efficiency and bandwidth of theidler pulse, the result of which is shown in Fig. 13 for Type-I and Type-II phase matchings.By increasing the GDD value of the seed pulse for better overlap between the pump and seedpulses, the conversion efficiency of the idler pulse is improved. We need to pay attention to two


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aspects for achieving broad bandwidth of the output idler pulse, which are different from thesignal case. On the one hand, the signs of the linear chirps of the pump and signal pulses shouldbe opposite so that we can obtain a broadband idler pulse at the beginning of DC-OPA; on theother hand, we must choose proper stretching parameters to avoid gain narrowing during theamplification of the idler pulse. We found an optimal range of GDD from 3500 fs2 to 5500 fs2

for the idler pulse, within which a relative high conversion efficiency and a broad bandwidth ofthe idler pulse can be achieved simultaneously. This means, the optimized pulse duration of theseed pulse is 35% to 55% of the pump pulse.

By using the DFG process, we can generate a self-CEP-stabilized idler pulse with a highpeak power. In recent works [34, 63, 64], a two-stage OPA scheme was used to generate aCEP-stabilized IR pulse. In this scheme, an idler pulse with a self-stabilized CEP obtainedfrom the first stage is used as the seed pulse for the second stage; consequently, the amplifiedsecond-stage signal pulse is also CEP-stabilized. Of course, we can utilize this scheme in DC-OPA. Furthermore, we chirped the pump and seed pulses to apply a higher input energy withoutdamaging the BBO crystal, so that the amplified signal and produced idler pulses will be morepowerful.

The shortest idler pulse we can obtain is also limited by the gain-narrowing effect and theacceptable bandwidth of the BBO crystal. In addition, we must take care of the absorptionwavelength of the BBO crystal located at about 2.1 μm, beyond which the transparency curvedecreases rapidly. Therefore, the long-wavelength part of a broadband idler pulse with a centralwavelength at 1.87 μm might fall into this absorptive region. When we apply pump and seedpulses of 10-fs TL, an idler pulse with a duration of 14 fs can be produced for optimizedconditions.

7. Summary

We proposed a novel OPA scheme called DC-OPA for producing high-power IR pulses witha few-cycle pulse duration. In this scheme, a pump pulse at 0.8 μm and a NIR seed pulseare generated employing a common Ti:sapphire laser system; thus, low-timing-jitter all-opticalsynchronization between the pump and seed pulses can easily be realized without the needof costly synchronization electronics. By introducing chirps to both pump and seed pulses, asufficiently high-energy pump pulse can be applied in an OPA. The possible detrimental effectsoriginating from the Kerr nonlinearity and parametric superfluorescence were also consideredand place a limit on the peak intensity of the pump pulse and the propagation length in the BBOcrystal for a targeted energy gain of 103. For an optimized GDD combination between the pumpand seed pulses, the conversion efficiency attained is > 40% with broadband signal and idlerspectra in the one-dimensional calculation. For a fully three-dimensional model, the calculatedtotal conversion efficiency is lower than for a one-dimension model, and a conversion efficiencyof 31% - 36% within the optimized GDD range can be achieved. For example, in Type-I phasematching, a 40-fs signal pulse was obtained after chirp optimization, and a 31-fs idler pulse withpassively stabilized CEP can be achieved under the additional requirement of opposite signsfor pump and seed chirps. A similar result was obtained for Type-II phase matching except forthe narrower acceptable bandwidth. If we utilize an even broader bandwidth pump/seed pulsesupporting 10-fs TL duration, DC-OPA in collinear configuration guarantees 18 fs for the signalpulse and 14 fs for the idler pulse as the shortest pulse durations.

DC-OPA, a novel BBO-OPA scheme pumped by a Ti:sapphire laser, is efficient and scalablein output energy of the IR pulses, which provides us with the design parameters of an ultrafastIR pulse source with an energy of a few hundred mJ. Meanwhile, 10-TW class (e.g., 40 fs, 0.4J) Ti:sapphire laser systems are commercially available. From the conversion efficiency of DC-OPA, if we apply all the laser energy (0.4 J) for DC-OPA under optimized GDD conditions, we


can expect to generate a signal energy of more than 80 mJ before recompression. In addition,we may also obtain a 70 mJ self-CEP-stabilized idler pulse. Both signal and idler pulses havefew-cycle pulse durations. Of course, the stretched pump and seed pulses ensure that the pump,signal, and idler intensities are below the damage threshold of the BBO crystal. Although ourestimation ignores optical losses from the compressor, we can still obtain more than sufficientTW IR power.

We believe that DC-OPA has great potential to markedly increase the IR pulse energy, whichwill pave the way for the generation and application of not only intense ultrafast coherent waterwindow x-rays but also high-intensity laser physics [51]. Especially, an intense water windowx-ray source can open the door to demonstrate direct seeding [10] of a FEL in the water window[7].

Acknowledgment

Q.Z. is grateful for the support of the International Program Associate (IPA) program ofRIKEN. O.D.M. acknowledges support from a Lise-Meitner Fellowship from the Austrian Sci-ence Fund (FWF), project M1094-N14.


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