Dual-chirped optical parametric amplification for ... - OSA Publishing

Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses 3, ¨ Qingbin Zhang1,2 , Eiji J. Takahashi1,∗ , Oliver D. Mucke 2 1,2 Peixiang Lu , and Katsumi Midorikawa 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 and Technology, Wuhan 430074, China 3 Photonics Institute, Vienna University of Technology, Gusshausstrasse 27-387, A-1040 Vienna, Austria ∗ [email protected]

Abstract: An ultrafast high-power infrared pulse source employing a dual-chirped optical parametric amplification (DC-OPA) scheme based on a Ti:sapphire pump laser system is theoretically investigated. By chirping both pump and seed pulses in an optimized way, high-energy pump pulses can be utilized for a DC-OPA process without exceeding the damage threshold of BBO crystals, and broadband signal and idler pulses at 1.4 μ m and 1.87 μ m can be generated with a total conversion efficiency approaching 40%. Furthermore, few-cycle idler pulses with a passively stabilized carrier-envelope phase (CEP) can be generated by the difference frequency generation process in a collinear configuration. DC-OPA, a BBO-OPA scheme pumped by a Ti:sapphire laser, is efficient and scalable in output energy of the infrared pulses, which provides us with the design parameters of an ultrafast infrared laser system with an energy up to a few hundred mJ. © 2011 Optical Society of America OCIS codes: (320.7110) Ultrafast nonlinear optics; (190.4970) Parametric oscillators and amplifiers; (190.2620) Harmonic generation and mixing.

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Received 22 Dec 2010; revised 4 Mar 2011; accepted 9 Mar 2011; published 30 Mar 2011

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 producing ultrashort 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:sapphire laser (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 nonlinear optics in the XUV region. The recent years have witnessed the birth of attosecond science, in which 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 observation of 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 community when using an HHG source for seeding a free-electron laser (FEL) to improve its temporal coherence [10]. To further develop other applications of HHG, one of most important issues is the extension of the wavelength range into the soft- and hard- x-ray region. The maximal harmonic photon energy Ec is given by the cutoff law Ec = I p + 3.17Up [1] , where Ip is the ionization potential of the target atom and Up [eV] = 9.38 × 10−14 I [W/cm2 ] (λ [μ m])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 #140098 - $15.00 USD

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and cutoff, and a promising route for generating harmonics of higher photon energy is to use a driving laser of longer wavelength. This has motivated HHG experiments with near-infrared (NIR) to mid-IR pulse sources. Using a 1.55-μ m driving field from an optical parametric amplifier (OPA), for example, Takahashi and coworkers have recently succeeded in generating harmonics with photon energies of 300 eV from Ne and 450 eV from He gas, which lies well in the water-window region [11, 12]. Recently, the intriguing prospects of using HHG for the realization of table-top x-ray sources in the sub-keV to keV region have stimulated increased research efforts in the development of few-cycle NIR OPA systems [13, 14, 15, 16, 17, 18, 19] and optical parametric chirped pulse amplification (OPCPA) systems [20, 21, 22]. The broad gain bandwidths of OPA and OPCPA have been widely exploited for few-cycle pulse generation in the NIR region. For CEP-stable IR sources, although pulse durations in the few-cycle regime can be achieved, the output energy of the IR pulses barely reaches the mJ level. To generate a harmonic beam of not only higher photon energy but also higher photon flux, NIR pulses having a sufficient output energy and ultrashort pulse duration are required to examine the energy scaling [23] of HHG under phasematching conditions. Although few-cycle OPCPA at 0.8 μ m with energy over 100 mJ has already been demonstrated [24], such 0.8-μ m systems are not in our interested wavelength range for long-wavelength HHG. Therefore, the scalability of the pulse energy at long-wavelength IR is 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 energy and the size of the nonlinear crystals. Although periodically poled nonlinear crystals such as LiNbO3 (PPLN) and stoichiometric LiTaO3 (PPSLT) are attractive media for obtaining a broadband IR pulse with high conversion efficiency, the acceptable pump intensity in the OPA is quite low owing to the damage threshold of the crystal and its AR coatings. Therefore, OPCPA with PPLN and PPSLT crystals might be suitable for generating ultrashort pulse durations at high repetition rates (>1 kHz). On the other hand, β -BaB2 O4 (BBO) is one of the most outstanding nonlinear optical crystals for obtaining broadband IR pulses, which has unique properties: wide transparency region (0.19 μ m - 3.5 μ m), wide phase-matching range (0.41 - 3.5 μ m), large nonlinear coefficient, and high damage threshold. In the OPA scheme with BBO, an output energy exceeding 7 mJ with a pulse width of 40 fs [25] was achieved at a signal wavelength near 1.4 μ m using a terawatt Ti:sapphire laser system (0.8 μ m). Because the OPA scheme in general does not require a pulse compressor, high-power NIR pulses can easily be obtained with high efficiency. By using BBO crystals with 0.8 μ m pumping, Brida et al. [17] also demonstrated sub-two-cycle near-IR pulses (8.5 fs) at the degeneracy wavelength (1.6 μ m). Further increase in the output power of OPA requires BBO crystals with a large aperture. However, the power scalability 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 crystals. To increase the acceptable pump energy in a parametric amplifier, OPCPA has attracted much attention as a promising route for the scaling of output power from the visible to the NIR region. Rudd et al. [26] reported a high-power OPCPA system that utilized a 10 Hz, 300 ps Nd:YAG pump laser system, a 1.575 μ m fiber oscillator and amplifier as seed source, and rubidium titanyl phosphate RTiOPO4 (RTP) and KTiOAsO4 (KTA) crystals. Although the output energy was ∼ 30 mJ at a wavelength of 1.55 μ m, the pulse duration was 260 fs due to the collinear 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 noncollinear OPCPA of 1.56-μ m pulses using KTA, in which 100-fs pulses from an erbium fiber laser were stretched to 100 ps and the KTA crystals were synchronously pumped by high-power 100-ps pulses from a Nd:YLF regenerative amplifier at 1.053 μ m. This work disclosed a promising route for power scaling of OPCPA to simultaneously obtain a high peak power and an ultra-

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short pulse duration less than 100 fs. Following this pioneering work, Mücke et al. [28, 29] have recently demonstrated the generation of CEP-stable multi-mJ 1.5-μ m ∼ 70-fs pulses from a 4-stage KTP/KTA OPCPA based on the fusion of a diode-pumped solid-state (DPSS) femtosecond 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/KTA crystals [28, 29], OPCPA in general encounters important technical challenges [30], such as the requirement of a specific pump laser, the problem of synchronization with an external pump laser, and the unwanted generation of parasitic superfluorescence accompanying the primary pulse in broadband high-parametric-gain configurations [31, 32]. On the other hand, a BBO OPA pumped by 800-nm pulses can easily produce a few-cycle IR pulse with a multi-mJ output energy. Since the maximum output is limited by the damage threshold of BBO crystals, however, the power scalability for the pump laser is inferior to that of OPCPA. Although Ti:sapphire laser 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 crystals. Moreover, the concept of OPCPA does not work if we employ a high-energy Ti:sapphire laser. If we only stretch the seed pulse in OPCPA, the intensity of the pump pulse from the high-energy Ti:sapphire laser is still far beyond the crystal damage threshold. If a Ti:sapphire laser with sufficient energy can be applied to pump an OPA while preventing damage to BBO crystals, we will conveniently obtain a high IR energy. In this paper, we propose and investigate in detail a novel OPA method, dual-chirped OPA (DC-OPA), which permits to simultaneously obtain high peak power and ultrashort pulse duration in the IR region. DC-OPA allows us to not only apply the powerful scalability of the IR energy while maintaining ultrashort pulse duration (< 50 fs), but also employ high pump energy in the OPA. Moreover, DC-OPA can also 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-angle noncollinear interaction reduces the conversion efficiency down to 2%, and the idler pulses are produced with an unavoidable angular dispersion. Besides, the amplified signal pulse is difficult to compress to the transform limit (TL), even after sufficient compensation of higher-order dispersion, owing to the divergence and pulse-front tilting of the seed pulse. In DC-OPA, we employ chirped pump and seed pulses, which interact in BBO crystals in a collinear configuration. By controlling the chirping value of the pump and seed pulses around , we achieve a relatively high conversion efficiency and broadband signal and idler pulses without exceeding the damage threshold of the BBO crystals. The rest of this paper is organized as follows. In Sec. 2, we explain the concept of the DCOPA. 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 DCOPA scheme are discussed. In Secs. 5 and 6, we discuss the characteristics of the amplified signal and idler pulses in the DC-OPA scheme. We also show how to achieve a high conversion efficiency and a broad bandwidth for signal and idler pulses in DC-OPA. In Sec. 7, we present a summary of our conclusions and discuss the prospects for high-power IR pulse sources based on 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) ultrashort pump and seed pulses are synchronized and interact in the nonlinear crystal, yielding amplified ultrashort signal and idler pulses. As we have discussed in the introduction, however, the maximum acceptable pump energy is limited by the aperture and damage threshold of the nonlinear crystal. Naturally, the concept of chirped pulse amplification (CPA) is transferred to OPA in or-

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der to solve this problem, forming a new scheme called OPCPA. Besides, OPCPA inherits the advantages 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 markedly reduced peak intensity makes it possible to use a much higher pump energy, and therefore to obtain a much higher amplified pulse energy than from an OPA. In this scheme, one no longer needs an ultrashort pump pulse, instead one can resort to the use of a high-energy laser system with a comparatively long pulse duration (for example, a picosecond Nd:YAG system) as the pump source. If the seed pulse temporally overlaps well with the pump pulse, good energy extraction can be achieved, and it is possible for the amplified signal and idler pulses to reach a high power after subsequent recompression. Note that seed and pump pulses are derived from different laser sources in many OPCPA experiments [26, 37]. Actually, an additional timing circuitry for synchronizing pump and seed pulses is necessary, because the synchronization between the pump and seed pulses is critical for efficient and stable amplification in OPCPA systems. Alternatively, low-jitter all-optical synchronization between pump and seed pulses can be achieved by injection seeding the pump laser with a part of the seed spectrum. Having in mind the objective of producing high energy and ultrashort pulses, the laser source itself is preferably powerful and broadband. The table-top Ti:sapphire laser system, which has developed into a rather mature stage in recent years, might be an attractive candidate. For a Ti:sapphire laser system, the DC-OPA scheme as shown in Fig. 1(c) is proposed. The DC-OPA is seeded with a chirped, broadband seed pulse and pumped by a stretched, broadband pump pulse. In our envisaged DC-OPA system, automatically synchronized pump and seed pulses are obtained because they come from a common source. The pump pulse can be spatially separated into two pulses on a beam splitter, one strong and the other one weaker. The strong one will be used as pump pulse, 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 configuration of OPA was proposed [38]. In the noncollinear geometry, the signal group velocity equals the projection of idler group velocity along the signal direction, resulting in the compensation of group-velocity mismatch (GVM) [36, 38, 39]. However, the idler pulse generated from noncollinear geometry is inherently accompanied by angular dispersion, which makes further application of the idler pulse difficult. Although we can obtain an ultrashort signal pulse duration using a noncollinear configuration, angular dispersion becomes a disadvantage for idler pulse compression. Therefore, a collinear configuration is better suited for the utilization of the idler pulses. The principle of optical parametric amplification is quite simple; in the difference frequency generation (DFG) process, the instantaneous angular frequency ωm (t), where m denotes the pump (p), signal (s), or idler (i), should satisfy the law of energy conservation, i.e.,

ωi (t) = ω p (t) − ωs (t).

(1)

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

ω p (t) = ω p + β pt,

(2)

ωs (t) = ωs + βst,

(3)

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

ωi (t) = (ω p − ωs ) + (β p − βs )t.

(4)

Equation (4) indicates that the chirp of the idler pulse is determined by the chirps of the pump and seed pulses. #140098 - $15.00 USD

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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 amplification of the signal and idler pulses. Figure 2 shows the possible chirp conventions of the pump and 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 frequency chirp. The seed pulses with positive and negative chirps shown in Figs. 2(a) and (b) interact with unchirped pump pulses, producing idler pulses with negative and positive chirps, respectively. Compared with the OPCPA scheme, the DC-OPA scheme has more degrees of freedom for 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 four different chirps. The lines represent ω p (t) and ωs (t). In Fig. 3(a), ω p (t) and ωs (t) are parallel to each other. This indicates that the pump and seed pulses have the same chirp, therefore the generated 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 of the idler pulses. In Figs. 3(c) and (d), both seed pulses are positively chirped but with different values. If the chirp of the seed is smaller than that of the pump, the produced idler pulse would be positively chirped, as shown in Fig. 3(c); in the opposite situation [Fig. 3(d)], the idler pulse has a negative chirp. In Figs. 3(e)-(h), the sign of the chirp for the pump pulse is changed from positive 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

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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 important 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 isolated on a sub-cycle time scale; thus, CEP control is very important in attosecond metrology. To investigate these CEP-dependent phenomena, a CEP-stabilized laser system is strongly desired. Baltuˇska et al. [43, 34] have proposed and demonstrated self-stabilization of the CEP using the DFG process. In this scheme, a white light continuum (WLC) generated by the pump pulse is used as the seed pulse, thus the seed pulse (and consequently the signal pulse) inherits the phase of the pump φ0 . In this case, the phases of the pump (φ p ), signal (φs ), and idler (φi ) pulses are given by

φ p = φ0 , φs = φ0 − π /2, φi = −π /2 + φ p − φs ,

(5) (6) (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 signal pulses, automatically cancels out in the idler pulse according to Eq. (7). In direct analogy, DCOPA can also produce a self-CEP-stabilized idler pulse.

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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-wave interactions. We assume that the Gaussian-type electric fields of the pump, signal and idler pulses in the DC-OPA processes are expressed as 1 Em (x, y, z,t) = Am (x, y, z,t)exp[ j(km z + ωmt)] + c.c, 2

(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 the Gaussian spatial and temporal shapes. The spatial shape is assumed to be exp[−(x2 + y2 )/w20 ], where w0 is the waist size. Because the laser beams are loosely focused and the Rayleigh range is much greater than the propagation length in the following calculations, we approximate the phase for the spatial component as that of the plane wave. For a chirped pulse, its linear chirp should be taken into account; thus, the temporal shape is described by Am (t) = exp(−αmt 2 )exp( jβmt 2 ),

(9)

where αm = 2ln2/Dt2 (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 wave equations: 3 ∂ A p (x, y, z,t) (− j)n−1 (n) ∂ n A p (x, y, z,t) ∂ ∂ +∑ kp + A p (x, y, z,t) = + ρ p ∂z n! ∂ tn ∂x ∂y n=1

χ (2) ω p 3χ (3) ω p 1 − κ p A p (x, y, z,t)− j As Ai e−iΔk(t)·z − j (γ pp|A p |2 +γ ps|As |2 +γ pi|Ai |2 )A p ,(10) 2 2n p c 2n p c 3 ∂ As (x, y, z,t) (− j)n−1 (n) ∂ n As (x, y, z,t) ∂ ∂ +∑ ks + As (x, y, z,t) = + ρ s ∂z n! ∂ tn ∂x ∂y n=1 3χ (3) ωs 1 χ (2) ωs A p A∗i eiΔk(t)·z − j (γss|As |2 +γ ps |A p |2 +γsi |Ai |2 )As , (11) − κs As (x, y, z,t) − j 2 2ns c 2ns c 3 ∂ Ai (x, y, z,t) (− j)n−1 (n) ∂ n Ai (x, y, z,t) ∂ ∂ +∑ ki + Ai (x, y, z,t) = + ρ i ∂z n! ∂ tn ∂x ∂y n=1 3χ (3) ωi 1 χ (2) ωi A p A∗s eiΔk(t)·z − j (γii |Ai |2 + γis |As |2 + γip |A p |2 )Ai , (12) − κi Ai (x, y, z,t) − j 2 2ni c 2ni c where c is the velocity of light in vacuum; nm (m = p, s and i) denote the refractive indices of 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 of spatial walk-off; κm are the absorption coefficients; χ (2) and χ (3) are the second- and thirdorder nonlinear coefficients, respectively; the time-dependent wave-vector mismatch is given by Δk(t) = k p (t) − ks (t) − ki (t); and γm,n (m, n = p, s and i) are the correction factors for a Kerr-effect nonlinear refractive index accounting for self-phase modulation (SPM, m = n) and cross-phase modulation (XPM, m = n). The γm,n depend on the polarizations of the light; it is equal to 1 if the light beams have the same polarization, otherwise it is 1/3 [45]. #140098 - $15.00 USD

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To integrate the nonlinear partial differential equations (10)-(12), we employ the split-step Fourier-transform algorithm [46]. The BBO crystal is divided into a large number of small segments, in each of which the propagation is calculated in the spectral domain. The propagation step is calculated in the reference frame of the pump for convenience. Then the obtained solution is back-Fourier-transformed into the time domain, where the nonlinear source terms are taken into account, and solved with a fourth-order Runge-Kutta method. After the treatments in the 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 equations 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 coupled 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-dimensional model 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 linear chirp and group delay dispersion (GDD) should be clarified in advance to avoid confusion. The linear chirp is a concept in the time domain, whereas the GDD is introduced in the frequency domain. We employ the Fourier transform to relate these two quantities. Starting from the frequency domain, it is often helpful to expand the spectral phase into a Taylor series [47]; 1 φm (ω ) = φ (ωm ) + φ (ωm )(ω − ωm ) + φ (ωm )(ω − ωm )2 + · · · . 2

(13)

The spectral phase of zero-order term is a phase constant in not only frequency domain but also the time domain, since this term has no influence on the Fourier transformation. The coefficient of first-order φ is nothing but a time shift in the time domain according to the Fourier shift theorem. We mainly concentrate on the second-order dispersion φ (also termed GDD), which accounts for the linear chirp of the laser pulse. Considering only GDD, the laser field with a Gaussian envelope in the frequency domain can be described approximately as m (ω ) = exp(−ηm (ω − ωm )2 )exp( j ζm (ω − ωm )2 ), A 2

(14)

where ηm = 2ln2/D2ω (ηm >0), and Dω denotes the FWHM of the pulse in the frequency dom (ω ) back into the time domain and dropmain; ζm represents the GDD value. Transforming A ping the term exp( jωmt), we obtain − ζm ηm2 2 2 (15) t exp j 2 t . Am (t) = exp − 2 4ηm + ζm2 8ηm + 2ζm2 Comparing Eqs. (9) and (15), αm and βm are given by

αm =

ηm2 , 4ηm2 + ζm2

(16)

βm =

−ζ m . 8ηm2 + 2ζm2

(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 μ m and 1.4 μ m, and #140098 - $15.00 USD

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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 durations. The chirp can be introduced by stretchers, such as bulk materials, prism pairs, grating pairs, or an acousto-optic programmable dispersive filter (AOPDF). As shown in Table 1, we stretch the pump and seed pulses with the absolute value of GDD varying from 0 to 10000 fs2 , 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 diameter of 5 mm. Although the chirped pump pulses have different pulse durations, we keep the peak intensity of each pump pulse at 100 GW/cm2 . Obviously, the available pump energy is higher for the chirped pump pulse owing to the longer pulse duration, which benefits the production of amplified signal and idler pulses with higher energies. Thus, we can easily scale the energy of the pump and seed pulses to the aperture of the BBO crystals. The parameters of the pump pulse marked by an asterisk∗ are employed to discuss the B-integral in Sec. 4. In the following calculations, both Type-I (pump extraordinary, signal ordinary, idler ordinary) and Type-II (pump extraordinary, signal extraordinary, idler ordinary) phase-matching configurations are considered. The BBO crystals are cut at θ = 20◦ and θ = 27◦ to satisfy the Type-I and Type-II phase-matching conditions, respectively, in the 1 - 2 μ m region. Table 1. Parameters of pump and seed pulses.

pump pump∗ seed

4.

wavelength (μ m) 0.8 0.8 1.4

|GDD| (fs2 ) 0 - 10000 10000 0 - 10000

pulse duration (fs) 35 - 792 792 35 - 792

intensity (GW/cm2 ) 100 50 0.04 - 0.9

energy 0.685 mJ - 15.5 mJ 7.75 mJ 6.25 μ J

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

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

2π λs

L 0

n2 (γss |As |2 + γsp |A p |2 + γsi |Ai |2 ),

(18)

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

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3.0 2

B−integral

2.5

100 GW/cm

50 GW/cm2

2.0 1.5 1.0 0.5 0.0 0

5 Crystal Length (mm)

10

Fig. 4. Operational limits for pump intensity and BBO crystal length in a DC-OPA as determined 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 concrete value of the B-integral for Type-I phase matching. The B-integral is calculated from the position-dependent values of the pump, signal and idler intensities, which are obtained from a solution of the coupled wave equations. In this calculation, pump pulses with intensities of 50 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 . The calculated B-integral is plotted in Fig. 4. It is shown that the B-integral is larger for a higher pump intensity and a thicker BBO crystal. As pointed out by Ross et al. [50, 51], the B-integral is required to be less than 1; therefore, the thickness of the BBO crystal is limited to 8 mm and 4 mm for pump intensities of 50 GW/cm2 and 100 GW/cm2 , respectively. Simulations for Type-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 consider the superfluorescence background, which originates from the parametric amplification of the vacuum or quantum noise in OPA [52]. The amplification of vacuum noise, also known as optical parametric generation (OPG), will occur at those wavelengths at which the parametric interaction is phase-matched, i.e., where spontaneous conversion of pump photons into a signal and idler photon pair takes place [52, 53]. The OPG process is very useful for many applications, such as in a high-repetition-rate femtosecond laser system [54, 55]; however, we must take 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 of the idler pulse may be destroyed by the OPG process, because the CEP relationship between the pump and OPG pulses is completely random [56]. Table 2. Types of pump and seed pulses

C-OPG OPG DC-OPA OPA

pump chirped pulse TL pulse chirped pulse TL pulse

seed (noise) (noise) chirped pulse TL pulse

We evaluate the energy evolution of the signal pulse during the C-OPG, OPG, DC-OPA and OPA processes as a function of BBO. The newly emerging C-OPG is an abbreviation for #140098 - $15.00 USD

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μJ

[Energy]

nJ

log

mJ

J (b) Type−II

C−OPG

mJ

OPG

μJ

DC−OPA

10

log10 [Energy]

J (a) Type−I

pJ 0


10

nJ pJ 0

OPA


10

Fig. 5. Energy evolution during C-OPG, OPG, DC-OPA and OPA processes inside a BBO crystal. TL pump: 35 fs, 685 μ J (intensity of 100 GW/cm2 ). Chirped pump: 792 fs (GDD value of -10000 fs2 ), 15.5 mJ (intensity of 100 GW/cm2 ). TL seed: 35 fs, 6.25 μ J (intensity of 0.9 GW/cm2 ). Chirped seed: 792 fs (GDD value of 10000 fs2 ), 6.25 μ J (intensity of 0.04 GW/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 for the four parametric processes in Table 2. The C-OPG/OPG process is initiated by a random noise field with an energy of approximately 1 pJ, while a seed pulse with an energy of 6.25 μ J is used for the DC-OPA/OPA process. In Fig. 5, we observe that the saturation of signal energies is slightly faster for Type-I phase matching compared to the Type-II case for all of the four processes, since the nonlinear coefficient is higher for Type-I phase matching in the BBO crystal. It is also found that the C-OPG process reaches saturation more rapidly than the OPG process, 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 the idler pulse, the output energy of the signal generated by C-OPG should be much lower than that of the DC-OPA. It is evident from Fig. 5 that the signal energy of C-OPG is rather low when the DC-OPA process reaches saturation. The signal energy amplified by DC-OPA is still more than 3 orders of magnitude higher than that amplified by C-OPG for propagation lengths of 3 mm under Type-I, and 5 mm under Type-II phase matching. Thus, we can suppress the amplification of noise by imposing proper restrictions on the propagation length. In a practical OPA/DC-OPA process, the amplification of noise and seed is inseparably entwined, so that the signal energies coming from C-OPG and DC-OPA are indistinguishable. Possible means of estimating the energy amplified from noise is to measure the signal output energy in absence of the 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 necessary to 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 that there is a trade-off between the conversion efficiency and bandwidth of the amplified signal pulse 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 gain profile to illustrate the gain-narrowing effect. The chirp situation corresponds to that shown in Fig. 3(b). During the parametric process, only the central part of the pump strongly couples with the signal and idler; therefore, the temporal gain profile (white dotted-dash line) is narrower than the profile of the pump pulse (yellow dashed line). Considering the temporally varying #140098 - $15.00 USD

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wave vector mismatch induced by the chirp, there would be further gain reduction in the wings of 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, the whole phase-matching bandwidth can be preserved. In contrast, the wings of the contour plot for 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 efficiently within the gain profile. The gain-induced narrowing will shape the signal pulse to a form with steeper leading and trailing edges. We keep in mind that the GDD of the shaped pulse is still unchanged. If we consider two stretched pulses having different durations but the same GDD value, the Fourier principle tells us that the shorter (longer) stretched pulse corresponds to an unstretched 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 signal pulse for Type-I and Type-II phase matching, respectively. Basically, the conversion efficiency is improved as the added GDD value increases. The conversion efficiency is mainly determined by 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 energy can be transferred from the pump pulse to the seed pulse when the pulse duration of the seed pulse is sufficiently long. We now address the bandwidth of the amplified signal pulse. The resulting bandwidth of the signal exhibits a different trend from the conversion efficiency. The signal bandwidth firstly increases, then saturates, and finally decreases with increasing GDD value of the seed pulse. This result seems to contradict our earlier findings: a short seed pulse is favorable for preserving the input bandwidth. However, this puzzle can be resolved by taking the group-velocity mismatch (GVM) between the interacting pulses into account. The GVM for 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 pointed out by Cerullo et al. [36], δsp and δip limit the interaction length over which parametric am-

1.8

1.0 1.8

λ (μm)

(a) 1.4

1.0 1.8

(b)

0.5 1.4

1.0 −1000

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(c) 0.5 1.4

0

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0.0 1.0 1000 −1000

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Intensity

Time (fs)

Fig. 6. Normalized pump pulse profiles (yellow dashed line) and corresponding temporal gain profiles (white dotted-dash line). The contour plots indicate seed pulses (corresponding to a 35-fs TL) for different GDD values of (a) 0 fs2 , (b) 4000 fs2 , and (c) 10000 fs2 in time-frequency analysis. The pump pulse is chirped with a GDD value of -10000 fs2 . The calculations are carried out for Type-I phase matching.

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60 40

0.0 0

5 3 2 Seed GDD (×10 fs )

10

100

0.4 (b)

80 0.2

60 40

0.0 0

5 Seed GDD (×103 fs2)

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80 0.2

Conversion Efficiency

(a)

Bandwidth (nm)


100

0.4

10

Fig. 7. Signal pulse conversion efficiency (blue, ) and bandwidth (red, ) obtainable from the DC-OPA for different values of the seed GDD (0-10000 fs2 ). The parameters of pump and seed pulses are chosen according to Table 1. (a) Type-I and (b) Type-II phase matching. The conversion efficiency and bandwidth are calculated for the gain saturation point in the BBO crystal.

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 temporal broadening and asymmetry of the temporal profile coming along with a gradual reduction in bandwidth. With the parameters for Fig. 7, δsp is estimated to be about 7 fs/mm for Type-I phase matching. If we employ a 35-fs TL pulse as the seed pulse, the relative delay between pump and signal pulses is about 20% of the input seed pulse duration after propagation through 1 mm of BBO. However, the relative shift is less than 3% of the pulse duration if we input a 318-fs seed pulse. Hence, GVM can be neglected for limited propagation length if we use a seed pulse with sufficiently long pulse duration. Note that the output bandwidth of a signal pulse is narrower for Type-II phase matching, especially for a short seed pulse. E.g., employing a TL seed pulse, the bandwidth of the output signal is 76 nm for Type-I phase matching (Fig. 7(a)), while it is only 48 nm for Type-II phase matching (Fig. 7(b)). We believe that this is because of the different acceptable bandwidths for the two types of phase matching. As previously reported in [36], the acceptable bandwidth increases as the absolute value of GVM between the signal and idler pulses decreases. The calculated |δsi | for Type-I phase matching is 8.5 fs, which is much smaller than that of 72 fs for Type-II phase matching. From the above discussion, we can conclude that the bandwidth of the output signal is mainly determined by the GVM for short TL seed pulses, and the gain-narrowing effect plays a most important role for long chirped seed pulses. As can be seen from Fig. 7, it is suitable to choose the seed pulse GDD value within the range from 3000 fs2 to 5000 fs2 for both Type-I and Type-II phase matching, in order to simultaneously achieve a high conversion efficiency and a broad bandwidth of the amplified signal pulse. In other words, the seed pulse duration is preferentially 30% to 50% of the pump pulse duration. Figure 8 shows the output signal conversion efficiency as a function of BBO thickness, where the pumping intensity is fixed to be 100 GW/cm2 . The GDD values of the pump and seed pulses are -10000 fs2 and 4000 fs2 , respectively. These design parameters are chosen for the simultaneous optimization of the conversion efficiency and bandwidth, according to the result shown in Fig. 7. The conversion efficiency increases with the increasing thickness of the BBO crystal until it reaches the gain saturation position, after which back-conversion occurs. Note that the conversion efficiency reaches its maximum faster for Type-I phase matching owing to the higher nonlinear coefficient. The back-conversion to some extent may broaden the bandwidth of the signal pulse. This can be understood by the fact that the gain is different for different wavelengths, resulting in spectral shaping. Similar results were found in previous works [58, 59].

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However, the spectrum may suffer strong modulation if the propagation length is far beyond the 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 intensity and phase. For Type-I phase matching as shown in Fig. 9(a), the amplified signal pulse at 1.4 μ m exhibits a smooth distribution with a bandwidth of 79 nm, which is just slightly narrower than the injected spectra of the seed pulse. The spectral phase shows a parabolic profile originating from the added GDD and accumulated phase in the BBO crystal. The spatial profile of the produced signal pulse is also calculated with our three-dimensional model, in order to check the beam quality. Figure 9(b) shows the temporal profile of the amplified signal pulse after phase compensation. The compensation is realized by adding the inverse GDD of the seed pulse to that of the amplified signal pulse. A signal pulse with a pulse duration of 40 fs is obtained, which approaches the TL of 36 fs supported by the 79 nm bandwidth. We also show the temporal phase that was compensated for without considering the rapidly changing phase induced by the carrier frequency. The temporal phase profile of the signal pulse is very flat, 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-II phase matching are shown in Fig. 10. They are similar to the Type-I case in Fig. 9. One detail that 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 of the 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 are chirped by GDD values of -10000 fs2 and 4000 fs2 , which corresponds to the situation shown in Fig. 3(b). Note that according to Eq. (17) a negative GDD value induces a positive chirp for a pulse with a standard Gaussian temporal envelope, and vice versa. Actually, several other choices for chirping the pump and seed pulses to achieve a relatively high conversion efficiency and a broad bandwidth are available as well, as long as the ratio of the seed pulse duration to the pump pulse duration are maintained at a proper value. We employ the schemes shown in Figs. 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 slight difference in the maximum bandwidth of the output signal pulse is found. The situations shown in Figs. 3 (a), (c), (e) and (g) cannot be applied for DC-OPA, because the duration of the seed pulse is comparable to or longer than that of the pump pulse. As we have explained earlier, a serious gain narrowing would occur in these situations. 0.3

Type−I Type−II

0.2 0.1 0.0 0.0

0.5 1.0 1.5 Crystal Length (mm)

2.0

Fig. 8. Dependence of the signal pulse conversion efficiency on the propagation length in the BBO crystal for (a) Type-I and (b) Type-II phase matching. The curves are calculated for GDD values of -10000 fs2 for the pump pulse and 4000 fs2 for the seed pulse.

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−5

y (mm)

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(a)

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2 0 −2 79 nm −2 0 2 x (mm)


1.0

−1.0 100

Fig. 9. (a) Spectrum of the injected seed pulse (yellow solid line), and spectral intensity of 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 amplified signal pulse after phase compensation. The curves are calculated for the saturation point shown in Fig. 8, for the case of Type-I phase matching. The GDD values for the pump and seed pulses are -10000 fs2 and 4000 fs2 , respectively.

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x (mm)


Before we continue, we want to mention a few pioneering works on chirp-compensation schemes [50, 60, 61, 62]. These OPA systems are designed to choose pump and seed chirps such that at each temporal coordinate the combination of pump and seed wavelengths are phase matched, resulting in a bandwidth enhancement of the output signal. This idea has been successfully demonstrated at pump wavelength of 0.4 μ m and seed wavelength of 0.7 - 0.8 μ m. Although our proposed system also chirped pump and seed pulses, it is operated at 0.8 μ m pump and 1.4 μ m seed. Around our interested wavelength, the phase-matching curve is flatter and we do not employ the ultra-broadband pump and seed (TL pulse duration above 30 fs), therefore the chirp-compensation mechanism does not work in our system. In the previous paragraph, we demonstrate that the signal output bandwidth almost keeps unchanged even for four different chirp combinations of pump and seed pulses, this result strongly supports our judgement. However, if we consider ultrabroadband input pump and seed pulses (TL pulse duration below 10 fs), the bandwidth narrowing induced by mismatching of pump and seed gradually plays a role, in this case we should take the chirp-compensation scheme into account for optimization of the output bandwidth. Moreover, the experimental works which use chirp-compensation scheme [60, 61] only report on few-μ J systems, and they do not show the potential benefit of chirp-compensation for minimizing limitations due to B-integral. Since our

−1.0 100

Fig. 10. (a) Spectrum of the injected seed (yellow solid line), and spectral intensity (blue solid 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: Calculated spatial 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.

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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. According to the above discussion, the pump and seed pulses are chirped to optimize the conversion efficiency and bandwidth. The bandwidth of the injected seed pulse is 288 nm; however, the bandwidth of the output signal pulse decreases to 210 nm. We attribute this narrowing of bandwidth to several factors: First, we can reduce the gain-narrowing effect, but not completely avoid it. Second, the inherent existence of an acceptable bandwidth for the BBO crystal itself prevents us from using a seed pulse with a broad bandwidth. Third, we do not optimize the chirp ratio for the ultrabroadband pump and seed pulses according to chirp-compensation scheme. 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 all orders. This is because the amplified temporal intensity profile of the pulse deviates from a standard Gaussian type, consequently resulting in a reduction of the compression factor. In this evaluation, the collinear DC-OPA enables us to generate a few-cycle IR pulse with the shortest duration of ∼17 fs at the signal wavelength. Certainly, if the amplifier is operated at the degeneracy wavelength (pump: 0.8 μ m, signal/idler: 1.6 μ m), it is helpful to improve the acceptable bandwidth to support a shorter pulse [17]. Furthermore, if we slightly detune the BBO crystal similar to [17], two different wavelengths would be simultaneously phase matched, generating output pulses of broader bandwidth. In this case, the shortest achievable pulse centered at 1.6 μ m 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 the idler pulses. Here, we employ a collinear DC-OPA scheme to avoid angular dispersion of the idler pulses. For the utilization of the idler pulses, bandwidth and conversion efficiency should be further investigated. During DC-OPA, idler pulses are generated as soon as pump and seed pulses are injected into the BBO crystal, and then they experience energy gain and bandwidth modulation. 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 obtaining broadband idler pulses after propagation in the BBO crystal. Figure 11 shows the initially generated bandwidth of the idler pulse by chirping the pump and seed pulses with different GDD values. The idler pulse centered at a wavelength of 1.87 μ m is generated by a DFG process from 0.8-μ m pump and 1.4-μ m seed pulses. For comparison, the ideal case of unchirped pump and seed pulses is also shown. The GDD value of the pump pulse is fixed at -10000 fs2 , and it is found that the bandwidth of the idler pulse gradually increases to a value close to that in the ideal case when the GDD value of the seed pulse varies from -10000 fs2 to 4000 fs2 . To explain this finding, we recall the second-order nonlinear interaction term, A p e jω p t (As e jωs t )∗ = e−(α p −αs )t e[ j(ω p −ωs )t+ j(β p −βs )t ] , 2

2

(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) = ω p (t)− ωs (t)). Using Eq. (4), it is found that the linear chirp for the idler pulse vanishes, when the pump and seed pulse are chirped with the same GDD value. However, the envelope of the idler pulse is a product of the stretched envelopes of the pump and seed pulses, which can be described 2 by e−(α p −αs )t according to Eq. (22). Therefore the initially generated idler pulse still has long pulse duration, which is independent of the chirp signs of the pump and seed pulses. As is wellknown, long pulses without chirp induce a narrow bandwidth. So we expect to obtain an output idler pulse with narrow bandwidth using pump and seed pulses with the same chirp value. On #140098 - $15.00 USD

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1.0

Pump GDD (fs2)

0.5

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1700

1800

1900

λ (nm)

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−10000

−10000

[Fig. 3(a)]

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[Fig. 3(d)]

−10000

0

−10000

4000

[Fig. 3(c)]

0

0

[Ideal case]

0

4000

[OPCPA]

2100

Fig. 11. Bandwidth of the generated idler pulse at the beginning of DC-OPA, in addition to one example for OPCPA. The analysis is performed for different GDD values of the pump and seed pulses.

the other hand, if the chirps for the pump and seed pulses have opposite signs, the absolute value of linear chirp for the idler pulse will increase corresponding to a smaller absolute value of GDD. It is known that a smaller absolute value of GDD is needed when we stretch a pulse with broader original bandwidth to a certain pulse duration. Thus we expect to produce an idler pulse with relatively broad bandwidth using pump and seed pulses with opposite signs of chirp. Then we return to the OPCPA scheme, considering pump with GDD of 0 fs2 interacting with a chirped seed pulse which results in a narrower bandwidth than the output bandwidth at optimized chirp combination for DC-OPA. We choose seed GDD values of -10000 fs2 , -4000 fs2 and 4000 fs2 for the conditions of DCOPA, and then check the spectral properties of the output idler pulse. The spectral phase shown in Fig. 12(a) is attractive, since it is almost flat, which means that we can obtain a TL pulse without compensation. Unfortunately, the corresponding bandwidth is very narrow as expected. In Fig. 12(b), the bandwidth of the idler pulse increases because the long idler pulse still contains chirp 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 seed pulse used in Fig. 12(b). In Figs. 12(a), (b) and (c), we obtain spectral phases of the idler pulses with 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 31 fs is obtained (see Fig. 12(d)). Moreover, comparing the spectral phases shown as green solid and green dashed lines in Fig. 12(c), it is found that the GDD sign of the idler pulse changes when we invert the GDD sign of the input pump and seed pulses. This feature is useful for conventionally compensating for the GDD of the idler pulse. If we use transmission compressors such as bulk materials and prism pairs, the powerful idler pulse in the DC-OPA scheme may suffer degradation in beam quality due to high-order nonlinear effects. An appropriate choice is to use reflective compressors such as grating pairs. As is well-known, parallel grating pairs can only provide negative GDD. Therefore, a high-power idler pulse with a positive GDD (see green 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 the idler 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 seed pulses, the conversion efficiency of the idler pulse is improved. We need to pay attention to two

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−1

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250

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200

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100 10

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250 (a)

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Fig. 12. Spectral intensity (blue solid line) and phase (green solid line) of the amplified idler pulse before phase compensation for different seed GDD values: (a) -10000 fs2 , (b) -4000 fs2 , and (c) 4000 fs2 . The GDD value of the pump pulse is fixed at -10000 fs2 . In particular, the spectral phase denoted by the green dashed line shown in (c) is calculated with pump GDD of 10000 fs2 and seed GDD of -4000 fs2 . (d) Temporal intensity (red solid line) and phase (green solid line) profiles of the idler pulse after compensation of the spectral phase shown in (c). The curves are calculated for the saturated gain position of DC-OPA in the BBO crystal.

Fig. 13. Idler pulse conversion efficiency (blue, ) and bandwidth (red, ) obtainable from the DC-OPA for different values of the seed GDD. (a) Type-I and (b) Type-II phase matching.

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aspects for achieving broad bandwidth of the output idler pulse, which are different from the signal case. On the one hand, the signs of the linear chirps of the pump and signal pulses should be opposite so that we can obtain a broadband idler pulse at the beginning of DC-OPA; on the other hand, we must choose proper stretching parameters to avoid gain narrowing during the amplification 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 of the idler pulse can be achieved simultaneously. This means, the optimized pulse duration of the seed 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 high peak power. In recent works [34, 63, 64], a two-stage OPA scheme was used to generate a CEP-stabilized IR pulse. In this scheme, an idler pulse with a self-stabilized CEP obtained from the first stage is used as the seed pulse for the second stage; consequently, the amplified second-stage signal pulse is also CEP-stabilized. Of course, we can utilize this scheme in DCOPA. Furthermore, we chirped the pump and seed pulses to apply a higher input energy without damaging the BBO crystal, so that the amplified signal and produced idler pulses will be more powerful. The shortest idler pulse we can obtain is also limited by the gain-narrowing effect and the acceptable bandwidth of the BBO crystal. In addition, we must take care of the absorption wavelength of the BBO crystal located at about 2.1 μ m, beyond which the transparency curve decreases rapidly. Therefore, the long-wavelength part of a broadband idler pulse with a central wavelength at 1.87 μ m might fall into this absorptive region. When we apply pump and seed pulses of 10-fs TL, an idler pulse with a duration of 14 fs can be produced for optimized conditions. 7.

Summary

We proposed a novel OPA scheme called DC-OPA for producing high-power IR pulses with a few-cycle pulse duration. In this scheme, a pump pulse at 0.8 μ m and a NIR seed pulse are generated employing a common Ti:sapphire laser system; thus, low-timing-jitter all-optical synchronization between the pump and seed pulses can easily be realized without the need of costly synchronization electronics. By introducing chirps to both pump and seed pulses, a sufficiently high-energy pump pulse can be applied in an OPA. The possible detrimental effects originating from the Kerr nonlinearity and parametric superfluorescence were also considered and place a limit on the peak intensity of the pump pulse and the propagation length in the BBO crystal for a targeted energy gain of 103 . For an optimized GDD combination between the pump and seed pulses, the conversion efficiency attained is > 40% with broadband signal and idler spectra in the one-dimensional calculation. For a fully three-dimensional model, the calculated total conversion efficiency is lower than for a one-dimension model, and a conversion efficiency of 31% - 36% within the optimized GDD range can be achieved. For example, in Type-I phase matching, a 40-fs signal pulse was obtained after chirp optimization, and a 31-fs idler pulse with passively stabilized CEP can be achieved under the additional requirement of opposite signs for pump and seed chirps. A similar result was obtained for Type-II phase matching except for the narrower acceptable bandwidth. If we utilize an even broader bandwidth pump/seed pulse supporting 10-fs TL duration, DC-OPA in collinear configuration guarantees 18 fs for the signal pulse 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 scalable in output energy of the IR pulses, which provides us with the design parameters of an ultrafast IR pulse source with an energy of a few hundred mJ. Meanwhile, 10-TW class (e.g., 40 fs, 0.4 J) Ti:sapphire laser systems are commercially available. From the conversion efficiency of DCOPA, if we apply all the laser energy (0.4 J) for DC-OPA under optimized GDD conditions, we

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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 have few-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 our estimation ignores optical losses from the compressor, we can still obtain more than sufficient TW IR power. We believe that DC-OPA has great potential to markedly increase the IR pulse energy, which will pave the way for the generation and application of not only intense ultrafast coherent water window x-rays but also high-intensity laser physics [51]. Especially, an intense water window x-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 of RIKEN. O.D.M. acknowledges support from a Lise-Meitner Fellowship from the Austrian Science Fund (FWF), project M1094-N14.

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