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Abstract
Quasi-parametric chirped pulse amplification (QPCPA) is a new scheme that enables the amplification of chirped signal pulses without back conversion by depleting the idler pulses. In this paper, we present a numerical study on the bandwidth, efficiency, and robustness of QPCPA. Self-locked phase among the interacting waves is found to be the underlying mechanism for the suppression of back conversion, which allows signal efficiency approaching to the quantum limit even under the phase-mismatch condition, and thus greatly increases the phase-mismatch tolerance of QPCPA. We demonstrate that QPCPA can break through the trade-off between the efficiency and bandwidth encountered in conventional optical parametric amplification, hence supporting highly efficient amplification of few-cycle pulses.
© 2017 Optical Society of America
1. Introduction
Researchers have been putting great efforts on increasing the laser peak-power as well as shortening the laser pulse-duration since the invention of laser. Laser peak-power up to gigawatt level was reached soon after, with the aid of the mode-locking technique. Little progress, however, was achieved in the following two decades due to the limitations of optical damage and nonlinear distortions in laser amplifiers until the concept of chirped-pulse amplification (CPA) was introduced [1]. The CPA scheme adopts spectral dispersion to stretch a femtosecond pulse before amplification, after which it is recompressed. In such a way, the laser peak intensity within the amplifier can be kept much lower than the thresholds of optical damage and nonlinear distortions. Laser peak-power of petawatt level can be generated now via CPA routinely [2]. Such intense lasers have enabled and promoted various frontier researches in ultrafast science and strong-field physics, such as the generation of attosecond pulses by high-harmonic generation [3], the observation of ultrafast dynamic behaviors in matter [4], and the laser-driven particle acceleration [5].
Amplifiers used in CPA are the traditional laser amplifiers based on the stimulated emission. Currently, Ti:sapphire-based CPA systems are the dominant driving lasers for ultrafast science and strong-field physics. Ti:sapphire laser is a typical energy-level gain medium system in which one of the transitions is non-radiative, as illustrated in Fig. 1(a). This kind of laser amplifier is characterized by a high efficiency from the pump to signal pulses, which can be up to 50% [6,7], while its gain bandwidth is limited by the relatively sharp energy levels associated with the lasing process. The gain bandwidth of the Ti:sapphire amplifier is about 75 nm at 800 nm, which limits the output pulse duration to a typical value of 20 fs [8]. In order to generate even shorter pulses, additional nonlinear compression methods can be utilized to convert a portion of Ti:sapphire CPA output into few-cycle pulses [9,10]. These methods, however, limit the output peak power to terawatt level.
Fig. 1 Schematic of three different amplification schemes. (a) CPA based on a nonparametric laser amplifier in which the medium participates in energy transfer. (b) OPCPA based on an optical parametric amplifier in which an idler pulse is generated and the nonlinear crystal doesn’t participate in energy transfer. (c) QPCPA based on a quasi-parametric amplifier in which an idler pulse is generated and then absorbed by the nonlinear crystal. The black solid (dashed) lines represent the real (virtual) energy levels.
After the invention of CPA, its counterpart termed as optical parametric chirped-pulse amplification (OPCPA) was proposed in 1992 [11]. In an OPCPA system, pump-to-signal transfer of energy is realized by an optical parametric amplifier instead of a laser amplifier as in a CPA system. The photon-energy diagram for optical parametric process [Fig. 1(b)] bears some similarities to that for the optically pumped energy-level laser [Fig. 1(a)]. The energy levels in the OPCPA case, however, are virtual and tunable, which are determined by the pump wavelength and phase-matching conditions. As a result, OPCPA has a very broad gain-bandwidth and can directly generate intense few-cycle pulses without the need of nonlinear compression [8]. For example, the OPCPA scheme has enabled the generation of sub-three-cycle (7.9 fs) pulse with a peak power of 16 TW [12]. Since the phase matching condition can be flexibly manipulated, OPCPA is widely used to produce intense few-cycle pulses at wavelengths that are inaccessible by CPA. Sub-two-cycle pulses with a peak power of 110 GW at 2.1 μm [13] and sub-seven-cycle pulses with a peak power of 90 GW at 3.9 μm [14] were demonstrated and these infrared pulses have exhibited the unique advantage in high-order harmonic generation [15]. Nevertheless, it is worth to note that the strict phase-matching requirement of OPCPA results in lower efficiency and less robustness comparing to CPA in practice. The conversion efficiency in few-cycle OPCPAs is typically 10%-20% [8,12–15]. Besides, OPCPA is sensitive to the various parameters determining phase-matching condition, and it also puts stricter requirements on pump laser in terms of peak power, beam quality as well as energy and pointing stabilities comparing to a CPA system [16].
Further progress in ultra-high peak-power few-cycle lasers drives the quest for ideal amplification schemes capable of broad bandwidth, high efficiency and robustness. Several OPCPA variants such as frequency-domain parametric amplification and adiabatic parametric amplification [17–20], have been proposed. However, the limited beam or crystal size used in the amplifier hampers the peak-power scaling up in these schemes. We recently reported a scheme termed as quasi-parametric chirped pulse amplification (QPCPA) [21], which swaps the conventional nonlinear crystal in OPCPA for a dedicated crystal that depletes the idler wave [Fig. 1(c)]. In principle, the idler absorption plays an equivalent role as the nonradiative transition in energy-level lasers, and so QPCPA can exhibit nonparametric characteristics such as no back-conversion, high efficiency, and robustness against phase-mismatch. While we can envisage that QPCPA is more broadband than OPCPA, no work to date has quantified this bandwidth issue. Understanding the bandwidth characteristic of QPCPA therefore becomes important to assess the applicability and potential of QPCPA devices in practical scenarios. This paper presents a theoretical study on QPCPA with an emphasis on its bandwidth characteristic. We reveal the underlying mechanisms for QPCPA in the view of phase. The evolutions of both bandwidth and efficiency within the QPCPA crystal are also given, which are quite different from those of OPCPA, bringing the promise of combining the major advantages of both CPA and OPCPA. Finally, we design a few-cycle QPCPA system based upon a Sm3+‒doped yttrium calcium oxyborate (Sm:YCOB) crystal, showing the potential in producing both the high peak power and average power.
2. Simulation model and small-signal analysis
Either OPCPA or QPCPA uses a strong narrowband pump pulse to amplify a broadband while fully chirped signal pulse. Under this typical condition, the local approximation is satisfied, so that all the three interacting waves can be regarded as quasi-monochromatic at each temporal slice [22]. As a result, the QPCPA can be temporally sliced into a group of independent narrowband quasi-parametric amplifiers with their own instantaneous frequencies of signal and idler correspondingly. Therefore, the efficiency and bandwidth of a few-cycle QPCPA can be well described by the nonlinear coupled-wave equations derived under the slowly varying envelope approximation. Following the procedure in [23], we write the equations by taking into account the idler absorption (with a coefficient of α),
where Aj, kj, and ωj are the envelope, wave-vector, and angular frequency of light field Ej = Ajexp[i(kjz-ωjt)] (j = p, s, and i, indicating the pump, signal and idler, respectively). nj is the frequency-dependent refractive index. deff and c are the effective nonlinear coefficient and light speed in vacuum. Δk = kp–ks–ki is the wave-vector mismatch among the three interacting waves. For α = 0, Eqs. (1)–(3) become the standard three-wave coupled equations for OPCPA [22]. It should be noted that, for the simulations in sections 3 and 4, the amplification behaviors of QPCPA are investigated under different values of Δk. This approach allows us to generalize the conclusions for different crystals. For the simulations in section 5, we expand Δk up to the third-order as usual. In all the simulations, we exclude the spatial dimensions, because the spatial overlap between the pump and signal can be ensured by using the Poynting vector walk-off compensation geometry [24] and the idler walk-off from the pump and signal in the noncollinear interaction configuration plays a similar role as the material absorption on the idler. Equations (1)–(3) are numerically solved by the symmetrized split-step Fourier method.To reveal the amplification behaviors in the small-signal regime, we analytically solve Eqs. (1)–(3) under the approximation of pump nondepletion (i.e., ∂Ap/∂z = 0). When the gain is reasonably large, the small-signal gain coefficient g [defined by A(z)~exp(g × z)] can be derived as,
where Γ = 2ωsωiIp0deff/(nsninpc3ε0) is referred to as the nonlinear drive, Ip0 is the initial pump intensity, and ε0 is the permittivity of vacuum. We firstly consider the case of perfect phase matching (Δk = 0). In this case, it has g = Γ for OPCPA (α = 0) and g<Γ for QPCPA (α≠0). This indicates that the idler absorption α will render the small-signal gain of QPCPA lower than that of OPCPA driven by the same Γ. Note that, the sign of g will not be altered by α. Thus, there is the potential for high conversion efficiency in QPCPA with a strong idler absorption provided that the nonlinear crystal is long enough.Next we consider the phase-mismatching case (Δk≠0). Specially, for OPCPA (α = 0), Eq. (4) reduces to
where Δκ = Δk/2Γ is the normalized wave-vector mismatch. Obviously, the condition of |Δκ| = 1 defines the two kinds of nonlinear evolutions in OPCPA: |Δκ|<1 (g is real) corresponds to the amplification regime, while |Δκ|>1 (g becomes imaginary) represents the nonlinearly cascading regime where the conversion from pump to signal is very weak with rapid alternation between OPCPA and its reverse processes [22]. Clearly, the spectral components fallen into the amplification regime of OPCPA are restricted within the scope of |Δκ|≤1. For QPCPA (α≠0), Eq. (4) can be rewrote to . Obviously, the idler absorption α will increase the tolerance of g against Δκ, and hence extend the gain spectrum into the regime of |Δκ|>1. Different from the OPCPA case, the amplification regime of QPCPA is not restricted within the scope of as implied by Eq. (4). Instead, efficient signal amplification is also achievable in the case of as long as the crystal is long enough. This will be verified in the following simulation investigations.Since the small-signal analyses are not applicable in the saturated amplification regime, the numerical simulations based on Eqs. (1)–(3) are necessary to fully manifest the amplification behaviors. As an example but without loss of generality, we simulate a QPCPA process pumped at 515 nm and seeded at 800 nm. When we study the general amplification behaviors of QPCPA in this manuscript (sections 3 and 4), we assume a pure YCOB crystal by neglecting the effect of doping rare-earth ions on the dispersion of YCOB crystal. The required idler absorption for QPCPA is introduced artificially such that it can be freely adjusted for the convenience of simulation. In section 5 where a few-cycle QPCPA system is simulated based on parameters that are physically accessible, we assume a Sm:YCOB crystal as used in our previous experiment [21]. The measured Sellmeier equations of Sm:YCOB crystal are used in section 5 to make the results closer to reality. Generally, we use an absorption coefficient α to represent the idler loss. The major parameters used for the simulations are Γ = 1000 m−1, Ip0/Is0 = 106 (Ip0 and Is0 are the initial intensities of the pump and signal, respectively), α = 0 for OPCPA, and α = Γ for QPCPA.
3. Self-locked phase in QPCPA
To understand the physical scenario of QPCPA, we decompose Eqs. (1)–(3) into the envelope equations and phase equations, respectively. We equate respective real and imaginary parts of Eqs. (1)–(3) after using the transformation Aj = ρj(z)exp[iφj(z)] and defining a phase factor θ = Δkz + φs(z) + φi(z)–φp(z). The obtained envelope equations are,
Obviously, the direction of energy flow among the three interacting waves depends on the factor of sinθ. When sinθ<0, the pump is consumed (dρp/dz<0) and the signal is amplified (dρs/dz>0). When sinθ>0, the energy flow is reversed, i.e., from the signal and idler waves to the pump wave, corresponding to the back-conversion process. Therefore, sinθ can be used as an indicator for the instantaneous energy-flow direction either in OPCPA or QPCPA [25]. Since there is no idler seeding, the initial idler phase φi(0) is self-selected to favor θ(0) = –π/2, which maximizes the initial signal gain.The evolutions of sinθ and normalized signal efficiency for OPCPA (α = 0) and QPCPA (α = Γ) under three different phase-mismatch (Δκ) conditions are simulated, and the results are summarized in Fig. 2. As pointed out previously, under perfect phase-matching condition (Δκ = 0), sinθ in OPCPA will not change until the pump is completely depleted after an interacting length of L0, at which φp(L0) will experience a π-phase jump to change sinθ from −1 to 1 for triggering the back conversion process [25]. In this case, the pump-to-signal conversion efficiency can reach the theoretical maximum at z = L0. In the presence of a phase mismatch (Δκ≠0), however, the evolution of sinθ exhibits different features. If the phase mismatch is very small (e.g., Δκ = 0.2), sinθ evolves quite similar to the case of phase matching [Fig. 2(a)]. If the phase mismatch is large (e.g., Δκ = 0.6), sinθ rapidly changes from −1 to a value between −1 and 0, and then stays at that value until a jump occurs, as shown in Fig. 2(b). Due to the decrease of |sinθ|, a longer interacting length is needed for reaching the peak efficiency. In addition, due to the phase contribution from the phase mismatch, sinθ will change its sign before the pump is completely depleted, thus the conversion efficiency is unable to reach its theoretical maximum, and will degrade further with increasing Δκ. Once Δκ>1, sinθ will evolve quite differently [Fig. 2(c)]. In this case, sinθ will rapidly change between −1 and 1 due to the fast phase accumulation. The fast alteration of the sign of sinθ results in a negligible conversion efficiency. This corresponds to the cascade nonlinearity regime of OPCPA, as implied by Eq. (5). Generally, sinθ in OPCPA (α = 0) varies periodically between −1 and 1 no matter whether phase-matching is satisfied or not, leading to periodic transitions between OPCPA and its back conversion, as shown in Figs. 2(a)–2(c). Such periodic transitions is just the inherent characteristic of a parametric process, which implies that the conversion efficiency cannot be improved by further lengthening crystal under the phase-mismatch condition.
Fig. 2 Phase factor sinθ (red solid curves) and normalized signal efficiency η/ηmax (blue dashed curves) versus propagation distance z for various normalized wave-vector mismatch Δκ. Propagation distance z is normalized to the nonlinear length LNL which is defined by LNL = π/2Γ [22]. ηmax is the theoretical maximum efficiency. (a)–(c) correspond to the OPCPA cases (α = 0), and (d)–(f) correspond to the QPCPA cases (α = Γ). In (a) and (d), Δκ = 0.2. In (b) and (e), Δκ = 0.6. In (c) and (f), Δκ = 1.2. In each case, Γ = 1000 m−1, Ip0/Is0 = 106. Note that, the signal efficiency in (c) has been enlarged by 105 times for visibility.
The sinθ and signal efficiency in QPCPA (α = Γ) present quite different evolution behaviors as shown in Figs. 2(d)–2(f). sinθ, at a small phase-mismatch of Δκ = 0.2, stays close to −1 within z/LNL<7. With further propagation (7<z/LNL<40), sinθ begins a rapid oscillation, gradually decaying toward a value of −0.25. This behavior breaks the symmetry of energy flow and prefers the forward conversion. Thus the signal efficiency trends toward the theoretical maximum value. For z/LNL>40, sinθ is self-locked to a fixed value of approximately −0.25, which thoroughly prevents the occurrence of back conversion and keeps on increasing the signal efficiency to the maximum. At a larger phase-mismatch of Δκ = 0.6, sinθ quickly changes from −1 to around −0.7, then stays at this value before z/LNL~7. It allows the signal efficiency climbing to ~70% of the maximum. With further propagation, sinθ begins oscillating with decaying amplitude. Note that, the value of sinθ in this case always keeps negative, so there is no drop in the signal efficiency. In the subsequent propagation, sinθ is self-locked to a value of −0.22, which also keeps on increasing the signal efficiency to the maximum. In the case of Δκ = 1.2, while the OPCPA shows a negligible conversion efficiency [Fig. 2(c)], the QPCPA still takes place a complete conversion, as shown in Fig. 2(f). In this case, sinθ stays around −0.33 for a long propagation distance, resulting in a low conversion rate from the pump to the signal. With further propagation, sinθ no longer oscillates but transits to a locked value of −0.17 smoothly. These unique behaviors of sinθ lead to a slow and monotonic increase of the signal efficiency.
From Figs. 2(d)–2(f), it is obvious that the sinθ in QPCPA can be self-locked to a negative value for all three phase-mismatch (Δκ) parameters used in the simulations, which completely inhibits the back conversion process and allows the achievement of the maximum signal efficiency. This phase characteristic of QPCPA is very similar to the phase-locked evolution in adiabatic three-wave mixing [26]. Although the locked-value of |sinθ| decreases with increasing Δκ, further simulations show that sinθ can still be locked to a negative value even for Δκ>100 provided that the crystal is long enough. For completeness, we also calculate the dependence of sinθ on α and Γ for a fixed Δκ of 0.6. The results presented in Fig. 3 clearly show that the locked value of sinθ is proportional to the ratio of α to Γ, i.e., sinθ≈−0.22α/Γ. A larger Γ can produce a faster energy transfer [Fig. 3(a)], but a stronger idler absorption α is undoubtedly beneficial to suppressing the back conversion [Fig. 3(b)].
Fig. 3 Phase factor sinθ versus normalized propagation distance in QPCPA for (a) different idler absorption coefficients and (b) different nonlinear drives. In each case, Δκ = 0.6, Ip0/Is0 = 106. In (a), Γ is fixed at 1000 m−1. In (b), α is fixed at 500 m−1. Note that, although various Γ is used in (b), we adopt the same LNL as those used in panel (a) and Fig. 1.
4. Bandwidth and efficiency evolutions
In the last Section, we revealed the effect of self-locked phase in QPCPA, which fundamentally ensures a complete pump depletion when Δκ≠0. In this section, we mainly focus on the evolutions of bandwidth and efficiency of QPCPA within the nonlinear crystal. The results for OPCPA are also presented for comparison. For the sake of generality, the tolerance of Δκ is adopted as a measure of the gain bandwidth, which can be easily converted to the spectral bandwidth according to the dispersion characteristics of the nonlinear crystal. Besides the case of Ip0/Is0 = 106, we also discuss the case of Ip0/Is0 = 103 to exhibit the influence of initial seeding condition on the bandwidth evolution.
The simulation results for OPCPA (α = 0) are shown in Figs. 4(a) and 4(b). One noteworthy feature is the periodic oscillation on signal efficiency, which is the inherent characteristic of a parametric process. Due to this periodicity, the theoretical maximum of the signal efficiency can be reached only at specific crystal lengths under the perfect phase-matching condition (Δκ = 0). In the presence of phase-mismatch (Δκ≠0), the achievable peak efficiency decreases with increasing Δκ. Another obvious feature is that the gain bandwidth of OPCPA is limited within |Δκ|<1 regardless of the seeding conditions. These results are consistent with Eq. (5) which indicates that the small-signal gain only holds when |Δκ|<1. Note that, different signal spectral components within |Δκ|<1 reach their peak efficiencies at different crystal lengths because of the Δκ-dependent parametric gain. Therefore, it is hard to optimally amplify all the signal components at a specific crystal length. A higher signal seeding can be helpful for increasing the gain bandwidth, because it shortens the crystal L needed to reach the peak conversion and thus releases the limitation on Δκ (|ΔκL|≤π). In practice, increasing the pump intensity will have the same effect as increasing the signal seeding, both are beneficial to the achievement of peak conversion using a shorter crystal.
Fig. 4 Conversion efficiency as a function of normalized phase mismatch and the normalized propagation distance. (a) and (b) represent the OPCPA (α = 0) cases with Ip0/Is0 = 106 and 103, respectively. (c) and (d) represent the QPCPA (α = Γ) cases with Ip0/Is0 = 106 and 103 respectively. In each case, Γ is fixed at 1000 m−1, and the efficiency is normalized to the theoretically maximum efficiency.
On the other hand, the simulation results for QPCPA (α = Γ) are quite different from those for OPCPA, as shown in Figs. 4(c) and 4(d). Firstly, there is no oscillation on signal efficiency in QPCPA. As described in Section 3, the self-locked QPCPA phase shuts off the back conversion and thus ensures a unidirectional signal increase until the theoretical maximum efficiency is reached, regardless of the phase-mismatch. Once the signal efficiency reaches the maximum, it will not change in the subsequent propagation within the nonlinear crystal. Secondly, the gain bandwidth of QPCPA can break through the limit of |Δκ| = 1 and keep growing with the propagation in nonlinear crystal. This result verifies the analysis for Eq. (4) in Section 2. Different from the OPCPA case where a shorter crystal benefits a broader gain bandwidth, a longer crystal is desirable contrarily in QPCPA. In addition, the method that can increase the conversion rate in OPCPA, e.g., increasing the signal seeding or using a stronger pump, still works on enhancing the gain bandwidth in QPCPA.
At this point, we emphasize that the essential difference between OPCPA and QPCPA can be manifested in a best way via the contour map of conversion efficiency in the domain of phase mismatch (Δκ) and crystal length (z). In OPCPA, signal amplifications only take place in several small and discrete areas in the Δκ−z domain [Figs. 4(a) and (b)], which reveals the bandwidth limitation and small tolerances in both the crystal length and pump intensity. In QPCPA, however, high-efficiency signal amplifications can take place in a large and single area in the Δκ−z domain [Figs. 4(c) and (d)], which clearly demonstrates that QPCPA is noncritical to the phase-mismatch, crystal length and pump intensity. Interestingly, Figs. 4(c) and (d) also suggest that the phase-mismatch tolerance and hence the bandwidth are linearly increased with the crystal length in QPCPA.
There is an inherent trade-off between the conversion efficiency and gain bandwidth in OPCPA due to the Δκ-dependent parametric gain. As shown by the blue curves in Fig. 5(a), it is impossible to simultaneously maximize the efficiency and bandwidth through adjusting the crystal length. For instance, the highest efficiency is achieved at z/LNL = 5 where the gain bandwidth is not the broadest. On one hand, the gain bandwidth can be well broadened by using a slightly longer crystal (e.g., z/LNL = 6), with a sacrifice of back-conversion at the center of spectrum which inevitably degrades the conversion efficiency. On the other hand, although the optimal efficiency for Δκ = 0 can be obtained at several specific crystal lengths as shown in Fig. 4(a), the gain bandwidth becomes narrower when a longer crystal is used [Fig. 5(b)]. Therefore, compromise between the efficiency and bandwidth has become an important design consideration in OPCPA [27,28]. The common practice is to optimize the efficiency-bandwidth product which determines the achievable peak power.
Fig. 5 (a) Normalized efficiency versus normalized phase mismatch Δκ at different crystal lengths for both QPCPA (α = Γ, red curves) and OPCPA (α = 0, blue curves). The normalized crystal length z/LNL has been marked near the corresponding curve. (b) Normalized efficiency versus Δκ at three crystal lengths that can support complete conversion for Δκ = 0 in OPCPA (α = 0). The plots for OPCPA and QPCPA in this figure are based on the results in Figs. 4(a) and 4(b), respectively. In each case, Γ = 1000 m−1 and Ip0/Is0 = 106.
QPCPA can break the trade-off between the conversion efficiency and gain bandwidth as shown by the red curves in Fig. 5(a). QPCPA exhibits a flat gain spectrum with the bandwidth increasing with the increase of crystal length. More importantly, all the spectral components within the gain spectrum can reach the theoretical maximum efficiency. The results in Fig. 5-show that the gain bandwidth of QPCPA is several times broader than that of OPCPA. The gain bandwidth of QPCPA is limited in principle only by the available crystal length if the seed conditions are fixed.
The simulation results presented in Figs. 4 and 5 suggest that QPCPA is more robust against phase-mismatch than OPCPA. Such robustness can bring about various advantages to QPCPA, one of which is the extension of gain bandwidth as shown above. Besides, QPCPA is expected to be insensitive to the parameters determining phase-matching, such as the crystal temperature and pump beam pointing. Compared to OPCPA, QPCPA’s robustness against the change of pump parameters originates from the large tolerance to phase mismatch. Different from CPA, however, QPCPA still poses high requirement on pump laser. In short, QPCPA outperforms OPCPA in terms of gain bandwidth, conversion efficiency and robustness, all of which benefit the amplification of few-cycle pulse towards high peak power.
5. Simulation on a few-cycle QPCPA system
To show the potential capabilities of QPCPA, we design and simulate a QPCPA system that is pumped by the second harmonic of a Yb:YAG thin-disk pump lasers and seeded by a broadband pulse from a Ti:sapphire ultrafast laser. Here we use the Sm:YCOB crystal with a Sm3+ doping level of 30% in our simulations as our previous experimental work [21]. In our simulations, a 40-mm-long Sm:YCOB crystal is used, which is available in practice. The refractive indices of this crystal have been precisely measured by using the minimum deviation method at a temperature of 300 K, which fit the following Sellmeier equations,
where λ is the wavelength expressed in micrometers. According to the Sellmeier equations, for a QPCPA system pumped by a 515 nm pulses (i.e., the second-harmonic of a Yb:YAG thin-disk laser), the noncollinear broadband “magic” phase-matching corresponds to a signal wavelength at 788.1 nm [29]. Under this phase-matching geometry, the phase-matching bandwidth for signal is governed by the third-order dispersion (TOD).A single-stage QPCPA system pumped by a Gaussian pump pulse is simulated. The seed signal has a bandwidth of 180 nm (Full width at half maximum, FWHM) centered at 788.1 nm, which is the typical spectrum from a broadband Ti:sapphire laser oscillator. This broadband signal pulse is chirped to a FWHM duration of 10 ps, same as the pump duration. A pump intensity of 50 GW/cm2 is used (corresponds to Γ~1450 m−1), which is well below the damage threshold of Sm:YCOB in the picosecond regime. The peak intensity of the seeding chirped signal is set to be 50 MW/cm2 (i.e., Ip0/Is0 = 103).
Firstly, we simulate the case with a constant absorption of α = 3/cm across the idler spectrum. The calculated results are summarized in Fig. 6. As shown in Fig. 6(b), the signal efficiency keeps an upward trend with the propagation in crystal although some oscillations present initially. The efficiency can reach as high as 62%, very close to the theoretical limit (65%). The initial oscillations are attributed to the limited idler absorption comparing to the large nonlinear drive Γ. From the view of spectrum [inset in Fig. 6(b)], initially it evolves similar to that of a OPCPA as shown in Fig. 4(b), then its oscillation behavior is gradually suppressed by the depletion of idler, and finally it enters a stable region in which all the spectral components are efficiently amplified without back conversion. The pump pulse is significantly depleted by ~95% and the chirped signal pulse is amplified to ~32.7 GW/cm2, as shown by the blue and red curves in Fig. 6(a).
Fig. 6 Simulation results for a QPCPA system with a uniform idler absorption of α = 3 cm−1. (a) Temporal profiles of input pump (black curve), residual pump after amplification (blue curve), and amplified chirped-signal (red curve). (b) Evolution of pump-to-signal conversion efficiency within the crystal. The dashed line shows the theoretical efficiency limit. Inset shows the spectrum evolution within the crystal. (c) Input (black curve) and output (red curve) signal spectrum, and the nonlinear spectral phase imposed on the signal (blue solid curve). The blue dashed curve represents the residual nonlinear spectral phase after compensating TOD by 1.95 × 103 fs3. (d) Dechirped pulse without (black curve) and with (blue curve) TOD compensation. The Red curve represents the Fourier transform limited pulse (red curve).
The spectral components spanning from 660 nm to 980 nm are effectively amplified by the QPCPA [red curve in Fig. 6(c)]. Such a broad spectrum may support a Fourier-transform-limited pulse with duration of 6.6 fs (sub-three cycles), as shown by the red curve in Fig. 6(d). Similar to OPCPA, QPCPA also imposes a nonlinear spectral phase on the signal pulse [blue solid curve in Fig. 6(c)], which will affect the pulse compression [black curve in Fig. 6(d)]. This nonlinear spectral phase exhibits an obvious feature of TOD. Indeed, after subtracting a TOD phase by 1.95 × 103 fs3, this nonlinear spectral phase can be well compensated [blue dashed curve in Fig. 6(c)], resulting in a compressed pulse with duration of 7.6 fs [blue curve in Fig. 6(d)]. In practice, by using an acousto-optic programmable dispersive filter or a liquid-crystal spatial light modulator to further compensate the residual higher-order dispersions, the amplified pulse may be well compressed to its Fourier limit. The results presented in Fig. 6 manifest that QPCPA can support sub-three-cycle pulse amplification with a close-to-optimal efficiency.
Next, we simulate a more practical case with a non-uniform absorption across the idler spectrum. A Gaussian absorption profile is assumed, with a peak absorption of α = 3/cm at 1486 nm corresponding to the magic phase-matching wavelength [Fig. 7(a)]. As shown in Figs. 7(b)–7(d), the simulation results of the signal efficiency, bandwidth, nonlinear spectral phase and the compressed pulse duration remain fairly consistent with those obtained in the situation with constant absorption. This is reasonable, because different spectral components experience different pump intensities [Fig. 6(a)]. As implied by Eq. (4), a lower pump intensity will relax the requirement for idler absorption value. The robustness of QPCPA against the idler absorption non-uniformity brings about great convenience for the practical implement of QPCPA
Fig. 7 Simulation results for a QPCPA system with a non-uniform idler absorption. (a) Gaussian absorption spectrum used in the simulation. (b) Evolution of pump-to-signal conversion efficiency within the crystal. Inset shows the spectrum evolution within the crystal. (c) Output signal spectra (black curve), and the nonlinear spectral phase imposed on the signal (blue curve). (d) Dechirped pulse (black curve) and the Fourier transform limited pulse (red curve). Other parameters used in calculations are same with those for Fig. 6.
As shown by Figs. 6 and 7, QPCPA can indeed support efficient and broadband signal amplification thanks to the obstruction of back conversion. Thermal effect induced by the idler absorption might be a problem in practice of QPCPA [30]. The increase of crystal temperature may change the refractive index of the nonlinear crystal and consequently the phase-matching condition, which can degrade the efficiency and bandwidth. To explore the temperature tolerance of QPCPA, temperature-dependent Sellmeier equations of the crystal are needed. As such Sellmeier equations of Sm:YCOB crystal are not available, we use the thermo-optic dispersion formulas of YCOB crystal instead [31]. We assume this substitution is acceptable because of a relatively low doping concentration of Sm3+. The simulation results presented in Fig. 8 show that the efficiency and bandwidth don’t vary too much comparing to those in Fig. 6, even with a crystal temperature increase of 200 K. This temperature tolerance is much larger than that in typical OPCPA systems, which is a direct result of the robustness of QPCPA to phase-mismatch.
Fig. 8 Performances of QPCPA when the temperatures deviate from the reference temperature for perfect phase matching. (a) Efficiency evolutions within the crystal under a temperature increase of 100 K (black curve) and 200 K (red curve), respectively. (b) Output signal spectra (solid curve) and nonlinear spectral phases (dashed curve) under a temperature increase of 100 K (black curve) and 200 K (red curve), respectively. Other parameters used in calculations are same with those for Fig. 6.
Finally, we discuss the average power scalability of QPCPA. The heating of the crystal is dominant by the absorption of the idler wave, so only the idler absorption is taken into account. For simplicity, we assume the idler beam has a flat-top spatial profile with a diameter of 2w and a Gaussian temporal profile with a FWHM duration of τ. The heat-flow along the longitudinal direction is neglected. The cylindrical nonlinear crystal is in thermal contact with a heat sink of temperature 300 K which is the reference temperature set for phase-matching. Owing to the absorption of idler power, the nonlinear crystal is heated with an effective heat-density [32]:
where I0 is the maximum intensity of idler, α is the absorption coefficient and f is the repetition rate of the laser. The heat distribution inside the Sm:YCOB crystal with a radius of r0 (r0> w) can be calculated as [32], where κ = 2.0 W m−1 K−1 is the heat conductivity of Sm:YCOB crystal [33]. In the simulation case of Fig. 6 with a pump intensity of 50 GW/cm2, the idler intensity within the crystal is shown in Fig. 9(a), which is high only within the initial part of the crystal. Here, the pump laser specifications used in the simulations are pulse energy of 400 mJ, pulse duration of 10 ps, repetition rate of 100 Hz, and beam radius of 0.5 cm (corresponding to pump intensity of 50 GW/cm2 and pump average power of 40 W). In this QPCPA, the temperature distributions at three different locations in the crystal are shown by the black curves in Fig. 9(b). The idler intensity of ~14 GW/cm2 at about 3.2 cm leads to a temperature increase of 190 K, which is still acceptable. At other locations, the temperature increases are well below the tolerance of 200 K. Therefore, in a QPCPA system with above pump parameters, the repetition rate as high as 100 Hz is acceptable and will not degrade the amplification performance, which may give a high signal average-power of 24 W. As suggested by Eq. (12), similar signal average power can be obtained at a higher repetition rate of 1 kHz if a shorter pump duration of 1 ps and the same pump intensity of 50 GW/cm2 are applied (in this case, the pump energy should drop to 40 mJ and its average power is still 40 W). Under the same pump energy, we also simulate the temperature distributions at peak idler intensities when the beam radius is 0.4 cm (red curve in Fig. 9) and 0.6 cm (blue curve in Fig. 9), respectively. The results shown in Fig. 9 imply that using a large pump beam can reduce the temperature gradient transversely, which may enable the QPCPA operating at a higher average power. For the QPCPA with a beam radius of 0.5 cm, the amplified signal pulse has an energy of 240 mJ corresponding to a peak power of 37 TW if it is compressed into the Fourier limit duration of 6.6 fs. For a low repetition rate or single-shot mode, the thermal effect of QPCPA is negligible, and its peak power can be scaled to hundreds of TW or even higher level. Moreover, Sm:YCOB crystal with a size over 4 inches is available [33], which may support the high-peak-power operation of QPCPA.Fig. 9 Radial temperature change in the Sm:YCOB crystal. (a) Evolution of idler intensity within the crystal when the beam radius w is 0.4 cm (red curve), 0.5 cm (black curve), and 0.6 cm (blue curve), respectively. A, B, and C indicate three different crystal locations when w = 0.5 cm, corresponding to the idler intensity of 14, 7, and 1.4 GW/cm2, respectively. D and E indicate the peak idler intensities when w = 0.4 and 0.6 cm, respectively. (b) Radial temperature changes when QPCPA operates in a repetition rate of f = 100 Hz. The parameters used in the simulations are r0 = 0.6 cm, τ = 10 ps, α = 3 cm−1, κ = 2.0 W m−1 K−1. The pump energy is fixed at 400 mJ. Other parameters are same with those of Fig. 6.
Here, we discuss the possibility to reduce the thermal effect encountered in QPCPA. As pointed out in this paper, the key to enable the QPCPA scheme is the depletion of the idler wave. Absorption by doping rare-earth ions into the nonlinear crystal is just one of the methods for depleting idler waves. This method adds thermal load to QPCPA and may hamper its operation at higher average power levels. Instead of material absorption, it is also possible to use the non-thermal methods to deplete the idler wave. On one hand, the idler wave can be spatially separated from the three-wave interaction region by using a noncollinear phase-matching configuration [34]. On the other hand, the idler wave can be consumed by integrating another nonlinear frequency-conversion process within the same crystal, such as second harmonic generation [35] and difference frequency generation [36]. These methods, free of thermal effect, may reduce the thermal load of QPCPA to the level of OPCPA. Furthermore, in a QPCPA with the non-thermal idler depletion, the robustness against phase mismatch will enable QPCPA working at a higher average power than OPCPA.
6. Discussion and conclusion
We have presented detailed simulations on QPCPA, which manifests the advantages of QPCPA over OPCPA in terms of the bandwidth, efficiency and robustness. The underlying fundamental mechanism of inhibiting back conversion has been revealed in the view of phase. The effect of self-locked phase in QPCPA can boost the conversion efficiency of a very broadband signal approaching to the quantum limit, while the high efficiency OPCPA is achievable only under the condition of narrowband signal. QPCPA cannot only overcome the parametric bandwidth limitation of OPCPA but also break through the trade-off between efficiency and bandwidth encountered in OPCPA. The robustness of QPCPA against phase-mismatch makes it insensitive to the factors determining phase-mismatch and allows more stable signal output.
Our simulation results have demonstrated that a long-crystal-based QPCPA can operate not only in the few-cycle regime, but also in an efficient and robust way. The QPCPA can allow complete pump-depletion over a signal bandwidth of more than 180 nm in few-centimeters-long nonlinear crystals by properly designing the absorption spectrum. The average power level of a few-cycle QPCPA may be comparable to that of the state-of-the-art few-cycle OPCPA, while its peak power can be much higher. QPCPA is a promising technique to generate intense few-cycle pulses that can drive various high-field physics experiments, such as high-harmonic generation and laser-driven particle acceleration. A few-cycle QPCPA system has been designed and analyzed, which verifies the feasibility of the technique in generating ultrashort high peak-power pulses.
Finally, we simply discuss the effect of the profile of idler absorption spectrum on the conversion efficiency of QPCPA, which is necessary for the practical implement of QPCPA. The measured absorption spectrum of the Sm:YCOB crystal is composed with several absorption peaks [Fig. 10(a)], which is quite different from the smooth absorption profile adopted in Fig. 7. However, our simulation shows that the smoothness of idler absorption spectrum has little influence on the attainable conversion efficiency, as shown in Fig. 10(b). In addition, the width of absorption spectrum does not affect the efficiency significantly. Because back conversion occurs more significantly in the central peak parts of signal and idler spectra, the crystal absorption spectrum matching with the idler spectral peak but with a width slightly narrower than the idler bandwidth can also effectively suppress the back-conversion process. Therefore, it is reasonable to simulate a practical QPCPA system by using Gaussian-profile absorption spectrum. Based on the theoretical results presented in this paper, an experimental demonstration of a high-power few-cycle QPCPA system is under the consideration, and a qualified Sm:YCOB crystal with a large thickness of >50 mm has already been grown in the laboratory.
Fig. 10 The effect of idler-absorption profile on the conversion efficiency of QPCPA. (a) The practical absorption spectrum of Sm:YCOB crystal (black curve) and two Gaussian-profile absorption spectra (blue and red curves) adopted in the simulation. (b) Evolution of pump-to-signal conversion efficiency within the crystal for the three idler absorption profiles shown in (a). The horizontal dashed line in (b) represents the theoretical efficiency limit. The pump pulse at 515 nm has an intensity of 50 GW/cm2. The seed pulse at 820 nm has a bandwidth of 100 nm and an intensity of 50 MW/cm2.
Funding
Science and Technology Commission of Shanghai (17YF1409100, 17ZR1414000, 15XD1502100); National Natural Science Foundation of China (NSFC) (61705128, 61727820); National Basic Research Program of China (2013CBA01505).
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