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Abstract
Quasi-parametric amplification (QPA), a variant of optical parametric amplification, can release the phase-matching requirement owing to the introduction of idler dissipation, and thus may support ultrabroad bandwidth. Here we establish the gain-dispersion equation for QPA, which reveals the interplay of signal gain, idler dissipation and phase mismatch. The idler dissipation dramatically enhances the gain bandwidth, which breaks the limit set by phase matching. We theoretically demonstrate that QPA with strong dissipation allows high-efficiency few-cycle pulse amplification in those nonlinear crystals without a magic phase-matching solution.
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1. Introduction
The past three decades have witnessed the tremendous development of ultrafast and ultra-intense lasers [1,2]. In recent years, there is a growing interest in the generation and applications of super-powerful ultrashort pulses with duration down to a few optical cycles [3–6]. Nowadays, high-power few-cycle pulses are mainly produced based on optical parametric chirped-pulse amplification (OPCPA) [7–10]. Nevertheless, optical parametric amplification (OPA) processes, including but not limited to OPCPA, has restricted bandwidth since the medium dispersion induces a frequency-dependent phase mismatch over the ultra-broadband spectrum of signal. Ultrabroad bandwidth can only be achieved in very thin nonlinear crystals (∼1 mm or less) by complicated noncollinear ‘magic’ phase-matching geometry [11,12] at high pump intensities close to the damage threshold [13]. The ‘magic’ phase-matching solution (also termed as spectrally noncritical phase-matching solution) exists only at specific wavelengths for certain crystals. Moreover, there exists an intrinsic trade-off between the gain bandwidth and conversion efficiency in OPA processes [14]. The frequency-dependent phase mismatch leads to a frequency-dependent gain, and thus a time-dependent pump depletion. As a consequence, the amplification at some space and time points falls into the back-conversion of energy from the signal to the pump earlier than other space and time points [15]. The nonuniform back-conversion leads to a reduction in the maximum conversion efficiency from pump to signal. Therefore, amplification of few-cycle pulses generally suffers from low conversion efficiency (10 ∼ 30%) [16–18], such as the amplification of a sub-9 fs, 810 nm pulse in a 1.1 mm-thick BBO crystal has a conversion efficiency no higher than 30% (corresponding to a pump-depletion rate < 50%) [18].
For OPA processes, the gain bandwidth is dominated by the phase-matching bandwidth. Several concepts have been proposed to optimize the phase matching over broadband signal spectrum. A straightforward method is to adjust the phase-matching condition for different spectral regions of the signal separately. Such techniques include different designs. i) multiple-pumping OPA [19–22], wherein the wavelength, wave-vector as well as the timing of each pump laser can be arranged independently to optimize the amplification of different spectral regions of signal. ii) Frequency-domain OPA (FOPA) [23,24], wherein the ultra-broadband signal is dispersed spatially via a 4-f arrangement, so that different spectral components can be injected into different birefringent phase-matching crystals with suited cutting angles. iii) Achromatic phase matching (APM) [25,26], wherein the angular dispersion of signal pulse is precisely manipulated thus the different spectral components propagate at their different phase-matching angles. The enhancement of gain bandwidth is achieved at the expense of system complexity. For either FOPA or APM schemes, several optics are used for spatiotemporal coupling and decoupling, which introduces extra energy loss and consequently reduces the conversion efficiency. FOPA has been demonstrated to deliver 1.43 mJ, 1.8 µm two-optical-cycle pulses with 12.8-mJ pump, corresponding to a pump-to-signal conversion efficiency of ∼14% [24]. APM technique has been demonstrated to generate 65-µJ, 5.8-fs amplified pulse with 1.2-mJ pump, at a pump-to-signal conversion efficiency of ∼5% [27]. A less complicated approach for achieving ultrabroad phase matching is based on quasi-phase-matching (QPM) with specialized grating designs, such as chirped QPM grating [28] and adiabatic QPM grating [29]. However, the very limited aperture of QPM crystals restricts their applications in the production of powerful few-optical-cycle lasers.
In recent years, quasi-parametric amplification (QPA) with idler dissipation has been proposed as an alternative to OPA, with the distinct advantages of unidirectional energy transfer and thereby high-efficiency as well as robustness [30,31]. In the first experimental demonstration of QPA based on a Sm3+-ion-doped yttrium calcium oxyborate crystal (i.e., Sm:YCOB) [30], the energy back-conversion from the signal to the pump is suppressed via distributed idler dissipation, which is achieved by the frequency-dependent absorption of the rare-earth ion Sm3+. Based on a 30%-doped Sm:YCOB, a pump depletion as high as 85% has been demonstrated using a spatiotemporal Gaussian pump laser [32]. In addition to rare-earth ion doping, idler dissipation can be introduced into OPA processes via other approaches, such as single waveguide with Bragg grating [33], asymmetric directional couplers with engineered spectral absorption [34,35], and frequency converting the idler wave [36,37]. These related works have demonstrated the high conversion efficiency of QPA and theoretically the narrowband signal amplification in the absence of perfect phase matching [35]. Nevertheless, the bandwidth characteristic of QPA remains unclear.
In this work, we study the bandwidth characteristic of QPA, in comparison with that of conventional OPA without idler dissipation. We deduce a gain-dispersion equation to quantify the complicated interplay of the signal gain, idler dissipation and phase mismatch, three main parameters that responsible for gain bandwidth. This equation reveals that, by introducing strong idler dissipation, the acceptance phase mismatch of QPA can be substantially expanded to extreme phase-mismatch region. Based on both analytical derivations and numerical simulations, we further demonstrate that QPA enables ultrabroad bandwidth beyond the phase-matching limit as well as breaks the trade-off between bandwidth and efficiency inherent to conventional OPAs. Detailed simulations of a chirped-pulse QPA system is presented, which supports the amplification of few-cycle pulses with conversion efficiency approaching to the quantum efficiency limit.
2. Gain-dispersion equation for QPA in the small-signal regime
Efficient energy conversion from the pump to the signal necessitates phase matching between the interacting waves. Before deriving the gain characteristics of QPA, we first consider OPA of a seed pulse with spectrum centered at signal frequency ωs and a pump pulse with center frequency ωp. If the idler is unseeded, prior to significant pump depletion, the signal gain follows the relation G = Is(z)/Is(0) = 1 + [Γ2/g2]sinh2(gz), which approximately reduces to G = exp(2gz)/4 in the regime of large gain (Γz >> 1). Γ is known as the nonlinear drive strength [38] satisfying Γ2 = 2ωsωideff2Ip/nsninpɛ0c3, and g is known as the parametric gain coefficient. This gain coefficient is determined collectively by the nonlinear drive strength and the phase mismatch among the three interacting waves (Δk = kp − ks − ki), according to the gain-dispersion equation
The presence of phase mismatch (Δk ≠ 0) leads to g < Γ. Once the phase mismatch is greater than twice the nonlinear drive strength, i.e., |Δk| > 2Γ, the gain coefficient g switches from a real value to a pure imaginary one, thus the power of signal oscillates along the nonlinear crystal without amplification. Therefore, |ΔkcOPA| = 2Γ is defined as the critical phase mismatch for OPA.
For QPA with distributed idler dissipation, the signal gain is determined not only by the nonlinear drive strength and phase mismatch, but also by the idler dissipation. To characterize the signal gain of QPA, we next derive the gain-dispersion equation in the presence of idler dispersion. Under slowly varying envelope approximations, the QPA processes (with idler dissipation coefficient α) can be described by the following nonlinear coupled-wave equations [39]
The solution of Eq. (3) can be expressed as a superposition of two exponential functions exp(iΔkz/2 + λ±z) with a pair of complex eigenvalues as
The first term of Eq. (4) illustrates the interplay among the phase mismatch Δk, idler dissipation α, and nonlinear drive Γ, while the term −α/4 accounts for the induced dissipation in the signal laser. It is worth noting that for conventional OPA (α = 0), the first term in Eq. (4) is either real or purely imaginary. But for QPA, attributed to the non-zero idler dissipation α, the first term becomes a complex value, whose modulus ρ and argument θ can be calculated as
The real part of the complex eigenvalue λ+ (λ+ = iλ + g) that represents the gain characteristic of QPA can thus be deduced as
By setting αN = 0 in Eqs. (5) and (7), g reduces into the parametric gain coefficient of OPA, as has been described in Eq. (1). It is worth noting that in OPA (αN = 0), g would become a pure imaginary value at large phase mismatch (|Δk| > |ΔkcOPA|). But in QPA with non-zero idler dissipation (αN ≠ 0), surprisingly, there is always g > 0, since
The remain positive g indicates that as long as there is idler dissipation, no matter how weak it is, the signal can be amplified even in the region of extreme phase mismatch (|Δk| >> |ΔkcOPA|). Hence, the limit of phase-matching condition no longer exists in QPA.
Specifically, due to the presence of idler dissipation, the gain-dispersion equation should be rewritten as
To quantitatively determine the phase-mismatch insensitivity, we calculate the variation of signal gain G against the phase mismatch ΔkN for QPA with different strength of idler dissipation αN. We quantify the gain reduction caused by phase mismatch by the gain ratio
Given the same pump intensity, although the presence of idler dissipation induces a reduction in the gain coefficient, it promises a broader gain-bandwidth. To illustrate the signal gain bandwidth regulated by idler dissipation, Fig. 1(a) and (b) present the variation of signal gain ratio R against the phase mismatch ΔkN and idler dissipation αN, calculated for the normalized gain parameter C = 1 and 3, respectively. The red dashed lines highlight the points that the signal gain falls by half, that is R(αN, δkN) = 1/2. We define the phase mismatch at R(αN, δkN) = 1/2 as the acceptance phase mismatch of QPA, denoted by δkN. One noteworthy feature is that δkN monotonically increases with αN, indicating that strong idler dissipation helps eliminate the phase-matching limit of OPA and expand the bandwidth. Specifically, for OPA with αN = 0, |δkN| = 1.5 when C = 1, as shown by the blue line in Fig. 1(c), while for QPA with αN = 20, the acceptance phase mismatch expands by five times to |δkN| = 7.3 as shown by the black line in Fig. 1(c). For the case of larger gain parameter (C = 3), a stronger idler dissipation (αN = 30) is required to expand the acceptance phase mismatch by 5 times compared to that in OPA.
Fig. 1. (a)(c) Two-dimensional plot of the gain ratio R as a function of normalized phase mismatch ΔkN and idler dissipation coefficient αN. The red dashed lines indicate the precise locations of R(αN, δkN) = 1/2. The blue dotted lines plot δkN(αN) calculated based on Eq. (14). Gain ratio R as a function of ΔkN when αN = 0 (blue), 10 (red) and 20 (black) in (c) and αN=0 (blue), 30 (red) and 60 (black) in (d).
We move to derive an approximate analytical formula to estimate the acceptance phase mismatch of QPA. Under the approximation of strong idler dissipation (α > Γ), the modulus ρ in Eq. (5) can be approximated into
wherein p = (αN/4)2, q = (ΔkN/2)2, and strong idler dissipation ensures that (1 + p + q)2 >> 4q. Consequently, the gain coefficient of QPA given by Eq. (6) becomesSubstitution of Eq. (13) into Eq. (11) leads to the approximate analytical solution of the acceptance phase mismatch (defined by R(αN, δkN) = 1/2) solution,
Table 1. Acceptance phase mismatch versus the idler dissipation αN = α/Γ and the normalized gain parameter C = gM z. With a given nonlinear drive strength Γ = 20 cm−1, αN = 10 corresponds to a dissipation coefficient α = 200 cm−1 and a maximum gain coefficient gM = Γ{[1 + (αN/4)2]1/2 − αN/4} = 4 cm−1. In this case, the normalized gain parameter C = 1 corresponds to a crystal length of z = 2.5 mm.
3. Gain bandwidth of QPA in the saturation regime
The analytical model introduced in the previous section allowed us to characterize the gain bandwidth of QPA in the regime of small-signal amplification. In order to interpret the phase-mismatch insensitivity of QPA in the saturation regime of amplification, we numerically integrated the nonlinear coupled equations [Eq. (2)]. Assuming the seeding ratio Is0/Ip0 = 10−3 and nonlinear drive strength Γ = 20 cm−1, and OPA processes with same inputs are simulated in comparison. The signal intensity Is increases proportional to the signal gain G along the crystal, i.e., Is/Is0 = G. The gain ratio is defined as R(ΔkN) = Is(ΔkN)/Is(0).
The variation of the gain ratio on the phase mismatch is calculated for OPA (αN = 0) and QPA (αN = 10), as respectively shown in Fig. 2(a) and 2(b). The phase mismatch ΔkN on the ordinate correspondingly relates to the signal wavelength. A wider distribution of positive R(ΔkN) across the ΔkN means a broader gain bandwidth. It should be noticed that R(ΔkN) = 1 at z = 0 reflects the input condition. The gain ratio evolution along the crystal confirms the phase-mismatch insensitivity of QPA predicted in section 2 and presents additional behavior pertaining to the pump-depletion regime. At the beginning of amplification (in the small-signal regime), gain-narrowing effect occurs for both OPA and QPA due to the progressively lower gain experienced by the wings of the signal spectrum caused by phase mismatch. Then, as the amplification proceeds to the weak saturation regime (where pump depletion becomes moderate but energy back-conversion has not occurred), the gain bandwidth turns into increase for both OPA and QPA due to saturation of gain at the center of the pulse and preferential amplification at the wings. Finally, when the amplification further proceeds into deep saturation regime (where pump depletion becomes significant), the bandwidth of OPA stops increasing and the central part of signal spectrum sinks due to the occurrence of energy back-conversion from signal to pump. While for QPA [Fig. 2(b)], the bandwidth continues to increase with the nonlinear interaction length z.
Fig. 2. The evolutions of gain ratio R(ΔkN) along the crystal. (a) correspond to R(ΔkN) for OPA (αN = 0) and (b) to QPA (αN = 10). Propagation distance z is normalized to the nonlinear length defined by LNL = π/2Γ. The gray dashed lines mark the position where R(δkN) = 1/2.
Quantitatively, in the regime of small-signal amplification, the presence of idler dissipation of αN = 10 enlarges the acceptance phase mismatch by ∼ 2 times compared to that in OPA, as listed in Table. 1 (C = 2). In the regime of saturation amplification, the gain at the center of signal spectrum saturates without back-conversion in QPA, and the amplification at the spectral wings is always dominated by a positive gain coefficient g, as has been indicated in Eq. (7). In consequence, this saturation behavior of QPA further enlarges the acceptance phase mismatch to 4 times of that in OPA.
4. Ultra-broadband and high-efficiency QPCPA
The analytical model as well as numerical simulations in the previous sections exclude the medium dispersion of the nonlinear crystal, allowing isolation of the dispersion dependence of the signal bandwidth. In this section, we numerically simulate a practical chirped-pulse QPA (QPCPA) system to demonstrate the ability of QPCPA in the amplification of few-cycle pulses, wherein media dispersion up to the 3rd-order are considered. We consider a QPCPA system pumped by the second harmonic of a Yb:YAG thin-disk picosecond pump laser (centered at 532 nm) and seeded by a few-cycle femtosecond signal pulse centered at 810 nm. Comparative numerical simulations are carried out for OPCPA and QPCPA by assuming the same pump (a 4th-order super-Gaussian pulse with duration 5 ps and intensity Ip0 = 40 GW/cm2), the same seed signal (a Gaussian chirped pulse with duration 3 ps, bandwidth 350 nm and seeding ratio Is0/Ip0 = 10−3), as well as the nonlinear crystal with the same medium dispersion (the Sellmeier function of Sm:YCOB [40] and type-I phase-matching geometry are assumed). The effective nonlinear coefficient is uniformly set as deff = 1.39 pm/V. In such a nonlinear crystal, the pump intensity Ip0 = 40 GW/cm2 leads to a nonlinear drive strength of Γ = 20 cm−1. The simulations of OPCPA and QPCPA are distinguished by setting αN = 0 and αN = 10 in Eq. (2), respectively. The energy conversion efficiency is defined as η = (Es − Es0) / Ep0, where Es, Es0 and Ep0 denote the pulse energy of the amplified signal, seed signal and incident pump, respectively. Its theoretical limit value is 65%, which means that all the pump energy is converted into signal energy and idler energy.
The Fig. 3(a) and 3(b) present the evolution of signal spectrum during amplification in OPCPA and QPCPA, respectively. The full width at half-maximum (FWHM) bandwidth of the amplified signal is depicted in Fig. 3(c) and 3(d) as the blue line, wherein the corresponding energy conversion efficiencyη is plotted as the red line. For OPCPA, η reaches the maximum (20%) at a crystal length of z = 2.5 mm [Fig. 3(c)]. This maximal efficiency point also corresponds to the maximum of the efficiency-bandwidth product. At this point, the center spectral components of signal falls into back conversion, as shown by the penultimate curve in Fig. 3(a). It’s for this reason that the energy conversion efficiency falls into reduction afterwards. Figure 3(c) explicitly shows that when OPCPA proceeds into the regime of saturated amplification (η > 1%), the conversion efficiency begins to increase quickly but the output signal bandwidth decreases, this phenomenon is known as the trade-off between the conversion efficiency and gain bandwidth inherent to OPA and OPCPA. For QPCPA, this trade-off no longer exists. As shown in Fig. 3(d), the energy conversion efficiency (red line) and signal bandwidth (blue line) increase simultaneously in the regime of saturation amplification. Ultimately, all the spectral components of signal can reach their respective maximum without back conversion. At the exit plane of the crystal (z = 20 mm), η reaches 53% with an output bandwidth of Δλs ∼320 nm (centered at λs = 810 nm), corresponding to a fractional bandwidth as large as Δλs /λs = 40%.
Fig. 3. Signal spectrum evolution along the crystal in (a) OPCPA with αN = 0 and (b) QPCPA with αN = 10. Evolution of energy conversion efficiency η (red line) and bandwidth (blue line) in (c) OPCPA with αN = 0 and (d) QPCPA with αN = 10.
Firstly, we simulate the case of OPCPA. The results are summarized in Fig. 4(a)-(c). Figure 4(a) shows the temporal profile of the amplified chirped signal pulse output from OPCPA at the maximum conversion efficiency point (i.e., the case of z = 2.5 mm marked in Fig. 3(c)). Only the central portion of signal within a duration of 1.5ps is amplified due to the limitation of phase-matching condition. Correspondingly, the bandwidth of the amplified signal is limited to 180 nm, as indicated by the red line in Fig. 4(b). The blue line in Fig. 4(b) plots the spectral phase of the amplified signal pulse after compensating the 2nd and 3rd-order spectral phase. The compressed pulse has a duration of 9.6 fs, which is approximately 1.18 times the Fourier-transform-limited duration (8.1 fs), as shown in Fig. 4(c).
Fig. 4. Comparison of output characteristics for OPCPA and QPCPA. (a)(d) The temporal profile of input/residual pump (black curves) and input/amplified chirped-signal (red curves) for OPCPA (α = 0) and QPCPA (α = 10), respectively. (b)(e) The input/amplified signal spectrum (black/red lines) and the residual nonlinear spectral phase (blue lines) after compensating the 2nd and 3rd-order spectral phase for OPCPA and QPCPA, respectively. (c)(f) The temporal profile of the amplified signal for OPCPA and QPCPA, respectively. The Fourier transform limited pulses correspond to the solid lines. The compressed pulses correspond to the dotted lines.
In comparison, Fig. 4(d)-(f) present the output characteristics of the saturated QPCPA with a crystal length of z = 20 mm (as marked in Fig. 3(d)). In time domain, the temporal duration shows the temporal profile of the amplified chirped signal pulse is as large as the pump pulse window [Fig. 4(d)], indicating no effect of gain-narrowing. We note that there exists a peak in the trailing edge of the amplified signal pulse. This peak results from the group velocity dispersion (GVD) of the amplified signal pulse during amplification. Due to GVD, the trailing edge of signal expands in time domain. In consequence, it can extract energy from the un-depleted pump wings. For this reason, this portion of signal obtains the highest gain in total. In principle, the front edge of signal can obtain higher gain analogously due to GVD. But in practice, group velocity mismatch (GVM) leads to temporal walk-off between the pump and signal. This temporal walk-off is more significant than the temporal expansion of signal induced by GVD. Therefore, the un-depleted front edge of the pump pulse walks out of the signal pulse window, as shown in Fig. 4(d) by the black solid line.
In spectral domain, the amplified signal has a bandwidth as large as 320 nm [Fig. 4(e)]. Similar to OPCPA, the signal also suffers from a nonlinear spectral phase shift in QPCPA. The blue line in Fig. 4(e) plots the spectral phase of the amplified signal after compensating the 2nd and 3rd-order spectral phase. The nonlinear spectral phase induced in QPCPA is at the same level as that in OPCPA [Fig. 4(b)]. As shown in Fig. 4(f), the amplified signal pulse of QPCPA has a duration of 5.3 fs, which is approximately 1.12 times the Fourier-transform-limited duration (4.7 fs). These results indicate that the use of longer crystals does not induce more severe spectral phase distortion of the signal pulse in QPCPA.
Finally, we briefly discuss the possibility for achieving the dissipation coefficient (i.e., αN = α/Γ ≥ 10) proposed in this paper. In our previous experiments [30], borate crystal yttrium calcium oxyborate (YCOB) with Sm3+ doping was used. Nowadays, we can grow pure SmCOB crystals, wherein Y3+ is completely substituted by Sm3+. Such a pure SmCOB crystal provides an absorption as strong as α = 20 cm−1. Further enhancement of the dissipation might be achieved by using nanofabricated structures, e.g., a waveguide exhibiting a deep Bragg grating on the top [35].
5. Conclusions
In conclusion, we investigated on the optical gain bandwidth of QPA using an analytical approach and numerical simulations. The results show that the introduction of the idler dissipation along the propagation modifies the dependence of gain on phase mismatch, which leads to an increased spectral acceptance. We develop an analytical formula of gain bandwidth for QPA, which can be directly used in the design of ultra-broadband QPA or QPCPA systems. Based on numerical simulations, we demonstrate that a QPCPA system with strong idler dissipation can support the amplification of sub-two optical cycles pulse with a high energy conversion efficiency approaching the quantum efficiency limit (a pump conversion efficiency of 100%).
Funding
National Key Research and Development Program of China (2023YFA1608501); National Natural Science Foundation of China (62122049, 62375165); Science and Technology Commission of Shanghai Municipality (21QA1404600, 22JC1401900).
Acknowledgment
We thank the support from Fundamental Research Funds for the Central Universities. J. Ma would like to thank the sponsorship of the Yangyang Development Fund and the Cyrus Tang Foundation through the Tang Scholar program.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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