Periodically intensity-modulated pulses by optical parametric amplification for multicycle tunable terahertz pulse generation

: The superposition of signal and idler pulses in dual-chirped optical parametric amplification is proposed for the efficient generation of intensity-modulated pulses with periodic modulation. Both the duration and the modulation period are easily and independently adjustable. Numerical simulations for a three-stage optical parametric amplifier system predicted an efficiency as high as ~50% for about 40 mJ of output pulse energy at a wavelength of 2 µm. Sources of such intensity-modulated pulses near 1.6 µm or 2 µm wavelength, pumped by Ti:sapphire or Yb-doped lasers, can be ideally suited for intense multicycle THz pulse generation with tunable frequency and bandwidth by optical rectification for example in organic, semiconductor, or lithium niobate materials.


Introduction
Intense, nearly single-cycle terahertz (THz) pulses can be efficiently generated by table-top sources [1][2][3][4][5][6]. Such pulses can be used in materials science [7] and for the acceleration of electrons [8][9][10] and protons [11,12]. The waveguide and resonator structures proposed for electron acceleration can be more efficiently driven by multicycle THz pulses. Dielectric grating structures were proposed for laser-driven electron acceleration [13], where using multicycle THz pulses, rather than optical pulses, can be of significant advantage. Various methods were proposed for multi-cycle or shaped THz pulse generation. Intensity-modulated laser pulses were used to drive photoconductive antennas [14]  Tilted-pulse-front pumping [29] of LiNbO 3 (LN) has been providing the highest THz pulse energies and efficiencies in the low-frequency part of the THz spectrum [4,30], especially advantageous for particle acceleration, whereby limitations of this technology also became apparent [31,32]. It was demonstrated recently that semiconductor nonlinear materials can provide an efficient alternative technology with scalability to highest THz pulse energies and field strengths [33-37]. These materials need to be pumped at an infrared wavelength sufficiently long to suppress two-and three-photon absorption [35,36], ideally delivered by efficient diode-pumped solid-state lasers [38,39]. While rapidly developing, such laser sources cannot yet deliver pulses ideally suited to drive single-or multi-cycle THz pulse generation. Another, but generally less efficient possibility is to use and infrared optical parametric amplifier (OPA) as the pump source.
Here we propose an efficient method to generate high-energy periodically intensitymodulated pump pulses for multi-cycle THz pulse generation. The method is based on the interference of signal and idler pulses generated in an optical parametric amplifier (OPA). A high degree of flexibility in modulated pulse length and modulation frequency is achieved by using suitably chirped pump, signal, and idler pulses in a dual-chirped OPA (DC-OPA).
Previous studies of the DC-OPA concept were aiming at the generation of high-power ultrabroadband optical [40][41][42][43][44] or THz [43] pulses compressible to few-cycle pulse durations. Importantly, in all these cases either the signal or the idler pulses were used, rather than the superposition of both. In addition, the instantaneous frequency difference between the chirped signal and idler pulses was not constant. In contrast, our goal is to generate pulses with periodic intensity modulation by the superposition of signal and idler, which requires a constant instantaneous frequency difference between the chirped signal and idler pulses. Such periodically modulated pulses can be used to drive THz generation by optical rectification in organic, semiconductor, or lithium niobate materials.
This paper is organized as follows. In Section 2 conditions are formulated to produce the periodic intensity modulation by the superposition of the chirped signal and idler pulses. Section 3 describes the theoretical model used for numerical simulations. Section 4 presents the conceptual design study of a high-energy DC-OPA pump source. The conclusion is drawn in Section 5.

Periodically intensity-modulated pulses by DC-OPA
Multi-cycle THz pulses with constant carrier frequency can be generated by optical rectification using intensity-modulated optical pump pulses with a constant modulation period. Such a pump pulse can be generated by superposing two Fourier limited (FL) pulses with a frequency offset, for example the signal and idler pulses from an OPA ( Fig. 1(a)). More flexibility can be achieved by using a pair of chirped pulses rather than FL pulses. Both the duration of the modulated pulse and the modulation period can be set by varying the chirp and the relative delay between the two pulses. As it will be shown below, in this case the signal and idler pulses from a suitably designed DC-OPA can be used ( Fig. 1(b)). ω , and i0 ω ,

Condition for periodic modulation
In parametric amplification energy conservation requires for the frequencies of pump (p), signal (s), and idler (i) that holds. The superposition of signal and idler will have a constant intensity modulation period holds. Our first goal is to determine the relation between the chirp of the pump and the signal under the condition of Eq. (2). The electric field of a FL Gaussian pulse can be written as where 0 E is the amplitude of the electric field, Δt is the pulse duration at full width at half maximum (FWHM) of the intensity, 0 ω is the carrier angular frequency, and j is the imaginary unit. The corresponding amplitude spectrum is given by After chirping and delaying the pulse, the electric field becomes where GD is the group delay, GDD is the group delay dispersion. Possible higher order dispersion terms were neglected here for the sake of simplicity. The duration of the chirped pulse (τ ) and the chirp parameter ( β , see also Fig. 1(b)) are given by The instantaneous frequency can be calculated as In the rest of the paper, quantities referring to the pump, signal, and idler pulses will be indexed by p, s, and i, respectively. From the condition of periodic modulation, Eq. (2), with the help of Eqs. (1) and (8) In case when the FL duration of the signal pulse is shorter or only slightly longer than that of the pump, the above expression simplifies to s p 2 GDD GDD = .

Tuning the modulation period
The instantaneous frequencies of the interacting waves can be expressed from Eqs. (8), (9), and (1) in the following way: Here, Obviously, the intensity modulation frequency is determined by the offset of the signal central frequency from degeneracy, the chirp of the pump, and the temporal delay between the (chirped) pump and the (chirped) signal seed. An easy way of tuning the modulation frequency is to change the group delay sp GD . An approximate limit on the tuneability is set due to the finite pulse durations and the reduction of the OPA efficiency for large pump-signal delays. Equation (12) can be cast into a more practical form by using Eqs. (6) and (7): gives the stretching factor for the pump.
In practical situations, a jitter between the pump and the signal pulses can result in a fluctuation of the modulation frequency. Figure 2 shows the calculated relative deviation in the intensity modulation frequency caused by a 40-fs jitter between pump and signal as function of the FL pump pulse duration and the pump stretching factor. The jitter-free ( sp 0 GD = ) modulation frequency was 0 Δ / 2 1THz ω π = . For a given FL pump pulse duration, a sufficiently large stretching factor can reduce the modulation frequency fluctuation below any desired limit. For example, in case of 200-fs FL pump pulses, a stretching factor of about 40 will limit the modulation frequency fluctuation below 1%.

Stretched pulse durations
In a typical practical situation, the FL pump pulse duration is given by the available pump laser. The necessary The derivation of this formula is outlined in the Appendix. Here, is the ratio of the signal and pump FL pulse durations. μ   is best. In this range, the pump stretching factor N can be freely varied while M remains nearly unchanged (Fig. 3), allowing for a large flexibility in the length of the modulated pulse. In addition, within this range, smaller μ values (with larger M ) ensure good tunability of the intensity modulation period (without a significant reduction of the OPA efficiency, see Section 3.2).

Theoretical model for simulations
In order to investigate the feasibility of our concept for the generation of intensity modulated pulses with a constant modulation period and to present a practical design study of a multistage OPA system we carried out numerical simulations. The coupled wave equations in the slowly varying envelope approximation were numerically solved. The time-dependent electric field is given by where l A is the slowly varying complex amplitude, l k is the wave vector, and p, s, i l = for the pump, signal, and idler, respectively. The one-dimensional coupled wave equations in the following form were numerically solved [45,46]:

Selection of the nonlinear crystals
For multi-cycle THz pulse generation the frequency of intensity modulation should be equal to the desired THz frequency. As this is much smaller than the optical frequencies, a nearlydegenerate OPA should be used. Ultrabroadband amplification was demonstrated in a 1-µm pumped degenerate OPA using LiNbO 3 (LN) crystals [49]. An even larger bandwidth can be expected in KNbO 3 (KN) [50]. Figure 5 shows the calculated phase mismatch as function of the signal wavelength for LN and KN with 5 mm thickness, both for collinear as well as noncollinear phase matching geometries, in the latter case for 0.5 α =° internal pump-signal noncollinearity angles. (Phase matching was set to 2.074 µm wavelength, corresponding to a 1 THz intensity modulation frequency.) In all cases excellent phase matching can be achieved over the laser bandwidth (shaded range in Fig. 5). The wave vector mismatch was approximated by its projection onto the pump propagation direction [51]. Refractive index data were taken from Refs [52]. and [53] for LN and KN, respectively. In all cases the signal and idler had parallel polarizations, perpendicular to that of the pump, enabling to use a polarization filter (beamsplitter) at the DC-OPA output.
Both  [57] were used for the nonlinear refractive index of LN and KN, respectively. In the investigated cases related self-and cross-phase modulation effects were not significant.
In the numerical simulations discussed below LN was chosen as the nonlinear crystal for all OPA stages, including the DC-OPA.

Simulation results
Numerical simulations were carried out for the setup shown in Fig. 4 using the model described in Section 3. We note that all losses from the stretcher and the Fresnel losses were neglected in this calculation. The losses in the stretcher depend on the grating type. Typically, for a reflection-grating stretcher, these are about 30%, and for a transmission-grating stretcher less than 10%. The Fresnel losses can be reduced below 1%.
As mentioned above, the FL pulse duration was 200 fs at 1.03 µm central wavelength for the pump and 125 fs at 2.06 µm central wavelength for the signal seed. The subsequent OPA stages were pumped by 50 µJ, 5 mJ, and 75 mJ energy. The most important input and output parameters of the two noncollinear preamplifier stages are listed in Table 1. The angle between the propagation directions of pump and signal was 0.5 α =° inside the LN crystal and the angle between the optic axis of the crystal and the propagation direction of the pump was 42.82 θ =°. The signal output of the first stage was used as input for the second stage.
We note that central frequency of the signal is not changing during the nonlinear interaction process because of the excellent phase matching. Prior to entering the third, DC-OPA stage both the signal and the pump pulses were stretched by grating stretchers (Fig. 4). Both the GDD as well as the 3rd-order dispersion were taken into account [58,59]. For the pump, a pair of 600-lines/mm gratings was assumed. The angle of incidence was 23 γ =°, deviating from the Littrow configuration by only 5°. As mentioned in Section 3.3, the pump stretching factor can be determined by the desired length of the modulated pulse. In the simulations, 5 ps and 10 ps were considered ( Table 2). For given FL pulse durations, once the pump stretching factor has been chosen as desired, the required signal stretching factor and the corresponding GDD can be determined from Eqs. (14) and (10)    The results of the simulations are summarized in Table 2 and Fig. 6. Three representative cases were simulated to demonstrate the feasibility and flexibility of the scheme: 1 THz modulation frequency with 5 ps and 10 ps pulse durations, and 0.5 THz modulation frequency with 10 ps pulse duration. The calculated pulse forms support the expectation that highquality shaped pulses can be generated by the proposed DC-OPA method with a highly uniform intensity modulation frequency. For a given pump laser the setup can be flexibly configured to deliver a broad range of pulse lengths and modulation frequencies, which can be tailored to the need of different applications. We note that saturation in the DC-OPA can result in a flat-top modulated pulse, as can be seen in Fig. 6(b). Furthermore, according to Fig. 2, a temporal jitter of 20 fs between the pump and the signal results in a very small fluctuation of the modulation frequency of less than 1% in all investigated cases ensuring highly stable pulse forms. In all cases a conversion efficiency of about 52% was predicted in the DC-OPA stage, and about 49% for the entire system including also the two preamplifier stages. Such uniquely high efficiencies are the result of utilizing both the signal and idler pulses from the DC-OPA. According to Table 2, the efficiency is essentially independent of tuning the modulation frequency. This holds as long as the desired modulation frequency remains much smaller than the bandwidth of the signal pulse (in other words, the pump-signal delay remains much smaller than the stretched signal duration).

Conclusion
Utilizing both signal and idler pulses in OPA for the generation of intensity-modulated pulses was proposed. A constant intensity modulation period was achieved by using chirped pump and signal pulses (dual-chirped OPA) for which design conditions were given by simple analytical expressions. Easy tunability of the modulation frequency is enabled by introducing a time delay between the pump and the signal pulses. A conceptual design study was presented for a complete three-stage OPA system. The numerical simulations predicted an efficiency as high as ~50% for about 40 mJ output pulse energy at a wavelength of 2 µm. Such flexibly shapeable pulse sources, operating near 1.6 µm or 2 µm and pumped by compact Ti:sapphire or Yb-doped lasers, can be ideally suited to drive organic or semiconductor THz sources. These can deliver intense multi-cycle THz pulses with tunable frequency, enabling new applications in materials science and the construction of compact particle accelerators.
In the next step, we can calculate M: