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Abstract
We theoretically investigate non-phasematched broadband THz amplification in dispersive chi(3) media. A short 100 fs pump pulse is interacting with a temporally matched second harmonic pulse and a weak THz signal through the four wave mixing process and a significant broadband THz amplification and reshaping is observed. The pulse evolution dynamics is explored by numerically solving a set of generalized Nonlinear Schroedinger equations. The influence of incident pulse chirp, pulse duration and the role of wavelength, THz seed frequency and losses are evaluated separately. It is found that a careful choice of incident parameters can provide a broadband THz output and/or a significant increase of THz peak power.
© 2017 Optical Society of America
1. Introduction
Generation of THz radiation stays at the frontline of research with ever increasing number of applications including but not limited to biomedical imaging, security, industrial quality control etc. and hence there is a great demand on reliable high power THz sources, see e.g [1–5]. In fact, lack of suitable THz source is one of the major challenges of the current state-of-the-art THz technology. Of the existing THz tabletop sources, perhaps the photoconductive antennas (PCA) and quantum cascade lasers (QCL) received the greatest attention in the recent years [3]. Although both PCA’s as well as QCL’s are widely employed, they are inherently limited by their available bandwidth and low efficiency, features that represent a limitation for e.g. THz spectroscopy. From the other alternatives, wave-mixing quickly became a popular way of easily producing broadband and potentially powerful THz output. However, wave-mixing in either second order media (e.g. ZnSe, ZnTe) or in plasma [6,7] does not present a direct competition to the solid state sources since it requires powerful amplified ultrafast laser sources which are bulky and expensive. On the other hand, electronic (non-optically driven) THz sources, such as Gunn diodes are usually strongly limited by their bandwidth emitting radiation in the sub-THz range, that is, in the range basically unsuitable for THz spectroscopy. Formerly, four-wave mixing (FWM) in conventional [8] and later in microstructure polymer optical fibers (MPOF) [9] has been proposed as a mean to generate THz radiation. The THz generation was shown to proceed with a modest power requirements provided by current telecom fiber lasers and the use of (MPOF) allowed for efficient phase-matching with interaction lengths exceeding 70 mm.
Here we theoretically investigate amplification of THz waves using non-phasematched four-wave mixing in a solid state. Instead of MPOF, we consider a bulk chi(3) medium and model the corresponding nonlinear dynamics by means of generalized nonlinear Schroedinger equation. The influence of incident pulse chirp, pulse duration and the role of wavelength and losses are evaluated separately.
2. Methods
The pulse evolution dynamics was solved by means of coupled generalized Nonlinear Schrӧdinger equations (NLSE) using the split-step Fourier method [10]. Various schemes can be conceived for the THz pulse generation as given in Fig. 1.
Fig. 1 FWM based THz generation. Reverse sum frequency generation (a), nondegenarate FWM (b) and degenerate FWM (c).
The scheme in Fig. 1(a) represents a reverse sum frequency generation process while scheme in Fig. 1(b) represents a common FWM scheme and Fig. 1(c) depicts merely the degenerate FWM case. Formally it can be expressed by a set of four coupled NLSEs, with linear polarization assumed for all four waves:
Second- and all higher-order dispersion terms were properly accounted for as was the self-phase (SPM) and cross-phase modulation (XPM). Self-Steepening and Raman terms were excluded from the analysis. A comprehensive description of individual terms can be found elsewhere [10]. Specifically:fjk and fijkl are overlap integrals and ϕ is the phase mismatch. By assuming ω1 = ω2, i.e. the degenerate case given in Fig. 1(c), the set is further reduced to three equations. The set was solved by a dedicated broadband solver that supports both the optical (field frequencies of the order of 1015 Hz) and terahertz fields.The cyclic olefin co/polymer (COC/P) TOPAS was chosen as a model material for the simulation as it shows excellent optical properties in the terahertz frequency range [11–13] and is consequently a common choice for studies including terahertz waveguides [14] and microstructure polymer optical fibers (MPOFs) [11,15]. The optical material dispersion curves of TOPAS were obtained from [16], whereas the THz dispersion data was extracted from [12,13]. All dynamics, both optical and THz proceeded in the normal dispersion regime. Optical Kerr index (n2) of TOPAS was acquired earlier by using the Z-Scan method [17] and the corresponding value of 2 × 10−20 m2W−1 was subsequently used throughout the simulations. Terahertz Kerr index of polymers including TOPAS is presently not known and consequently, a value of 1 × 10−20 m2W−1 was applied, the choice being rationalized by following reasoning. Recently, terahertz n2 of As2S3 and As2Se3 chalcogenide glasses were measured and the values closely matched the near infrared optical n2 [18]. Hence we assumed the same order of magnitude for both optical and THz n2.
3. Results
3.1 The role of pump wavelength
A set of four different pump/idler wavelength combinations was examined, including 800/400 nm, 1030/515 nm, 1300/650 nm and the 1550/775 nm case. Their choice was rationalized by the availability of modern femtosecond laser systems, i.e. Ti:Saphire in the first case, Yb: doped materials/fiber lasers in the second and Er: doped fiber lasers in the last case. The 1300/650 nm combination was chosen merely as an intermediate case.
800nm
First, we’ve examined the case of 2-color 800nm/400nm wave-mixing with the following input parameters: P800 = 3.8 × 109 W, P400 = 1 × 108 W, PTHz = 5W, W800 = W400 = WTHz = 700 μm, ΔT800 = ΔT400 = 130 fs, ΔTTHz = 1 ps where Px is the peak power at respective wavelengths/frequencies, Wx are the corresponding FWHM beam diameters and ΔTx the FWHM pulse durations. The parameters were taken such as to fit realistic experimental conditions under loose focusing. Any losses were neglected in the simulation. As can be seen on Figs. 2(a)-2(c), an incident THz seed pulse centered at 3 THz gets transformed and spectrally broadened after propagating 2 mm into the sample. More specifically, within the first 2 mm of the propagation, all three pulses at 800nm, 400nm and the THz pulse, each propagating with a different group velocity, catch up and a temporal overlap is achieved for a propagation distance not exceeding 1 mm. Thereafter, the pulses walk-off again, interaction ceases and the THz propagation is lossless and only dominated by dispersion as can be seen in Fig. 2(b). Both the pump pulse Fig. 2(d) and the idler pulse Fig. 2(e) thereby undergo an evolution typical for the normal dispersion regime dominated by the interplay between the dispersion, self-phase modulation (SPM) and cross-phase modulation (XPM). The resulting features e.g. multiple peak pump spectra Fig. 2(d) were characteristic for all subsequent simulations. Both the THz temporal pulse profile and corresponding spectrum after the propagation distance of 3.6 mm are given in Fig. 2(c). The distance was chosen such as to correspond to the highest temporal THz peak power. Note that pulse splitting is observed in the retarded reference frame given in Fig. 2(b) with a significant part of energy confined to a dominant temporal peak Fig. 2(c). We observed a net energy gain of 3 and the original THz pulse underwent strong temporal reshaping resulting into short sub-cycle feature in the temporal domain. In the spectral domain the amplification/reshaping results into a well defined broadband THz pedestal extending from the low frequency limit and exceeding the 10 THz mark.
Fig. 2 Pump, idler and THz seed propagation dynamics for a 800 nm pump pulse. THz spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c). The original THz pulse seed temporal profile and spectrum (dotted). Pump (d) and idler (e) spectral scaling with the propagation distance.
1030nm
Next, the excitation wavelength was switched over from 800nm/400nm case to the 1030nm/515nm wave-mixing while keeping all the other parameters unchanged. Assuming lossless propagation as in the previous case, no qualitative change in the behavior was observed. As can be seen both in Figs. 3(a) and 3(c) the THz spectrum is more flat, while at the same time, a factor of five higher THz powers are achieved. A closer examination of Fig. 3(b) reveals that the temporal features in the retarded reference frame are also less extended and we conclude, that this is explained by the reduced material dispersion in the 1030nm/515nm case as compared to 800nm/400nm.
Fig. 3 THz seed pulse propagation dynamics for a 1030 nm pump pulse. Spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c).
1300 and 1550 nm
The remaining 1300/650 nm a 1550/775 nm cases given in Figs. 4 and 5 respectively closely follow the trend observed in the former i.e. even less extended temporal evolution and an improved THz yield. The best THz peak power is achieved for the 1550/775nm combination with a value of 330 W exceeding the original output power at 800/400 nm by a factor of 10 while keeping all other input parameters unchanged.
Fig. 4 THz seed pulse propagation dynamics for a 1300 nm pump pulse. Spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c).
Fig. 5 THz seed pulse propagation dynamics for a 1550 nm pump pulse. Spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c).
3.2 THz seed frequency
In both of the previous simulations, the THz seed frequency was set such as to fit 3 THz. As these might not be the case under actual experimental conditions, we numerically examined the influence of THz seed central frequency on the temporal/spectral transformation while fixing the pump wavelength at 800 nm. As follows from Fig. 6 it turns out the spectral detuning would create a broadband pedestal that covers basically the same spectral range as given in Fig. 2 albeit with a reduced power. With other words, although the seed is now red shifted with respect to the original THz frequency, the ultrafast dynamics runs along the same lines as in the previous cases. The case in Fig. 6 is rather to be treated as an extreme case, we did observe only very little energy transfered to the new (temporally narrow) THz peak and the majority remained in the original THz pulse. Indeed, a qualitative comparison of Figs. 2(b) and 6(b) shows an essential part of the energy remaining within the 1 THz seed during the propagation while a minor part is channeled to the temporally short (spectrally broadband) THz peak. A direct comparison of the THz output power for the 3 THz seed and 1 THz seed can be drawn from the comparison of Figs. 2(c) and 6(c). The THz output peak power increases from 6 W to almost 35 W and the corresponding spectrum develops the characteristic extended pedestal. Blueshifting the THz seed even further, thus going to the 4 THz seed frequency results into pronounced spectral broadening Fig. 7(c) and a further enhancement of the THz peak power achieving almost 110 W. At the same time, the net energy gain scaled from the factor of 1.1 for the 1 THz seed to 2.2 for the 4 THz seed.
Fig. 6 THz seed pulse propagation dynamics for a 1 THz seed pulse. Spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c).
Fig. 7 THz seed pulse propagation dynamics for a 4 THz seed pulse. Spectral (a) and temporal (b) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c).
3.3 Pulse duration
Next we kept the incident peak power for all three waves of interest fixed and change the pulse duration, thereby increasing the energy content in the individual pulse. All other parameters were kept unchanged. Two different cases were explored, tuning of the pump pulse duration and tuning of the idler pulse duration. The representative data are given in Figs. 8 and 9. As follows from Figs. 8(a) and 8(d), as we increase the pump pulse duration, there is an efficient energy flow into the blue (high frequency) THz part of the spectrum. This ultimately results into an efficient spectral broadening and spectra that are both extended and flat as given in Figs. 8(c) and 8(f). Temporally, only very limited pulse splitting is observed as compared to e.g. Figure 2 and a major part of the energy remains confined to the main channel, see Figs. 8(b) and 8(e). Temporal dynamics for long pump pulses (500 fs and 1000 fs) shows the phase-matching occurring several times over a prolonged interaction length Figs. 8(b) and 8(e), whereas for shorter pump durations, phase-matching occurs only once, over a limited propagation distance (not shown). On the other hand, spectral scaling, Figs. 8(a) and 8(d), shows the phase-matching first occurs for the red shifted THz frequencies and only later for the blue shifted ones. Since for long pump pulses (500 fs and 1000 fs), the idler pulse is much shorter than the pump pulse it acts as a probe for the spectral phase modulation of the pump beam. At the same time, the timing of idler is set such as to allow for the interaction between pump and idler to take place after propagating a certain distance into the material. Consequently, the idler is first caught up by the fastest i.e. red components of the pump producing the red shifted THz frequencies and only later are the blue components phase-matched.
Fig. 8 THz seed pulse propagation dynamics for a 500 fs (upper row) and 1000 fs (lower row) pump pulse. Spectral (a, d) and temporal (b, e) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c, f).
Fig. 9 THz seed pulse propagation dynamics for a 500 fs (upper row) and 1000 fs (lower row) idler pulse. Spectral (a, d) and temporal (b, e) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c, f).
Interestingly, by increasing the pulse duration of idler, there is an enhancement of the low frequency (red shifted) THz part of the spectrum extending from approximately 0.5 to 2.5 THz, Figs. 9(a) and 9(d). Pronounced temporal splitting is observed in the retarded reference frame as seen in Figs. 9(b) and 9(e) giving rise to the oscillatory features both in the temporal domain as well as in spectra Figs. 9(c) and 9(f).
3.4 Chirped pulses
Since the propagation of chirped ultrafast pulses in dispersive media might lead to pulse broadening or compression, it naturally occurs to investigate the pulse chirp influence upon THz wavemixing. All simulation settings were kept unchanged except for linear chirp defined through the chirp parameter C (see [10] for details), a dimensionless quantity which satisfies the following relation:
where Δω is the spectral half-width (at 1/e – intensity point) and T0 is the temporal half-width (at 1/e – intensity point). Again, as in the previous case, two distinct situations were examined, with either the pump or idler signal being chirped. Four representative cases for the chirp parameter of + 5 and −5 for the pump pulses and idler pulses are given in Figs. 10 and 11 respectively. When a negatively prechirped pump pulse propagates through the medium in the normal dispersion regime as is the case for an 800 nm pulse, the pulse initially undergoes temporal narrowing followed by temporal broadening with the shortest pulse duration achieved approximately in the middle of propagation distance. The negative chirps consistently correspond to a short dominant THz temporal peak being induced by the FWM with the peak power exceeding that of the unchirped case given in Fig. 2. On the other hand, for positive pump pulse chirps only temporal broadening is observed during the pump pulsepropagation and the resulting THz pulse shows extensive modulation and a peak power reduced with respect to the unchirped case. Moreover, positive chirping produced almost negligible net energy gain of 1.03, whereas negative chirping resulted into a net energy gain of 2.9. The dynamics for idler pulse chirping is qualitatively similar to that of pump pulse chirping, however since the idler is approximately an order of magnitude less powerful, the effects are less pronounced.Fig. 10 THz seed pulse propagation dynamics for a pump pulse chirp parameter of + 5 (upper row) and −5 (lower row). Spectral (a, d) and temporal (b, e) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c, f). See text for further details.
Fig. 11 THz seed pulse propagation dynamics for an idler pulse chirp parameter of + 5 (upper row) and −5 (lower row). Spectral (a, d) and temporal (b, e) scaling with the propagation distance and the resulting THz pulse (red) and spectrum (blue) (c,f). See text for further details.
As can be seen from Figs. 11(a) and 11(c) and Figs. 11(d) and 11(f) the broadband spectral pedestal is largely suppressed for either positive or negative idler chirps. There was no net energy gain observed in either of the two cases but extensive reshaping has taken place. In contrast to the pump chirping, idler pulse chirping seems to open a new dissipative channel for the energy as can be seen in the upper right corner of Figs. 11(b) and 11(e) and produces a strong temporal interference in the resulting THz pulse, Figs. 11(c) and 11(f).
3.5 The role of losses
So far, we have examined the lossless case. Obviously, in real life scenarios losses have to be accounted for as they might considerably change the situation. The data for TOPAS optical losses were taken from [19] while the THz material loss was estimated from the data given in [11,13] and our own measurements [20]. The inclusion of realistic losses didn’t induce any significant changes to the dynamics observed in the lossless cases.
4. Discussion
In order to obtain a useful recipe for the parameter scaling we evaluate the cases given in the results section in terms of THz peak power and energy. The data given in Fig. 12(a) thus represent the THz peak power/pulse energy scaling with pump wavelength. Both the pulse energy and peak power scale up monotonically with increasing pump wavelength. This is explained by a reduced material dispersion for longer wavelengths that is effectively limiting temporal spreading of both the pump and idler pulse thus resulting into a more efficient energy transfer into the THz seed pulse. Another feature of interest is the THz seed frequency scaling of both the THz pulse energy and peak power, Fig. 12(b). Again, an increase of the THz pulse peak power/energy is observed with increasing THz seed frequency.
Fig. 12 THz peak power (blue) and THz pulse energy (red) as a function of: pump wavelength (a), THz seed frequency (b), pump pulse duration (c) and idler pulse duration (d).
Next we examined the THz pulse peak power/energy scaling upon pump/idler pulse duration, Figs. 12(c) and 12(d). If the pump pulse duration along with pulse energy is scaled up, the THz peak power/pulse energy first increase but start to saturate after passing the 500 fs tag. This is a direct consequence of the phase-matching occurring over prolonged interaction lengths leading to a strong temporal modulation of the THz pulse as described earlier, see Fig. 8 for comparison. In contrast, the idler pulse duration scaling shows a monotonic increase of THz pulse energy/peak power and no signs of saturation. This is expected as the temporal dynamics progresses differently in this case preserving the main THz pulse even for longer idler pulses in contrast to the previous case, see Fig. 9 for comparison.
We have shown by numerically solving a set of nonlinear Schroedinger equations that FWM between a strong optical pump, a frequency doubled idler and a weak THz seed can result into a significant amplification and spectral broadening of the THz pulses. The non-phase-matched interaction in a dispersive chi(3) medium, a cyclic olefin polymer, produced a short localized temporal THz spike and a corresponding broadband THz spectral pedestal exceeding 10 THz bandwidth. An optimum recipe for the amplification has been identified with respect to signal and idler duration, chirp, THz seed frequency and signal wavelength. Although this study was aimed at cyclic olefin polymers/copolymers with the possible outreach towards polymer fibers, waveguides and microstructure polymer optical fibers it would be of interest to explore the THz FWM in e.g. transparent semiconductors, metamaterials and quantum dots considering their high chi(3) optical nonlinearity.
Funding
NATO Science for Peace and Security program (SPS) (984698), Slovak Ministry of Education’s Research Funding Agency (VEGA) (1/0400/16).
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