Phase-Insensitive Scattering of Terahertz Radiation

The nonlinear interaction between Near-Infrared (NIR) and Terahertz pulses is principally investigated as a means for the detection of radiation in the hardly accessible THz spectral region. Most studies have targeted second-order nonlinear processes, given their higher efficiencies, and only a limited number have addressed third-order nonlinear interactions, mainly investigating four-wave mixing in air for broadband THz detection. We have studied the nonlinear interaction between THz and NIR pulses in solid-state media (specifically diamond), and we show how the former can be frequency-shifted up to UV frequencies by the scattering from the nonlinear polarisation induced by the latter. Such UV emission differs from the well-known electric-field-induced second harmonic (EFISH) one, as it is generated via a phase-insensitive scattering, rather than a sum- or difference-frequency four-wave-mixing process.

Resonant radiation (RR) is emitted by solitons propagating in a waveguide or by filamenting pulses in bulk media. Recent studies have highlighted the possibility to stimulate RR also in weaker pulses that co-propogate with a pump pulse. We numerically and experimentally demonstrate that RR radiation can be stimulated employing a THz seed co-propagating in diamond with an intense 800 nm pulse. This way we predict and observe the stimulated emission of RR at 425 nm, thus bridging a spectral gap of more than six octaves and allowing the detection of THz pulses by means of a silicon-based device.
Resonant radiation (RR), also referred to as optical Cherenkov or dispersive wave radiation is a process in which a laser pulse propagating in an optical medium with Kerr nonlinearity and higher order group velocity dispersion (GVD) is scattered through a resonant-like process to a shifted frequency [1][2][3]. The emission of RR was first analyzed and understood in the context of fibre solitons, i.e. in the form of an instability that develops on pulses propagating in the negative GVD regime and manifesting itself as a strongly blue shifted peak in the positive GVD spectral region. Since its first observation 30 years ago, progress has been made in controlling, optimizing and understanding the details of the process. Several investigations aimed at enhancing the conversion efficiencies of both blue and red shifted RR and at the discovery of new forms of RR, e.g. "spatial" RR in periodic waveguides [4] and RR with source terms that lie in the negative frequency part of the input pulse spectrum and hence called Negative Resonant Radiation (NRR) [5,6]. Remarkably, this last one exhibits links with a process similar to Hawking radiation from artificial black holes [7,10]. The frequency of the RR is determined by requiring that the phases of the input pulse and of the RR remain equal in propagation, thus leading to a wave-vector matching condition between the dispersive radiation and the pulse, i.e. (see e.g. [8]): where ω IN and ω RR are the pulse and RR frequencies, v is the velocity of the pulse and K NL = ω IN n 2 I/c is a nonlinear correction term (n 2 is the nonlinear Kerr coefficient) that may be small or even negligible at low intensities. Interestingly, Eq. (1) can also be derived via a first Born approximation description, whereby the input pulse creates a scattering potential via the Kerr effect and then self-scatters from this potential. Momentum conservation, in Eq. (1), governs the scattering process [11]. It has been recently observed that resonant radiation may also be induced in a weak, linearly propagating pulse when this propagates in combination with a second in-tense soliton or pump pulse [6]: if the two pulses have different frequencies and thus different speeds, the weak pulse may catch up with the soliton and the interaction will lead to a Cross-Pulse induced RR (XP-RR) emission, from the weak pulse. Similar effects have been reported in numerical simulations in fibre geometries where the interaction of a weak pulse with a soliton can lead to a transistor-like effect [12], trapping of the weak radiation between two solitons [13,14] and enhanced supercontinuum generation [15,16]. In Ref. [7] a 1D numerical simulation showed that it could be possible to observe RR from a far-infrared or Terahertz (THz) laser pulse. Terahertz pulses are typically too weak to directly induce nonlinear effects in dielectric media yet, when propagating in the presence of a second intense near-infrared pump pulse, the abovementioned XP-RR may take place. Equation (1) predicts that the XP-RR will appear as a peak in the UV spectral region, thus bridging a spectral gap of more than 6 octaves. Remarkably, this extreme up-frequency shifting mechanism may have a relevant impact on the issue of far-infrared and THz detection. Indeed detection at these frequencies suffers from the lack of detectors, whereas the ultraviolet light can be easily record by the widely available Silicon based devices (photodiodes or CCD camera), as discussed e.g. in Ref. [17].
Resonant radiation also appears in an apparently unrelated area, i.e. in ultrashort laser pulse filamentation in bulk media. Indeed, ultrashort pulse filaments also typically exhibit a frequency up-shifted peak whose (on axis) frequency can also be found by a Born approximation calculation and is determined by an equation that is identical to Eq. (1) [18]. This therefore opens a range of opportunities for studying RR beyond the limitations of a fibre or waveguide geometry.
In this letter we show with full 3D+1 numerical simulations and experiments performed in a bulk diamond sample that an intense pump pulse of wavelength λ p =800 nm may create a Kerr perturbation which will indeed induce XP-RR on a THz pulse that appears shifted, as pre- dicted, by more than 6 octaves, to 425 nm, at the limit of the visible spectrum. This XP-RR is thus easily detected using standard spectrometers for the visible region and appears to be more intense than a typical four-wavemixing process between the pump and THz frequencies.
Numerical model: Our 3D+1 numerical model relies on two coupled nonlinear envelope equations (NEE), describing the propagation of the intense pump laser pulse with frequency ω 0 coupled with a second THz pulse (at frequency ω t ). The coupled NEE are written for the Fourier transformed envelopes of the fundamental pulsê E ω0 (δω, r, z) and the THz pulseÊ ωt (δω t , r, z) and read as: − k 0 ω t and the frequency variable δω, for both pulses, is fixed by the numerical grid but represents the departure from ω 0 and ω t in Eq. (2) and Eq. (3), respectively (δω t ≡ ω − ω t ).
As input for our simulation we use a 800 nm pump pulse with duration of 40 fs and input energy of 10 µJ. The propagation of the pump pulse alone is shown in Fig. 1(a). The dielectric nonlinear medium is diamond which does not exhibit any resonance over a very broad bandwidth and is transparent also at THz wavelengths. The probe has a wavelength of 30 µm, duration of 90 fs and energy of 2 µJ in accordance with the experimentally available values as described below. We verified that the THz pulse does not excite any Kerr effect when propagating alone [see Fig. 1(b)]. Furthermore we introduced a relative delay of 110 fs for the THz pulse in order to compensate for the difference in group velocities of the two pulses: this delay was optimized for the THz pulse to catch up the pump pulse after a propagation distance of 300 − 400 µm, corresponding to a position where the pump pulse generated the steepest shock front [see Fig. 1(a) and (c), the latter showing the pump and THz propagating together], which increases the efficiency of the resonant scattering process [6]. Figure 2(a) shows in detail the spectral evolution of the 800 nm pump pulse over the full propagation distance. The input pulse initially self-steepens leading to the formation of a steep front at z = 0.3 mm: at that moment the delayed probe pulse encounters the shock front and A broadband (20 THz) seed pulse is generated by field ionization of nitrogen in an asymmetric field composed by 1800 nm and 900 nm radiation. The generated THz field is collimated by a gold-coated off-axis parabolic mirror and filtered by a gold mesh long pass filter in order to remove every frequency component above 20 THz. The THz pulse is then focused in a diamond single crystal sample collinear and temporally overlapped with an intense 800 nm pump pulse. The generated UV radiation is collected by a lens and detected by an imaging spectrometer coupled to a CDD.
the interaction between the pulse induces XP-RR scattering from the THz radiation as shown in Fig. 2(b). The position of the XP-RR peak can be predicted from Eq. (1) and is graphically shown in Fig. 2(c), where we plot the dispersion D = k(ω) − ω/v and the RR position is given by D = k(ω 0 ) − ω 0 /v (solid red line), in agreement with Eq. (1) with K NL = 0 (and D = −[k(ω 0 ) − ω 0 /v] for the NRR emission, dashed red line) [19]. There is a an excellent correspondence between the predicted and numerically observed spectral peaks, thus further confirming the RR origin of the peak at ∼ 425 nm. It is interesting to note that, in accordance with previous results reporting the existence of NRR at further blue-shifted frequencies, we should also expect a NRR process in our experimental conditions. However, due to the extremely red-shifted wavelength of the input THz pulse, the position of the NRR spectral peak, predicted graphically in Fig. 2(c) by the intersection of the dashed line with the dispersion curve, lies extremely close to the standard RR peak. The RR and NRR peaks are therefore expected to overlap and form a single indistinguishable peak, thus explaining the absence of an additional and isolated NRR peak in the numerical result shown in Fig. 2(b).
Experiments: We performed experiments to verify the existence of stimulated resonant radiation -XP-RR and, following the theory described in Ref. [7] we employed a THz pulse as input field. This long wavelength radiation is efficiently generated by a two-colour gas ionization stimulated by a mid-infrared pump at 1.8 µm and its second harmonic (generated by a 100 µm thick, free standing Beta Barium Borate -BBO-crystal), as described in Ref. [21] (see Fig. 3). This way a single cycle THz field of ∼ 90 fs duration, ∼ 1 µJ energy and ∼ 10 THz carrier frequency is generated (30 µm wavelength). A long pass filter with cutoff at 20 THz and 10 4 isolation is employed to remove from the THz pulse the high frequency components. The THz pulse is then focused onto a 500 µm, <100> cut diamond single crystal sample and temporally overlapped with a more tightly focused, 40 fs, 800 nm pump pulse. The radiation at the output of the crystal was measured by means of a spectrometer (Newport MS260i) coupled to a CCD camera (QSI 620). The measured spectra are shown in Fig. 4(a) and (b) for two different pump pulse energies (3 and 10 µJ respectively). Different neutral density filters have been employed while acquiring data on different portions of the spectrum in order to enhance the overall dynamic range. For the low energy case, the phenomenology is that of Electric Field Induced Second Harmonic generation (EFISH), typically employed for broadband THz detection in gas [22,23], and in solid state media [17,24,25]. In this case two lobes appear in the spectrum around the frequency of the second harmonic of the 800 nm pump pulse [26], as visible in Fig. 4(a) (red dashed curve), corresponding to the two four wave mixing components: ω TFISH = 2ω 0 ±ω t . Under these conditions, in the absence of input THz seed, the pump pulse undergoes minimal nonlinear reshaping and the output spectrum (blue curve) is still close to the input spectrum [black dashed curve in Fig. 4(a)]. On the other hand, when the pump intensity is increased a strong spectral reshaping occurs, as can be seen by the onset of super continuum generation indicated by SC on the blue curve in Fig. 4(b). Supercontinuum generation is a signature of strong nonlinear effects and indicates self-steepening associated to the large extension of the spectrum to the blue [20]. In this condition a second, more intense peak centred at 425 nm appears. This signal cannot be explained by the TFISH mechanism owing to its large shift with respect to 400 nm, which would require to consider a THz frequency of nearly 45 THz, out from the transmission window of the employed longpass filter. On the other hand such signal is in very good agreement with the numerical simulations presented above. We therefore interpret it as the evidence of XP-RR occurring on the THz pulse. We note that the two lobes of the SH signal in Fig. 4(a) are not so clearly distinguishable in Fig. 4(b) due to spectral broadening of the 800 nm pulse.
In conclusion, frequency conversion through a resonant transfer of energy from an input laser pulse to a typically blue-shifted peak is a well-studied process in nonlinear optics and has attracted substantial interest. Here we have shown that he same process can be observed when a seed is scattered off an intense pump. Employing a THz seed pulse, energy conversion from the far-infrared region to the near-UV, i.e. over more than six octaves, can be achieved, with applications for THz detection and imaging [17].