# Phase-Insensitive Scattering of Terahertz Radiation

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## Abstract

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## 1. Introduction

## 2. Experiments

#### Excluded Possible Origins of the UV Signal

**Fluorescence.**We note that although diamond exhibits fluorescence in the analysed UV spectral regions, all the recorded UV signals only appear when the THz field is spatially and temporally overlapped to the 790 nm intense pulse. This observation rules out the possibility that the ≃430 nm signal is originating from a fluorescence process in diamond.**Residual THz pump field.**The multimesh filters used to remove the 1800 nm field from the THz beam path provide more than ${10}^{4}$ extinction. Given the low <${10}^{-4}$ efficiency of the THz generation process, it is essential to exclude the possibility that a residual 1800 nm field contributes to the onset of the ≃430 nm signal. To this end, we have first inserted a 1 mm thick ${\mathrm{CaF}}_{2}$ window in the THz beam path and searched for evidence of UV radiation. In this case, no EFISH or 430 nm signal were observed. Since ${\mathrm{CaF}}_{2}$ is transparent to IR radiation but opaque to THz, we concluded that THz is essential to generate the 430 nm signal. Further, we observed that the UV radiation was visible upon removing the ${\mathrm{CaF}}_{2}$ window and inserting a paper filter, which both absorbs and scatters the 1800 nm radiation. We therefore exclude any possible role of a residual 1800 nm field in the generation of the 430 nm signal.**EFISH.**We note that the 430 nm peak cannot be explained by the EFISH mechanism described above. Indeed, its large shift from 395 nm—the wavelength of the pump second harmonic—cannot be justified by the broadening of the NIR pump pulse spectrum.**Raman.**Diamond is a well-known Raman-active crystal and has a large ${\mathrm{\u2206}}_{R}$∼1350 ${\mathrm{cm}}^{-1}$ shift. The Raman-mediated wave mixing, ${\omega}_{p}+{\omega}_{p}-2\phantom{\rule{0.166667em}{0ex}}\pi \phantom{\rule{0.166667em}{0ex}}c\phantom{\rule{0.166667em}{0ex}}{\mathrm{\u2206}}_{R}$, gives a product at 417 nm (c is the speed of light). This process is independent of the presence of the THz field, whereas in our measurement, the 430 nm signal only appeared when the THz field was injected in the crystal. Another Raman-mediated four-wave-mixing process that may lead to a signal around 417 nm is ${\omega}_{p}+({\omega}_{p}-2\phantom{\rule{0.166667em}{0ex}}\pi \phantom{\rule{0.166667em}{0ex}}c\phantom{\rule{0.166667em}{0ex}}{\mathrm{\u2206}}_{R})\pm {\omega}_{\mathrm{seed}}$. This would require the presence of a Raman peak at ${\omega}_{p}-2\phantom{\rule{0.166667em}{0ex}}\pi \phantom{\rule{0.166667em}{0ex}}c\phantom{\rule{0.166667em}{0ex}}{\mathrm{\u2206}}_{R}$, corresponding to a ∼884 nm wavelength, which does not appear in the recorded spectra. Finally, we note that the EFISH signal has energies <1 pJ, and is too low to directly excite a Raman red-shifted peak.

## 3. Model

#### 3.1. Nonlinear Maxwell Equation Via Pseudospectral Space Domain Algorithm (1D + 1)

#### 3.2. Coupled Nonlinear Envelope Equations

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Federici, J.F.; Schulkin, B.; Huang, F.; Gary, D.; Barat, R.; Oliveira, F.; Zimdars, D. THz imaging and sensing for security applications—Explosives, weapons and drugs. Semicond. Sci. Technol.
**2005**, 20, S266–S280. [Google Scholar] [CrossRef] - Wallin, S.; Pettersson, A.; Östmark, H.; Hobro, A. Laser-based standoff detection of explosives: A critical review. Anal. Bioanal. Chem.
**2009**, 395, 259–274. [Google Scholar] [CrossRef] [PubMed] - Tonouchi, M. Cutting-edge terahertz technology. Nat. Photonics
**2007**, 1, 97–105. [Google Scholar] [CrossRef] - Zhang, X.C.; Xu, J. Introduction to THz Wave Photonics; Springer US: Boston, MA, USA, 2010; pp. 1–246. [Google Scholar]
- Ohlhoff, C.; Meyer, C.; Lupke, G.; Loffler, T.; Pfeifer, T.; Roskos, H.G.; Kurz, H. Optical second-harmonic probe for silicon millimeter-wave circuits. Appl. Phys. Lett.
**1996**, 68, 1699. [Google Scholar] [CrossRef] - Nahata, A.; Heinz, T.F. Detection of freely propagating terahertz radiation by use of optical second-harmonic generation. Opt. Lett.
**1998**, 23, 67. [Google Scholar] [CrossRef] [PubMed] - Dai, J.; Xie, X.; Zhang, X.C. Detection of Broadband Terahertz Waves with a Laser-Induced Plasma in Gases. Phys. Rev. Lett.
**2006**, 97, 103903. [Google Scholar] [CrossRef] [PubMed] - Karpowicz, N.; Dai, J.; Lu, X.; Chen, Y.; Yamaguchi, M.; Zhao, H.; Zhang, X.C.; Zhang, L.; Zhang, C.; Price-Gallagher, M.; et al. Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”. Appl. Phys. Lett.
**2008**, 92, 011131. [Google Scholar] [CrossRef] - Clerici, M.; Faccio, D.; Caspani, L.; Peccianti, M.; Yaakobi, O.; Schmidt, B.E.; Shalaby, M.; Vidal, F.; Légaré, F.; Ozaki, T.; et al. Spectrally resolved wave-mixing between near- and far-infrared pulses in gas. New J. Phys.
**2013**, 15, 125011. [Google Scholar] [CrossRef] - Li, C.Y.; Seletskiy, D.V.; Yang, Z.; Sheik-Bahae, M. Broadband field-resolved terahertz detection via laser induced air plasma with controlled optical bias. Opt. Express
**2015**, 23, 11436–11443. [Google Scholar] [CrossRef] [PubMed] - Clerici, M.; Caspani, L.; Rubino, E.; Peccianti, M.; Cassataro, M.; Busacca, A.; Ozaki, T.; Faccio, D.; Morandotti, R. Counterpropagating frequency mixing with terahertz waves in diamond. Opt. Lett.
**2013**, 38, 178–180. [Google Scholar] [CrossRef] [PubMed][Green Version] - Efimov, A.; Yulin, A.V.; Skryabin, D.V.; Knight, J.C.; Joly, N.; Omenetto, F.G.; Taylor, A.J.; Russell, P. Interaction of an Optical Soliton with a Dispersive Wave. Phys. Rev. Lett.
**2005**, 95, 213902. [Google Scholar] [CrossRef] [PubMed] - Efimov, A.; Taylor, A.J.; Yulin, A.V.; Skryabin, D.V.; Knight, J.C. Phase-sensitive scattering of a continuous wave on a soliton. Opt. Lett.
**2006**, 31, 1624–1626. [Google Scholar] [CrossRef] [PubMed] - Petev, M.; Westerberg, N.; Moss, D.; Rubino, E.; Rimoldi, C.; Cacciatori, S.L.; Belgiorno, F.; Faccio, D. Blackbody Emission from Light Interacting with an Effective Moving Dispersive Medium. Phys. Rev. Lett.
**2013**, 111, 043902. [Google Scholar] [CrossRef] [PubMed] - Clerici, M.; Peccianti, M.; Schmidt, B.E.; Caspani, L.; Shalaby, M.; Giguère, M.; Lotti, A.; Couairon, A.; Légaré, F.; Ozaki, T.; et al. Wavelength Scaling of Terahertz Generation by Gas Ionization. Phys. Rev. Lett.
**2013**, 110, 253901. [Google Scholar] [CrossRef] [PubMed][Green Version] - Couairon, A.; Mysyrowicz, A. Femtosecond filamentation in transparent media. Phys. Rep.
**2007**, 441, 47–189. [Google Scholar] [CrossRef] - Kolesik, M.; Wright, E.M.; Moloney, J.V. Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media. Opt. Express
**2005**, 13, 10729–10741. [Google Scholar] [CrossRef] [PubMed] - Kolesik, M.; Tartara, L.; Moloney, J.V. Effective three-wave-mixing picture and first Born approximation for femtosecond supercontinua from microstructured fibers. Phys. Rev. A
**2010**, 82, 045802. [Google Scholar] [CrossRef] - Philbin, T.G.; Kuklewicz, C.; Robertson, S.; Hill, S.; Konig, F.; Leonhardt, U. Fiber-Optical Analog of the Event Horizon. Science
**2008**, 319, 1367–1370. [Google Scholar] [CrossRef] [PubMed] - Webb, K.E.; Erkintalo, M.; Xu, Y.; Broderick, N.G.R.; Dudley, J.M.; Genty, G.; Murdoch, S.G. Nonlinear optics of fibre event horizons. Nat. Commun.
**2014**, 5, 4969. [Google Scholar] [CrossRef] [PubMed] - Belgiorno, F.; Cacciatori, S.L.; Clerici, M.; Gorini, V.; Ortenzi, G.; Rizzi, L.; Rubino, E.; Sala, V.G.; Faccio, D. Hawking Radiation from Ultrashort Laser Pulse Filaments. Phys. Rev. Lett.
**2010**, 105, 203901. [Google Scholar] [CrossRef] [PubMed] - Agrawal, G. Nonlinear Fiber Optics; Academic Press: New York, NY, USA, 2001; p. 467. [Google Scholar]
- Dudley, J.M.; Genty, G.; Coen, S. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys.
**2006**, 78, 1135–1184. [Google Scholar] [CrossRef] - Gordon, J.P. Dispersive perturbations of solitons of the nonlinear Schrödinger equation. J. Opt. Soc. Am. B
**1992**, 9, 91–97. [Google Scholar] [CrossRef] - Yulin, A.V.; Skryabin, D.V.; Russell, P.S.J. Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion. Opt. Lett.
**2004**, 29, 2411–2413. [Google Scholar] [CrossRef] [PubMed] - Efimov, A.; Taylor, A.J.; Omenetto, F.G.; Yulin, A.V.; Joly, N.Y.; Biancalana, F.; Skryabin, D.V.; Knight, J.C.; Russell, P.S. Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: Experiment and modelling. Opt. Express
**2004**, 12, 6498–6507. [Google Scholar] [CrossRef] [PubMed] - Skryabin, D.V.; Yulin, A.V. Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers. Phys. Rev. E
**2005**, 72, 016619. [Google Scholar] [CrossRef] [PubMed] - Zaitsev, A.M. Optical Properties of Diamond; Springer Berlin Heidelberg: Berlin/Heidelberg, Germany, 2001; p. 502. [Google Scholar]
- Tyrrell, J.C.A.; Kinsler, P.; New, G.H.C. Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion. J. Mod. Opt.
**2005**, 52, 973–986. [Google Scholar] [CrossRef] - Conforti, M.; Marini, A.; Tran, T.X.; Faccio, D.; Biancalana, F. Interaction between optical fields and their conjugates in nonlinear media. Opt. Express
**2013**, 21, 31239–31252. [Google Scholar] [CrossRef] [PubMed] - Couairon, A.; Brambilla, E.; Corti, T.; Majus, D.; de J. Ramírez-Góngora, O.; Kolesik, M. Practitioner’s guide to laser pulse propagation models and simulation. Eur. Phys. J. Spec. Top.
**2011**, 199, 5–76. [Google Scholar] [CrossRef] - Boyd, R.W. Nonlinear Optics; Academic Press: New York, NY, USA, 2008. [Google Scholar]
- Rubino, E.; Lotti, A.; Belgiorno, F.; Cacciatori, S.L.; Couairon, A.; Leonhardt, U.; Faccio, D. Soliton-induced relativistic-scattering and amplification. Sci. Rep.
**2012**, 2, 932. [Google Scholar] [CrossRef] [PubMed][Green Version] - Conforti, M.; Baronio, F.; Trillo, S. Resonant radiation shed by dispersive shock waves. Phys. Rev. A
**2014**, 89, 013807. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Experimental setup. A broadband (20 THz) seed pulse is generated by field ionisation of nitrogen in an asymmetric field composed of 1800 nm and 900 nm radiation. The THz electric field recorded via air-biased coherent detection (ABCD) and its spectrum are shown in (

**b**,

**c**), respectively. The THz field is collimated by a gold-coated off-axis parabolic mirror and filtered by two gold mesh long pass filters in order to remove every frequency component above 20 THz. The THz pulse is then focused in a diamond single crystal sample collinearly, and is temporally overlapped with an intense 790 nm pump pulse. The generated UV radiation is collected by a lens and detected by an imaging spectrometer coupled to a charge-coupled device (CCD).

**Figure 2.**(

**a**) Experimental measurement showing the electric field-induced second harmonic (EFISH) generation signal typically observed for low pump intensities (red, shaded curve, S stands for power spectrum). The blue curve shows the pump pulse spectrum after the propagation in the crystal without any injected THz seed. The black shaded curve shows a section of the input spectrum (a very limited spectral reshaping of the pump pulse is evident); (

**b**) as in (a), but for a higher pump pulse intensity. The occurrence of an additional signal at ∼430 nm is clearly visible in the red shaded curve.

**Figure 3.**(

**a**) Numerically simulated spectra for the total (pump and THz) field—spectra are shown at the sample output ($z=5$ mm). The light blue shaded area is for low pump intensity, while the thick black curve is for high pump intensities (S stands for power spectrum). (

**b**) Same as in (a), but comparing—for the high energy case—the results from a full simulation (black solid curve) with those from a simulation accounting only for the ${P}_{nl1}\propto \mathrm{Re}\left(\right)open="["\; close="]">{\left|\u0190\right|}^{2}\u0190$ (red shaded curve). (

**c**) Same as (b), but comparing the full simulation (black solid curve) with one including only the ${P}_{nl2}\propto \mathrm{Re}\left(\right)open="["\; close="]">{\u0190}^{3}$ term (green shaded curve). PI: phase insensitive scattering.

**Figure 4.**(

**a**) Numerically simulated evolution of the seed field spectrum along the full 5 mm propagation distance based on Equation (4) (colormap is in logarithmic scale). (

**b**) Dispersion, $D\left({\omega}_{\mathrm{scatter}}\right)=k\left({\omega}_{\mathrm{scatter}}\right)-{\omega}_{\mathrm{scatter}}/{v}_{p}$, for diamond calculated for the speed of the shock front, ${v}_{p}=\sim 1.223\times {10}^{8}$ m/s (red curve). The horizontal line shows $D\left({\omega}_{\mathrm{seed}}\right)=k\left({\omega}_{\mathrm{seed}}\right)-{\omega}_{\mathrm{seed}}/{v}_{p}$: the intersections with $D\left({\omega}_{\mathrm{scatter}}\right)$ gives the spectral location of the phase-insensitive scattering peak. The blue-shaded box shows the THz input spectrum at $1/{e}^{2}$. For the sake of clarity, the spectral position of the 790 nm pump pulse is also shown with a vertical red dashed line.

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## Share and Cite

**MDPI and ACS Style**

Petev, M.; NiclasWesterberg; Rubino, E.; Moss, D.; Couairon, A.; Légaré, F.; Morandotti, R.; Faccio, D.; Clerici, M.
Phase-Insensitive Scattering of Terahertz Radiation. *Photonics* **2017**, *4*, 7.
https://doi.org/10.3390/photonics4010007

**AMA Style**

Petev M, NiclasWesterberg, Rubino E, Moss D, Couairon A, Légaré F, Morandotti R, Faccio D, Clerici M.
Phase-Insensitive Scattering of Terahertz Radiation. *Photonics*. 2017; 4(1):7.
https://doi.org/10.3390/photonics4010007

**Chicago/Turabian Style**

Petev, Mihail, NiclasWesterberg, Eleonora Rubino, Daniel Moss, Arnaud Couairon, François Légaré, Roberto Morandotti, Daniele Faccio, and Matteo Clerici.
2017. "Phase-Insensitive Scattering of Terahertz Radiation" *Photonics* 4, no. 1: 7.
https://doi.org/10.3390/photonics4010007