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Article Cite This: ACS Photonics 2019, 6, 1515−1523

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Terahertz Hyper-Raman Time-Domain Spectroscopy Andrea Rubano,*,† Sen Mou,† Lorenzo Marrucci,†,‡ and Domenico Paparo*,‡ †

Dipartimento di Fisica “E. Pancini”, Università “Federico II”, Monte S. Angelo, via Cintia, 80126 Napoli, Italy ISASI, Institute of Applied Sciences and Intelligent Systems, Consiglio Nazionale delle Ricerche, via Campi Flegrei 34, 80078 Pozzuoli, Italy

‡

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S Supporting Information *

ABSTRACT: A new spectroscopic method has been demonstrated on the benchmark crystal α-SiO2. The new technique makes use of femtosecond optical pulses and intense, sub-ps, broadband terahertz (THz) pulses to generate a THz-optical four wave mixing in the investigated material. The spectrum of the generated signal is resolved in wavelength and displays two pronounced frequency sidebands close to the optical second harmonic central frequency 2ωL, where ωL is the optical central frequency of the fundamental beam. The two sidebands develop around the central frequency at the (anti-) Stokes side of ωs;a = 2ωL ∓ ωT, where ωT is the THz central frequency, thus resembling the spectrum of standard hyper-Raman scattering, and hence, we named this effect “THz Hyper-Raman”THYR. Due to the large laser and THz bandwidths, it is not possible to resolve the THYR signal in the frequency domain. Nonetheless, by taking advantage of the same principle at work in THz time-domain spectroscopy, it is possible to follow the evolution of the THYR signal in time and access the frequency domain again by Fourier Transform. In this way we were able to observe pronounced oscillations in time of the THYR signal whose frequencies correspond to a large variety of material excitations including Γ-point phonons, polaritons, and phonons out of the Γ-point, which are usually observed only by neutron scattering techniques. To complement the richness of these observations, we will show that the selection rules of the THYR process allow the simultaneous observation of both IR- and Raman-active material modes, thus highlighting the potential of this innovative experimental method. KEYWORDS: THz Nonlinear Optics, THz Four-Wave-Mixing, THz Time-Domain Spectroscopy, Coherent Hyper-Raman Spectroscopy, Crystalline Quartz, Polariton

T

magnons, excitons, etc.). A strong absorption can sometimes hide a significant portion of the spectral range; the index of refraction or the dielectric function are sometimes difficult to interpret when the target material has a very complex structure; the bulk background can hide the contribution from the surface, which may be the real target of the experiment; or some particular excitations are out-of-reach because of symmetry, i.e., they just do not show up in the linear parameters. Most of these hurdles can be overcome by accessing the nonlinear optical parameters. Recent advances in THz technology allowed obtaining nonlinear optical effects with THz photons,2−13 showing remarkable results. In particular, in refs 11 and 12, optical four-wave mixing (FWM) in the THz range has been used for investigating liquids or solid interfaces leading to interesting results, although heavily limited because of the lack of intense THz pulses and of a spectral analysis of the FWM signal. In the present work, we greatly expand the spectroscopic capability of this technique by combining intense and broadband THz pulses with a detailed analysis of the spectral content of the generated signal. This has allowed us to detect

he Terahertz electromagnetic spectrum ranges in the frequency window from ∼100 GHz to ∼30 THz, i.e., in between the microwave and the mid-infrared domains. This spectral region was poorly explored for a long time because of the lack of powerful sources and efficient detectors. Yet, thanks to the constant improvement of pulsed and continuous THz sources during the last decades, today available table-top THz spectrometers are becoming more and more widespread. Taking advantage of femtosecond lasers, it is now possible to generate almost single-cycle THz pulses, i.e., electromagnetic pulses with only few periods duration (∼0.1 ps at 10 THz). The same laser pulses can also directly sample the THz electric field by means of the electro-optic sampling (EOS) technique, leading to the development of the THz time-domain spectroscopy (THz-TDS).1 In this spectroscopic technique the complex optical parameters (optical conductivity, refractive index, or dielectric function) of a target material are inferred by measuring the THz waveform in time-domain through EOS, Fourier-transforming it, and then comparing it with the spectrum acquired without the sample. This technique, thus, delivers the real and imaginary parts of the optical parameters in the investigated spectral range without the need of Kramers−Kroenig-based numerical methods. However, it is not always easy to access the target observables, such as the low-frequency excitations (phonons, © 2019 American Chemical Society

Received: February 14, 2019 Published: May 28, 2019 1515

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with a single technique a large variety of material excitations in the benchmark α-SiO2 that include Γ-point phonons, polaritons, and phonons out of the Γ-point, which are usually observed by means of neutron scattering techniques. This new method is supported by a theoretical analysis, presented in the Discussion, which provides a convincing interpretation of our observations, thus unveiling the spectroscopic potential of this tool that we name “coherent THz HYper-Raman” (THYR) spectroscopy.

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RESULTS Let us here briefly recall the main concept behind the coherent hyper-Raman effect that we exploit in our experiment. A more detailed derivation is provided in the Discussion, where we show that it may be classified as a FWM nonlinear optical process. Standard hyper-Raman spectroscopy is a modified version of the Raman technique where the scattered light displays a frequency spectrum with components that are close to the double frequency or second harmonic (SHG) of the fundamental laser frequency ωL.14 These components are the Stokes (anti-Stokes) frequencies ωs;a. The energy conservation law for the hyper-Raman effect can be written as ℏωs;a = 2ℏωL ∓ ℏΩ

Figure 1. (a) Intense THz pulses (green curve) are sent to the sample together with IR pulses at 800 nm wavelength (red curve) at adjustable delay Δt. After removal of the fundamental IR light (not shown), the THYR signal (violet curve) is decomposed spectrally by a reflection grating, and its spectral components intensity is measured as a function of both Δt and λ. (b) THz pulse measured by EOS in timedomain (upper panel, light blue curve) and its correspondent Fourier transform (lower panel, red curve). (c) Photon energy diagram of the THYR effect.

(1)

where ℏΩ is a low-energy excitation of the material under investigation. The main advantage of hyper-Raman spectroscopy is that it may provide information on low-energy modes that are suppressed in Raman spectra because of symmetry selection rules.14 Usually the hyper-Raman signal is very weak, but it may be strongly enhanced in the presence of additional optical fields oscillating at Ω, resonant hence with the material low-energy modes. To this aim it would be highly desirable to have an intense THz pulse, which may impulsively excite the material low-energy modes and then mix with the visible light so to generate a stimulated hyper-Raman signal. This is exactly what happens in the THYR effect, where the THz wave at ωT may directly couple to the Ω of the material, thus enhancing the THYR signal when ωT = Ω. The energy level diagrams of the THYR process are depicted in Figure 1c. In the THYR technique, the THz photon at ωT mixes with the two photons of the visible pulse at ωL and generates new spectral components at lower (Stokes) and higher (anti-Stokes) frequencies as compared to the SHG central frequency 2ωL. We stress here that the THYR effect is different from the well-known THz field-induced second harmonic effect (TFISH), which, for instance, is used to measure the THz field in a coherent fashion just as done with more standard EOS techniques.15 In this case, the material response is assumed (and it is, actually) instantaneous, so that the “reconstructed” THz wave is a faithful representation of the true THz wave. This difference will become obvious in the Discussion, where we present a detailed theoretical analysis of the THYR effect. We also mention that the novelty of our method is not limited to a mere technical upgrade consisting in the implementation of a spectrometer along the detection line. This has been already done, for instance, in work.13 However, in the latter work this technical improvement is not exploited for assessing a new spectroscopic technique, as we do, since the authors were only interested in nonresonant nonlinear optical effects occurring in samples that are transparent in the THz range of frequencies. An example of THYR measurement is shown in Figure 2, both in a 3D (a) and 2D (b) representation for a Y-cut 0.5

mm-thick α-SiO2 single crystal. The experimental details are reported in the Methods.

Figure 2. Example of a THYR|| measurement of a 0.5 mm Y-cut αSiO2 sample in a 3D (a) and 2D (b) representation.

It is apparent that the observed signal spans a wide spectral range going from about 380 to 440 nm, and it is composed of three distinct bands. Taking ∼400 nm as a reference value of the laser SHG wavelength, we can identify three bands from left to right: (i) “first-order anti-Stokes band” (ASB1, 380−400 nm), (ii) “first-order Stokes band” (SB1, 400−420 nm), and (iii) “second-order Stokes band” (SB2, 420−440 nm). The soundness of this nomenclature will emerge in the following. One additional band, the “second-order anti-Stokes band” (ASB2, ∼360−380 nm) may be seen (data not shown) by replacing the filters in order to allow the shortest wavelength to be detected. Let us now comment on the most relevant qualitative features that appear in our data. A first observation is that a significant amount of signal is still present for Δt > 1 ps, i.e., an order of magnitude larger than the THz pulse duration. This 1516

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Figure 3. (a) Normalized time-integrated spectra of the THYR|| (blue curve) and THYR⊥ signals (red curve), together with an SHG spectrum recorded without the THz field applied. (b,c) Measured THz (b) and IR (c) pulse energy dependence at two different wavelengths, 405 nm (red dots) and 440 nm (black dots). Solid curves are (linear or quadratic) best-fit curves.

Figure 4. Behavior of the THYR|| signal as a function of the delay Δt and correspondent Fourier spectrum. Left panel: time-domain data extracted at single wavelength λ. The labels on the left indicate the wavelength in nm. Red curves are best-fit curves for the nonresonant contribution. Right panel: Fourier transforms of the time-domain signals after subtraction of the nonresonant contribution. The labels close to each peak indicate the value of the resonance frequency and its symmetry when available.

in Figure 3a, where the time-integrated THYR signals are shown together with the SHG signal obtained in the absence of the THz pulse. Looking at Figure 3a, we notice that the THYR signal is much broader than the SHG signal in both parallel and perpendicular cases, but with a prominent spectral weight onto the Stokes side for the former and anti-Stokes for the latter. Most importantly, if we calculate the Stokes shift between the ∼400 nm central SHG wavelength and an extreme wavelength (for instance, ∼440 nm, where the signal is still clearly detected), we find a value close to 70 THz, which is much larger than our THz bandwidth (Figure 1a). Although the highest frequencies of the air-plasma spectrum can be hard to detect via the EOS technique, it is commonly accepted16 that

delayed signal appears, however, mainly in the SB1 band, while the ASB1 band has a much shorter decay and the SB2 and ASB2 bands present an almost instantaneous response. Moreover, we find that the THYR signal is vanishing in Zcut α-SiO2 samples and detectable, although weaker, in X-cut samples (data not shown). In the latter case, the results are qualitatively similar, and therefore, our discussion will be limited to the case of the Y-cut samples. By rotating the THz and IR polarizations with respect to the α-SiO2 optical axis z, all possible combinations were investigated. We found a nonvanishing signal for only two cases: z || IR || THz and z || IR ⊥ THz. For simplicity we will refer to these two cases with the labels “parallel” || and “perpendicular” ⊥, respectively. A direct comparison between the THYR|| and THYR⊥ spectra is given 1517

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Figure 5. Behavior of the THYR⊥ signal as a function of the delay Δt and correspondent Fourier spectrum. Left panel: time-domain data extracted at different wavelengths λ. The labels on the left indicate the wavelength in nm. Red curves are best-fit curves for the nonresonant contribution. Right panel: Fourier transforms of the time-domain signals after subtraction of the nonresonant contribution. The labels close to each peak indicate the value of the resonance frequency and its symmetry when available.

the THz bandwidth generated with this technique is roughly ∼30 THz, which is consistent with the value theoretically expected for a 35 fs laser bandwidth. In addition, the presence of a thick silicon filter in the THz path ensures that frequencies higher than ∼30 THz would be strongly dumped by Si−O−Si vibrational modes.17 In other words, ASB1 and SB1 bands are explainable by a one THz-photon process such as described in eq 1 with Ω = ωT, but the same is not true for the SB2 band. A possible explanation for the latter is that it results from a 2IR-/ 2THz-photons interaction of the form: ℏωs;a = 2ℏωL ∓ 2ℏωT. Moreover, looking at the temporal behavior of the signal for large wavelengths, it is apparent that the SB2 band has a time duration comparable with the THz pulse, and therefore, the process responsible for this band has a different nature from that responsible for the SB1 band. The same holds true for the ASB1 and ASB2 bands. In order to clarify this point, the THYR signal has been measured as a function of both the THz and IR pulse energies at several positions on the 2D map of Figure 2b, and the results are exemplarily shown in Figure 3b,c. We found that the THYR signal scales linearly with the THz pulse energy and quadratically with the IR pulse energy, in the ASB1 and SB1 bands, and it scales quadratically with both THz and IR pulse energies in the ASB2 and SB2 bands, thus nicely confirming our proposed interpretation involving one or two THz photons, respectively. Now we focus our attention on the temporal behavior of the THYR signal. We extract from the 2D spectrograms time-cut curves at several wavelengths for both parallel and perpendicular cases. The obtained curves are reported in Figures 4 and 5, which represent the main results of our work. When meaningful, these curves were analyzed by means of a fitting procedure explained in more detail in the Supporting Information (SI). The results of these fits represent a nonresonant contribution to the THYR signal (including a

very fast response that can be identified with the standard TFISH signal) and are indicated by the red curves in Figures 4 and 5. The obtained fit-functions are then subtracted from the data, and a fast Fourier transform is applied to the residual oscillatory signals. The so-obtained spectra are shown in the panels (b) of Figures 4 and 5. The spectral resolution is about 0.1 THz. Let us make some general considerations. The same peaks are seen (with different amplitudes) for different detected wavelengths, and there is no apparent relationship between the optical wavelength of the THYR signal and the phonon frequencies. This is not surprising since both the IR and the THz bandwidths are large, and there are many combinations of different frequencies in them that can launch the same phonon. More interesting is to observe that in general each phonon is observed either in the Stokes or in the anti-Stokes side, but never in both, with the remarkable exception of the peak at 9− 10 THz, which will be discussed later. This is due to the difference between the Stokes and anti-Stokes nonlinear susceptibilities, as derived in the next section. To better appreciate the power of the new technique, we compare these spectra with the absorption spectrum, obtained by means of THz-TDS, and the Raman spectrum reported in Figure 6. The experimental details on these measurements may be found in the Methods, while the procedure for extracting the THz absorption spectrum is explained in the SI From this comparison it is evident that the THYR spectra are very different from Raman and THz-TDS ones and may provide complementary information. In the THz absorption spectrum, we can distinguish only one phonon peak, at about 8 THz, and then the signal drops dramatically above 10 THz (see also Figure S2 of SI), despite the larger bandwidth of our detector as shown in Figure 1b. In particular, between 0 and 7 THz we cannot observe the resonances at 2, 5.2, and 7 THz 1518

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and 7.0 THz, and the Raman one at 4.2 THz, are particularly interesting because they belong to the M and A points of the Brillouin Zone. Usually, optical spectroscopies cannot detect modes far from the Γ point because of the very small momentum of optical photons, which can induce only vertical transitions, and therefore affects mainly the states close to the Γ point. However, if two phonons are simultaneously created with opposite momenta, the process is allowed. The twophonon process at 4.2 THz was already reported in ref 22. The fact that all these oscillations are observed so clearly in the THYR spectra is quite remarkable. Finally, we note that the oscillation at 9−10 THz in the ⊥ spectrum has no clear correspondence in the known phonon curves of α-quartz, although it is close to a resonance around 10.7 THz corresponding to a phonon of A2 symmetry.21 Interestingly, its value is not constant with the wavelength, but it moves, starting from 9 THz at 410 nm to about 10 THz at 370 nm, with a clear trend as shown in Figure 7. A similar

Figure 6. Raman (black curve) and THz absorption (red curve) spectra. The labels close to each peak indicate the values of the resonance frequency and its symmetry, when available.

that are clearly visible in Figures 4 and 5. The loss of information above 10 THz is caused by the presence of many phonon resonances in the interval between 10 and 13 THz, as reported in ref 18 and highlighted in Figure S2 by the blue marks. In contrast, these resonances are detected by means of THYR, as shown in Figures 4 and 5. However, it is worth noting that in our THYR spectra the resonance at 8 THz is absent. Turning our attention to the Raman spectrum, we notice that the resonances at 2, 5.2, 13.3, 14.3, and 16.3 THz, and all the peaks between 9 and 10 THz, recorded in the THYR spectra, are absent or barely visible here. However, our Raman spectrum displays phonon modes that are absent in the THYR spectra, i.e., those associated with the peaks at 4.2, 8.6, 11.7, and 15.4. Finally, the resonance at 6.85 is consistent with that observed at 7 THz in our THYR spectra (Figure 4).

■

Figure 7. Behavior of the polariton frequency associated with the A2phonon mode at 10.7 THz as a function of the THYR wavelength. The solid red and black circles are the central frequencies of the resonances displayed in Figure 4 for the || geometry and Figure 5 for the ⊥ geometry, respectively. The blue and green shaded areas refer to the anti-Stokes and Stokes regions of the THYR spectrum, respectively, and are obtained by solving simultaneously eqs 8 and 9. These areas are drawn by adjusting the angle θ within the interval 1−3° with no further adjustment of parameters.

DISCUSSION Resonance Assignment. In order to interpret the measured spectra, let us now compare our results with literature. We recall here Tab. I of ref 18, which summarizes the results of refs 19−21. In Table 1 of the present manuscript Table 1. List of THYR Resonances (in THz) and Their Closest Known Correspondence from Literature (Tab. I of Ref 18) Ω

lit.

mode

B.Z. point

2.0 5.2 7.0 13.4 14.3 16.3

2.1 5.1 7.1 13.5 14.4 16.4

T2 E T1 ET EL A2L

M(1/a,0,0) A(0,0,1/c) M(1/a,0,0) Γ(0,0,0) Γ(0,0,0) Γ(0,0,0)

resonance around 9.6 THz is visible in the || spectrum for 385−395 nm wavelengths too, although it is not possible to highlight any trend given the rapid vanishing at different wavelengths. The resonance at 8.6 THz in the Raman spectrum is also unidentified, and it might have the same origin as the THYR oscillations between 9 and 10 THz. We come back to this point later. In ref 23, the authors report on the observation of polaritons in α-quartz by means of Raman scattering. The latter are mixed excitations resulting from a strong coupling between phonons and photons, which may result in a frequency shift of the phonon mode. This interaction leads to spectra where the Stokes and the anti-Stokes sides are strongly asymmetric and strongly dependent on the relative direction of the THz polarization, as we observe in panels (b) of Figures 4 and 5.23 Their appearance is largely expected in polar crystals, such as quartz, when the energies and momenta of transverse optical phonons and electromagnetic waves are nearly equal.24 In the

the observed THYR resonances are shown together with the closest known vibrational resonances from literature (as observed with various techniques), and the corresponding symmetries and Brillouin Zone points. According to Tab. I in ref 18, the A2L mode at 16.3 THz is IR-only active, and the E modes at 13.4 and 14.3 THz are IR and Raman active. We note that the IR-only active mode cannot be observed with Raman spectroscopy, although in Figure 6 a barely visible peak appears around 16.8 THz. The three THYR modes at 2.0, 5.2, 1519

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Lorentz-model links q(t) to the driving force (see SI for details). AL,T are the i-th components of the complex field i amplitudes at ωL and ωT, and Δt is the temporal delay between the THz and visible pulse (for Δt > 0, the THz pulse is anticipated). μi is the i-th component of a possible permanent dipole of the material, αij is the material hyper-polarizability, and βijh is the susceptibility tensor accounting for the hyperRaman effect. We note that, even in the absence of a permanent dipole moment, ∂qμi may be different from zero, giving rise to a coupling with IR-active modes. One important result of eqs 4 and 5 is the presence of the oscillatory terms exp(−iωTΔt) and exp(−i2ωTΔt) that explain the observed oscillations of the THYR signal as a function of the delay Δt and that provide hence the foundation of the THYR time-domain spectroscopy (THYR-TDS). Equations 4 and 5 clearly show the difference between the THYR-TDS and the aforementioned TFISH effect. In the TFISH effect and in its precursor, i.e., the electric-field induced second-harmonic (EFISH) effect, the SHG signal is created (or modulated) by the presence of a static (EFISH) or quasi-static (TFISH) electric field that breaks the inversion symmetry (in case of centrosymmetric materials) or modifies an existing noncentral symmetry. The effect is classified as belonging to the wide family of four wave mixing processes. However, unlike our case, the static (or very slowly varying) nature of the applied field, makes possible to treat it as an “effective χ⃡ (2)” effect. In the case of EFISH, this can be done with no approximation, but in the case of TFISH one usually assumes that, since the THz field modulation is slow compared to the fs-optical pulses, the same description applies as for EFISH, i.e., that of an “effective” second order effect. An analogous simplification is not possible in the case of THYR. Finally, by comparing eqs 2 and 3 with eqs 4 and 5, we obtain the following relationships for the nonlinear optical susceptibilities:

following, by adapting to our case a simple phenomenological model developed in ref 25 for describing Raman-stimulated scattering, we will explain the observed behavior of the resonance in the 9−10 THz interval in terms of the formation of a polariton associated with the aforementioned phonon having A2 symmetry and a frequency of about 10.7 THz.21 THYR Semiclassical Model. Now we discuss the theoretical basis of the THYR effect. Here, we report the main results while a more detailed derivation is provided in the SI. Let us consider two input electromagnetic pulses, having spectral ranges that include frequencies ωL and ωT , respectively. Among several other mixing terms, they will induce a nonlinear polarization oscillating at the four-wave mixing frequencies ωs;a = 2ωL ∓ ωT having an amplitude given by (3) PiNL(ωs) = χijhk (2ωL − ωT)Ej(ωL)Eh(ωL)Ek*(ωT) + c . c . (3) PiNL(ωa) = χijhk (2ωL + ωT)Ej(ωL)Eh(ωL)Ek (ωT) + c . c .

(2) (3)

Where χ⃡ is the third-order susceptibility tensor describing all FWM optical processes. As seen, in our experiment we observe also spectral components that may be ascribed to higher order effects involving two THz photons. The amplitude of the optical nonlinear polarization generated in these processes is given by (4) PiNL(ωs) = χijhkl (2ωL − 2ωT)Ej(ωL)Eh(ωL)Ek*(ωT) El*(ωT) + c . c . (4) PiNL(ωa) = χijhkl (2ωL + 2ωT)Ej(ωL)Eh(ωL)Ek (ωT)El(ωT)

+ c. c. (3)

Following the approach and notation presented, for instance, in ref 26, we can determine the relationship between the nonlinear susceptibilities χ⃡ (3) and χ⃡ (4), and the macroscopic material constants that account for the induced dipole moment, the linear hyper-polarizability, and the hyperRaman susceptibility tensor. This is useful for understanding the symmetry selection rules that hold for our technique. The expressions of the nonlinear polarizations are

(3) χijhk (2ωL ± ωT) =

(4) (2ωL ± 2ωT) = χijhkl

(4)

R(ωT) ∂βijh ∂μk L L T −iωTΔt A j Ah A k e 4π ∂q ∂q R(2ωT) ∂βijh ∂αkl L L T T A j Ah A k Al δ(ωs − 2ωL − ωT) + 4π ∂q ∂q

Pi(ωa , Δt ) =

e−i2ωTΔt δ(ωa − 2ωL − 2ωT) + c . c .

R(2ωT) ∂βijh ∂αkl ∂q ∂q 4π

(6)

We note that ∂qμk ≠ 0 and ∂qαkl ≠ 0 couple to IR- and Raman-active modes, respectively. However, ∂qβijh is a fourthrank tensor characterized by selection rules that are different and may include those driving IR and Raman processes.14 Therefore, the THYR technique may allow a simultaneous investigation of low-energy excitations that are complementarily present in IR and Raman spectra. This possibility is confirmed by our measurements that display a signal also at 2ωL ∓ 2ωT. Phase Matching Considerations and Polaritons. In all FWM processes, an important element is the phase matching condition. In a THYR process, this condition is written as follows (here, we limit ourselves to the one THz photon process):

R(ωT) ∂βijh ∂μk L L T −iωTΔt Pi(ωs , Δt ) = A j Ah A k * e 4π ∂q ∂q R(2ωT) ∂βijh ∂αkl L L T T Aj Ah Ak *Al * δ(ωs − 2ωL + ωT) + 4π ∂q ∂q e−i2ωTΔt δ(ωs − 2ωL + 2ωT) + c . c .

R(ωT) ∂βijh ∂μk 4π ∂q ∂q

ks⃗ = 2kL⃗ − k ⃗T

(5)

ka⃗ = 2kL⃗ + k ⃗T

where δ(ω) is the Dirac function that accounts for the energy conservation leading to the Stokes (ωs) and anti-Stokes (ωa) components of the spectrum; q(t) is the oscillator material degree of freedom, and R(ω) is a material function that in a

(7)

where k ⃗ L,s,a,T are the fundamental laser, Stokes, anti-Stokes, and THz wave-vectors, respectively. In a standard all-optical FWM, 1520

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observations, summarized by Figure 7, which shows the peak trend (black and red solid points for the || and ⊥ geometry, respectively). By slightly adjusting the angle θ we obtain the shaded areas in Figure 7 that fully contain the measured peak frequencies and explain the decreasing trend of these frequencies as a function of the THYR-signal wavelength. The blue and green shaded areas refer to the anti-Stokes and Stokes regions of the spectrum, respectively. The upper and lower limits of these areas are obtained by adjusting the angle θ within the interval 1−3°, fully compatible with the cone of the THYR signal generated in our geometry. We note that the latter is the only adjustable parameter of our analysis. We also highlight that smaller angles, or even zero angles may be necessary if one properly accounts for the aforementioned anisotropy of the dielectric tensor. Finally, we note that the wavelength used in our linear Raman experiment is 532 nm, and hence, the trend observed in Figure 7 is also consistent with the interpretation that the peak observed at 8.6 THz is part of the same polariton spectrum. Although a more quantitative analysis will be performed to validate the claim, this qualitative analysis strongly supports the picture of a polariton resonance associated with the peak in the 9−10 THz range. To the best of our knowledge, the A2symmetry polariton of α-quartz has never been observed before.

eq 7 defines precise angles at which the output beams can be observed. In our case, kT/kL ≈ 10−6 for a 10 THz photon frequency, and hence, 2kL ≈ ks ≈ ka. This means that the effect is seen both on the Stokes and anti-Stokes sides almost collinear to the second-harmonic transmitted laser beam, in agreement with our observations. This is a very important point that establishes a significant difference between the angle-resolved Raman-scattering technique,24 used for observing polaritons in α-quartz, and the THYR technique. In the former, because of the incoherent emission over the solid angle, the detection angle may be significantly varied, including the geometries at 90° and 180° (backscattering) with respect to the input beam direction. In contrast, in the THYR technique, the detection geometry is fixed by the forward coherent emission that limits the accessible angle to the small cone set by the Gaussian beam divergence and to small imperfections in the collinearity between the IR and THz beams. We have estimated an emission cone of about 3°, and hence, our geometry is analogous to the forward geometry used in ref 14. The selection over different k-vectors, achieved in the standard Raman technique by varying the angle, is obtained in the THYR technique by analyzing the THYR signal at different wavelengths. The polariton momentum must satisfy the phase-matching condition as stated in eq 7 and the polariton dispersion relation. For a single polariton resonance, the latter is given by25,27 ε=

kT

2

c 2ωT 2

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METHODS The experimental layout is schematically shown in Figure 1a, while a more detailed plot is displayed in Figure S3 of SI. An intense and broadband THz pulse is generated via air-plasma technique using a regenerative Ti:Sa amplifier, which delivers 3.8 W power at 800 nm central wavelength and 1 kHz repetition rate. In this technique, broadband and ultrashort THz pulses are generated in a laser-induced plasma formed by ionized air via FWM of the fundamental beam with its second harmonic generated by a Beta Barium Borate (BBO) nonlinear crystal (see Figure S3 of SI for details).30 The THz radiation generated in this way is filtered by means of a high resistivity thick silicon wafer in order to remove the unwanted radiation. The power is measured by means of a commercial THz power meter to be about 140 μW, and therefore, the pulse energy is estimated to be 140 nJ. This value can be no more than 10% accurate because of the difficult removal of the thermal background. The pulse time-duration is measured with standard EOS technique by means of a LAPC crystal31 to be ∼0.1 ps, as shown in Figure 1a. With a beam waist in the focal plane of ∼0.5 mm, a rough estimation of the peak electric field is ∼1 MV/cm. The angle between the polarization plane of the fundamental beam and the BBO optical axis is adjusted to have a THz beam with an approximate circular polarization. In this way, the output energy and linear polarization of the THz pulse can be adjusted by means of two broadband wire-grids. The smallest detectable energy is found to be 30 nJ. By extrapolation with a cos2 law, the THz pulse energy can be determined in the range 0−70 nJ by measuring the relative angle between the two wiregrids. The THz pulse is sent at normal incidence on a commercial α-SiO2 sample together with a fs IR-pulse (∼35 fs of duration, ∼800 nm of central wavelength) which can be delayed with respect to the THz pulse by a given time Δt. Energy and polarization of the IR pulse can be adjusted by means of standard optical elements.

2

= ε∞ +

SΩ Ω2 − ω T 2

(8)

where ε is the dielectric constant and ε∞ is the high-frequency dielectric constant. S and Ω are, respectively, the dimensionless oscillator strength and the proper frequency of the phonon mode associated with the observed polariton. The phonon mode closest to the 9−10 THz range is that of symmetry A2 with a proper frequency around 10.7 THz,21 for which the values of Ω and S are tabulated in refs 21, 24, and 28. In eq 8, we have assumed that the dielectric function may be reduced to a single scalar value at the THz frequencies, but this is not true in general and the specific expression of the corresponding tensor strongly depends on the phonon symmetry and the experimental geometry. Although a more precise model will be provided elsewhere, the qualitative results of our model are not affected by the tensorial nature of the dielectric function. Equation 8 must be solved simultaneously with eqs 7, which may be rewritten: ω + ωT zy2 ij k T yz jj zz = (ωs + ωT)2 nejijj s zz + ωs 2no 2(ωs) 2 k { k c { i ωs + ωT zy j zzcos θ − 2ωs(ωs + ωT)no(ωs)nejj 2 k { 2

2 jij k T zyz = (ω − ω )2 n ijj ωa − ωT yzz + ω 2n 2(ω ) j z a T ej a o a z 2 k { k c { i ω − ωT yz zzcos θ − 2ωa(ωa − ωT)no(ωa)nejjj a 2 k { 2

(9)

where we have introduced the ordinary, n0, and extraordinary, ne, refractive indices of quartz, and θ indicates the angle between k ⃗ L and k ⃗ a,s. By inserting in eqs 8 and 9 suitable values of ε∞, Ω, and S21,24,28 and of the refractive indices,29 we may explain our 1521

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The generated THYR signal is filtered to remove unwanted residuals at the fundamental laser frequency. Two different filters with a thickness of 3 mm each are used alternatively: Schott-BG39 for better accessing the Stokes-side and SchottUG11 for better accessing the anti-Stokes-side. The BG39 has a transmission peak of about 91% at 500 nm, while in the band of our interest the transmission increases monotonously from about 48% at 370 nm to 83% at 450 nm.32 The UG11 filter has a maximum of transmission of about 59% at 330 nm, while the transmission is about 45% at 290 and 360 nm.32 The filtered THYR signal is then dispersed by a blazed-angle grating to analyze its spectral components separately. The signal is thus measured as a function of its wavelength lambda λ and the time-delay Δt. In order to remove any offset signal, such as the SHG signal generated by the fundamental beam only, the THYR signal is measured as the difference between a THz-ON and a THzOFF subsequent pulses by means of a mechanical chopper, which cuts every second pulse on the THz line. This procedure gives as result the 2D map exemplarily shown in Figure 1a. Several commercial α-SiO 2 samples of different axes orientation (X-, Y- , and Z-cut) and thicknesses (50, 500, and 1000 μm) have been measured. The signal is found to be vanishing in all Z-cut samples, while all other samples show qualitatively comparable results. All measurements are performed at room temperature in nitrogen or air atmosphere. The absorption spectrum in the THz range reported in Figure 6 was measured by using the same THYR apparatus, after turning it into a standard THz-TDS spectrometer.1 A detailed scheme of this setup is reported in Figure S3 of SI By means of this technique, we measured the complex refractive index of a 50 μm Y-cut crystal. We used a transmission geometry, and hence, it was not possible to perform the same measurement on the thickest samples given their strong absorption. Despite the small thickness of the sample, we could measure the THz signal only up to 10 THz because of the strong absorption of the sample above this frequency, as better explained in the SI The Raman spectrum was obtained with a commercial spectrometer (WITek Alpha 300) based on a doubled Nd:YAG laser generating CW light with a wavelength of 532 nm. The beam impinges on the sample at normal incidence and the scattered light is detected in reflection, i.e., a backscattering geometry is employed.

We believe that this new technique significantly expands the potential of THz spectroscopy that in the last two decades has already become an invaluable tool in sensing, spectroscopy, and many fields of material science and testing. In particular, by exploiting the collinear geometry and the fact that the THz information is encoded as sidebands of the SHG beam, we anticipate the possibility of using this new technique in combination with scanning probe microscopy for enhancing the spatial resolution of THz spectroscopy at the subnanometer scale. Another research direction may be the application of this technique to the investigation of solute vibrational modes in aqueous solutions where the water resonances usually dominate the THz linear spectra, thus concealing the solute resonances.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.9b00265.

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THYR semiclassical model, analysis of the THYR signal temporal evolution, THz-TDS spectra, and THz apparatus (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: rubano@fisica.unina.it. *E-mail: [email protected]. ORCID

Domenico Paparo: 0000-0002-7745-230X Author Contributions

A.R. and D.P. conceived the experiment. A.R. and S.M. performed the experiment. A.R., D.P., and L.M. developed the theoretical analysis. A.R. and D.P. wrote the paper. All authors discussed the results and contributed to the manuscript. D.P. supervised the work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Prof. Tobias Kampfrath for providing us with the LAPC crystal, Prof. G. Rusciano and Prof. A. Sasso for the measurement of the Raman spectrum, and Arianna Ceraso for helping us to develop a LABVIEW VI to interface the monochromator with the computer. This work was financially supported by ‘Ministero Istruzione Università e Ricerca’ and ‘Consiglio Nazionale delle Ricerche’.

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SUMMARY In conclusion, we have demonstrated a new spectroscopic technique in the THz interval of frequencies that exploits THzstimulated hyper-Raman scattering. Since the THYR process is driven by the hyper-Raman susceptibility, the THYR spectra may be more informative than standard linear THz and IR spectroscopies. In particular, THYR spectroscopy may allow the simultaneous measurements of both IR- and Raman-active material modes laying in the THz portion of the electromagnetic spectrum. We have applied this technique to the benchmark material α-quartz, showing indeed that the THYR signal generated from this material carries information on a large variety of lowenergy excitations that include polaritons and phonons far from the Γ-point. A direct comparison with Raman and THzTDS spectra has shown that the THYR spectra are highly complementary to those obtained by means of these more standard spectroscopies.

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REFERENCES

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