Cubic Nonlinearity of Tellurite and Chalcogenide Glasses: Terahertz-Field-Induced Second Harmonic Generation vs. Optical Kerr Effect

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The study demonstrates the possibility of using terahertz pulses for measuring third-order nonlinear susceptibility responsible for electric-field-induced second harmonic generation.

Abstract

Third-order nonlinear susceptibilities ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ responsible for electric-field-induced second harmonic generation and the optical Kerr effect were measured and directly compared for tellurite and chalcogenide glasses. The nonlinear coefficients were found by measuring the second harmonic radiation from samples under the action of an external field of terahertz pulses and by the classical z-scan technique, respectively. The influence of ambient air and helium gas on second harmonic generation was analyzed. It was demonstrated that both susceptibilities ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ have close absolute values, which shows the dominant role of nonresonant electronic nonlinearity in the studied glasses.

1. Introduction

Optical glasses are the main materials for manufacturing linear optical elements (waveguides, splitters). However, recently, primarily in view of the advance of fiber optics, optical glasses have also been used as nonlinear elements in various optical devices, modulators, optical switches, and frequency converters. Compared to the crystals typically utilized in nonlinear optics, glasses are more cost-effective and involve easy-to-use engineering materials, which is especially important in passive or active fiber networks. The third-order nonlinear coefficient of glasses, for example, the Kerr coefficient n₂, lies in a wide range of values [1,2]: from ~10⁻¹⁶ cm²/W for silicate and fluoride glasses to 10⁻¹⁴ cm²/W for tellurite and ~10⁻¹³ cm²/W for chalcogenide glasses. The glasses of the latter two families, by virtue of such a large nonlinearity and a broad transmission window, are particularly promising materials in near- and mid-IR nonlinear photonics, including supercontinuum [3,4] and frequency comb [5] generation, the development of fiber optical parametric oscillators [6], and others.

Actually, due to macroscopic centrosymmetry, second-order nonlinear phenomena are prohibited in the bulk of the glasses, which limits their nonlinear applications compared to crystals. To overcome this limitation and break the centrosymmetry, different poling techniques have been successfully used [2], including thermal [7,8,9,10] and optical [11,12,13] poling, which have become proven and efficient approaches to induce second-order optical properties in glasses. In both techniques, the most likely mechanism for the occurrence of quadratic nonlinearity is associated with the formation of a built-in (“frozen”) static electric field $E_{froz}$ due to the separation and subsequent freezing of charges on the application of a strong external DC field [14,15,16]. In the case of thermal poling, such charge separation occurs due to the displacement of cations (for example, the alkali ions Na⁺ or Li⁺ typically present in silica glass as impurities [10]) from an anode at a high temperature; in the case of optical poling, due to the displacement of photocarriers excited by short-wavelength optical radiation (UV, blue or green light). As a result, an effective second-order nonlinear susceptibility ${\chi }_{eff}^{\left(2\right)}=3{\chi }^{\left(3\right)}{E}_{froz}$ arises through the electro-optical effect from the third-order nonlinear susceptibility ${\chi }_{}^{\left(3\right)}$ [17]. In a particular (most applicable) case in which poled glasses are used for second-harmonic generation (SHG), two optical frequencies experience nonlinear mixing with the zero frequency of the DC field corresponding to ${\chi }^{\left(3\right)}={\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$. However, in most studies of glass poling, ${\chi }^{\left(3\right)}$ is usually taken from (or assumed to be close to) the all-optical third-order nonlinear susceptibility related to the optical Kerr effect (OKE) (${\chi }^{\left(3\right)}={\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$) or third-harmonic generation (THG) (${\chi }^{\left(3\right)}={\chi }^{\left(3\right)}\left(-3\omega ;\omega ,\omega ,\omega \right)$). In other words, the weak dispersion of ${\chi }^{\left(3\right)}$ is generally implied [9,18,19]. For silicate glasses, this assumption seems to be reasonable in view of the experimentally demonstrated small contribution of nuclear nonlinearity compared to that of electronic nonlinearity [20,21]. Additionally, the direct experimental measurement of the third-order nonlinear susceptibility for some silicate glasses by electric-field-induced SHG (EFISHG) [22,23] in comparison with DC or the optical Kerr effect [24] (including z-scan measurements [25]) and THG [26] showed that the corresponding ${\chi }^{\left(3\right)}$ coefficients are, indeed, rather close to each other. On the other hand, for other promising glass families with large nonlinearity, for example, tellurite and chalcogenide glasses, the use of all-optical ${\chi }^{\left(3\right)}$ in electric-field-induced ${\chi }_{eff}^{\left(2\right)}$ remains open. Moreover, the coefficient ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ inherent for ${\chi }_{eff}^{\left(2\right)}$ has not been measured for tellurite and chalcogenide glasses up until now.

A possible advancement of the EFISHG technique is to use terahertz electromagnetic pulses instead of a DC electric field. This so-called terahertz-field-induced SHG (TFISHG) technique was used for the investigation (primarily the transmission geometry) of the nonlinear response of different materials, including ferroelectric layers [27,28], electro-optic crystals [29,30], SiO₂, and optical adhesives [31,32]. Superstrong (above the DC-field breakdown threshold) terahertz fields (~20 MV/cm) were employed for studying SHG in p-doped Si under below-band-gap optical excitation [33]. TFISHG in reflection geometry was used to characterize the Si(111) surface under above-band-gap optical excitation with a moderate terahertz field (~300 kV/cm) [34]. The specific feature of the TFISHG measurements, shown in Refs. [31,34], is the influence of ambient gas (air) on the detected second harmonic (SH) signal (this influence was also discussed in regards to EFISHG [35]). In the previous studies, due to a rather strong SH signal from the sample or/and some overlapping geometry of the fundamental optical and terahertz pulses, the SHG from ambient gas could be neglected or strongly suppressed by using helium gas instead of air [31,34]. However, in the case of a small SH signal from the sample (due to its small nonlinearity), even rather weak generation from the ambient gas may be significant.

In this paper, third-order nonlinear susceptibilities ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ of different samples of tellurite and chalcogenide glass families were compared using the TFISHG and z-scan [36] techniques. All measurements were carried out using femtosecond laser pulses at the 790 nm fundamental wavelength. In view of the opacity of the samples for SH radiation (at a 395 nm wavelength) as well as of the strong absorption of terahertz radiation (especially in tellurite glasses), the TFISHG measurements were performed in reflection geometry. The generated SH radiation was detected in the reflection direction with respect to the incident fundamental optical pulses. Terahertz pulses were also assumed to be incident on the sample from the side of the laser pulses. In such a setup, the SH radiation generated from the sample can be rather small and comparable to (or even smaller than) the SH generated before the sample in the ambient gas. By changing the time delay between the femtosecond optical and terahertz pulses and using reference signals from fused quartz, we investigated the relationship between the gas and sample contributions to the total SH signals. As a result, the nonlinear susceptibilities ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ (responsible for SHG) for tellurite and chalcogenide glasses were measured for the first time to the best of our knowledge. Their direct comparison with ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ (responsible for the nonlinear self-action) allowed us to assess the applicability of the commonly used assumption of Kleiman symmetry for glasses [18].

2. Experimental Measurements

An amplified Ti:sapphire laser system at ${\lambda }_{\omega }=790$ nm central wavelength (1 kHz, 70 fs, 1 mJ) was used for measuring nonlinear coefficients by the TFISHG and z-scan techniques. We used the following as samples: fused quartz (SiO₂), two tellurite glasses doped with oxides of tungsten, lanthanum and bismuth (or aluminum) with the composition (mol%) 71.2TeO₂-23.7WO₃-4La₂O₃-1.1Bi₂O₃ (TWLB) and 70.6TeO₂-23.5WO₃-4La₂O₃-1.9Al₂O₃ (TWLA), and three chalcogenide glasses: As₂S₃ and commercially available infrared optical glass IKS23 and IKS25 (produced by the S.I. Vavilov State Optical Institute). IKS23 has a composition of As-S-Se, in which Se is a small additive and its optical transparency lies in the range of 0.6–9 µm. IKS25 has a composition of As-Se-Sb-Sn, in which Sb and Sn are small additives, and its optical transparency lies in the range of 1–17 µm.

The z-scan scheme had a classical configuration (see Figure 1a). The optical radiation attenuated by two wedges and calibrated neutral filters was focused (by a lens with F = 40 cm) on a sample placed on a motorized stage. The transmitted radiation was collected into a photodiode, which produced a signal registered by a lock-in amplifier. A variable diaphragm (aperture) was placed in front of the photodiode to perform open- and close-aperture z-scan measurements (in the latter case, the aperture was set to cut about 50% of the transmitted optical energy). For each sample, a series of measurements was recorded for different incident intensities. All were normalized and then a “background” trace with small optical power (which contained linear distortions introduced by the sample) was subtracted from the “signal” traces with high power. After division of the closed aperture by the open aperture traces, the nonlinear phase $\delta {\phi }_{NL}$ was extracted from the difference in peak and valley normalized transmittances $\delta T_{pv}$ using Equation (1) [36]:

$$\delta {\phi }_{NL}=\frac{\sqrt{2}\cdot \delta T_{pv}}{0.406\cdot {(1-s)}^{0.25}},$$

(1)

where s = 0.5. A nonlinear refractive index was then obtained from Equation (2):

$$n_2=\frac{{\lambda }_{\omega }}{2\pi }\frac{\delta {\phi }_{NL}}{L_{eff}I},$$

(2)

where $L_{eff}=\left(1-\exp (-\alpha L\right))/\alpha$, L is sample thickness (in our samples $L_{eff}\approx L$), $\alpha$ is linear absorption coefficient, and I is the intensity of laser radiation in the material placed into focus. A two-photon absorption (2 PA) coefficient $\beta$ was calculated from the valley of normalized transmittances $\delta T$ of an open aperture z-scan trace by solving Equation (3):

$$\delta T\approx -{\displaystyle \sum }_{m=1}^{\infty }\frac{{(-\beta L_{eff}I)}^m}{{(1+m)}^{3/2}},$$

(3)

With known n₂ and β, real and imaginary parts of the nonlinear susceptibility ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ were obtained from Equations (4) and (5) [37]:

$$Re{\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)=\frac{c{n}_{\omega }^2}{12{\pi }^2}{n}_2,$$

(4)

$$Im{\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)=\frac{{\lambda }_{\omega }c{n}_{\omega }^2}{48{\pi }^3}\beta ,$$

(5)

where $n_{\omega }$ is the sample refractive index. To ensure accuracy of our measurements, we used as a reference signal a close aperture z-scan trace in SiO₂ for which the nonlinearity n₂ is known to be 2.5 × 10⁻¹⁶ cm²/W [25,26,38].

Figure 2 shows z-scan traces for two tellurite glasses (TWLA and TWLB) and IKS23 (in this part of the experiment, IKS25 was not considered, as it is opaque at the 790 nm wavelength, the As₂S₃ sample was rather thick for our experimental setup, and we used data from the literature). In the TWLB sample, we observed a somewhat stronger nonlinear effect than that in TWLA, both in open- and closed-aperture z-scan traces. The IKS23 sample exhibited a significantly higher nonlinear response considering that the intensity used was 15 times less. The nonlinear coefficients calculated using Equations (1)–(5) are listed in Table 1 (see below). The magnitudes of n₂ and β for tellurite glasses are about (5–6) × 10⁻¹⁵ cm²/W and (6–8) × 10⁻² cm/GW, respectively, and lie in a reasonable range of values [39,40]. The nonlinear coefficients of IKS23 (n₂ = 1.45 × 10⁻¹³ cm²/W, and β = 7.8 cm/GW) are more than 2–3 times higher than those of the literature data for As₂S₃ at a wavelength of ~0.8 µm (n₂ = 0.65 × 10⁻¹³ cm²/W [41], and β = 1–4 cm/GW [42]) despite their similar compositions (see also the review in Ref. [43]). The main explanation for this discrepancy is the expected dependence of the nonlinearity in IKS23 on the probe laser intensity. Indeed, for As₂S₃ and As₂Se₃, a significant increase in n₂ and β with decreasing probe intensity below 1 GW/cm² and their saturation for the intensity above 1 GW/cm² was discovered in Ref. [44] and proved in Refs. [45,46]. In our experiment, the incident optical intensity on IKS23 was ~0.8 GW/cm², while in Ref. [41] the probe intensity exceeded several tens of GW/cm² (in Ref. [42], the probe intensity varied from 0.8 to 14 GW/cm²). This fact explains the higher nonlinear coefficients measured in our experiment. A small addition of Se in IKS23 should also somewhat increase n₂ and β due to the nonlinear response in As₂Se₃ being about 10 times stronger than that in As₂S₃ [44].

To measure the nonlinear susceptibility ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ responsible for the SHG, we used the TFISHG technique in reflection geometry [34] (in this part of the experiment, we used all samples). The schematic diagram of the experimental setup is shown in Figure 1b (for more detail, see Ref. [34]). The probe femtosecond laser pulses of the fundamental harmonic (with energy ~650 nJ) were weakly focused on the sample at the angle $\alpha ={45}^{°}$ collinear with the strongly focused terahertz pulses, which were generated by the tilted-pulse-front technique in a LiNbO₃ crystal (not shown in Figure 1b). The diameter of the optical beam near the sample was ~160 µm and the peak intensity of the pulse was ~25 GW/cm². The sample was placed near the focus of the terahertz beam (the size of the beam was ~500 μm and the peak amplitude of terahertz electric field was ~170 kV/cm). The duration of the terahertz pulse was about 1 ps, so the terahertz field can be considered as quasi-static relative to the femtosecond fundamental optical pulse (terahertz field waveform was measured in ambient air by a conventional time-domain spectroscopy technique using a 200 µm GaP crystal). The time delay between the optical and terahertz pulses was controlled by a motorized delay line placed in the part of the scheme responsible for the terahertz generation.

As a result of the nonlinear interaction of the femtosecond optical radiation with the glass sample in the presence of the terahertz field, the pulses of second harmonic optical radiation at λ_2ω = 395 nm wavelength were generated in the transmission and reflection directions. The duration of the generated SH pulses was within the duration of the fundamental optical pulses (for the Gaussian pulse, the SH pulse duration is estimated to be ~50 fs). The energy of the SH pulses depended on the time delay between fundamental optical and terahertz pulses. Bearing in mind the strong absorption of the SH radiation in the bulk of most samples (except for SiO₂), we performed measurements in reflection geometry. In the case of SiO₂, the SH radiation generated in the transmission direction reached the exit sample surface and produced an additional SH beam due to Fresnel reflection. This radiation was also registered separately (by using a diaphragm shown in Figure 1b) and was used to calibrate our measurements. In view of the low nonlinear conversion efficiency, the generated SH photons were detected by a Hamamatsu R4220P photomultiplier tube (PMT) operated in the photon counting mode (with quantum efficiency ~20% at the 395 nm wavelength). The generated SH energy (signal) S corresponds to the number of the detected PMT pulses N from 15,000 laser pulses (in case of a high SH signal, when N > 5000 pulses, we used a calibrated neutral filter to attenuate the signal). Taking into account a possible single PMT response to the arrival of several SH photons, the SH signal was calculated by $S=N\left(1+N/15,000\right)$. The fundamental optical and external radiations were filtered by BG39 and band-pass (BP) filters placed before the PMT. The polarizations of the terahertz and fundamental fields were set parallel and normal to the drawing plane (corresponding to the incident s-polarization), the second optical harmonic was measured with the same s-polarization.

An essential feature of the TFISHG scheme used is that during the joint propagation of optical and terahertz radiation in the gas surrounding the sample, a concomitant SH radiation is generated (using a vacuum chamber evidently solves the problem, but significantly complicates the experiment). Despite the typically small gas nonlinearity, due to the large coherence length, the magnitude of the second harmonic from the ambient gas can be comparable to or even exceed the SH signal from the sample. To take into account the gas contribution to the total SH signal correctly, the sample and a part of the experimental setup were placed in a hand-made plastic container filled with helium or ambient air. The choice of helium as an additional gas is explained by its small nonlinearity which is approximately 20 times less than that of air [47,48]. Hence, it was expected that in the helium atmosphere, the measured SH signal will correspond to the signal from the sample. However, as it is shown below, this is not always so.

Figure 3a–d shows normalized SH signals (normalized coefficients are depicted for the corresponding signals) from the exit facet of the SiO₂ sample and from the entrance facet of the SiO₂, TWLA, and As₂S₃ samples, respectively, as a function of time delay between the optical and terahertz pulses. The data were obtained in air and helium atmosphere. It is seen that the SH signal measured in air is always ahead of the SH signal from the same sample in helium. This is a result of faster THz pulse propagation between the parabolic mirrors PM1 and PM2 (see the inset in Figure 1b) when the box was filled with helium. Several other features can be identified in Figure 3. First, the magnitudes of SH signals for different glass families differ significantly (the smallest signal was obtained from SiO₂, and the largest from chalcogenide glasses), which is consistent with their markedly different nonlinearities [2]. Second, the amplitudes of SH signals differ in air and helium atmosphere, which demonstrates the influence of the SH generated in the ambient gas. Third, in some cases, the time-delay profiles of the SH signal qualitatively trace the square of the terahertz field waveform (dashed line in Figure 3a) in both gas atmospheres, but in some cases, this rule is violated (see Figure 3c).

3. Discussion

To explain the obtained experimental results of the TFISHG and retrieve ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$, we represent the total SH signal $S_{entr}^{samp,gas}\left(\tau \right)$ detected from the entrance facet of the sample as a superposition of the contributions from the sample and the ambient gas in the following form:

$$S_{entr}^{samp,gas}\left(\tau \right)=C_{entr}{\left|A_{ref}^{samp}\left(\tau \right)+r_{2\omega }^{samp}{A}_{ins}^{gas}\left(\tau \right)\right|}^2.$$

(6)

Here, $C_{entr}$ is a proportionality coefficient, $A_{ref}^{samp}\left(\tau \right)$ is the complex amplitude of the SH field generated in the reflection direction from the corresponding sample, $A_{ins}^{gas}\left(\tau \right)$ is the complex amplitude of the SH field generated in gas (air or helium) before the sample, and $r_{2\omega }^{samp}$ is the gas sample Fresnel reflection coefficient of the SH field. In general, Equation (6) should also contain $A_{ref}^{gas}\left(\tau \right)$, which is the complex amplitude of the SH field generated in the gas after the sample, but due to the decrease in the fundamental optical and terahertz fields after the reflection, $A_{ref}^{gas}\left(\tau \right)$ is typically more than 10 times less than $r_{2\omega }^{samp}{A}_{ins}^{gas}\left(\tau \right)$. So, in most cases, we will ignore $A_{ref}^{gas}\left(\tau \right)$. The SH radiation emitted into SiO₂ and then reflected from the exit facet (with subsequent transmission through the entrance facet) gives the SH signal:

$$S_{exit}^{Si{O}_2,gas}\left(\tau \right)=C_{exit}^{Si{O}_2}{R}_{}^{Si{O}_2}{T}_{}^{Si{O}_2}{\left|A_{tr}^{Si{O}_2}\left(\tau \right)+t_{2\omega }^{Si{O}_2}{A}_{ins}^{gas}\left(\tau \right)\right|}^2,$$

(7)

where $C_{exit}^{Si{O}_2}=C_{entr}{n}_{2\omega }^{Si{O}_2}\cos ({\beta }_{2\omega }^{Si{O}_2})/\cos (\alpha )\approx 1.8C_{entr}$, $n_{2\omega }^{samp}$, and ${\beta }_{2\omega }^{samp}$ are the sample (SiO₂) refractive index and the transmission angle of SH in the sample (SiO₂), respectively; $A_{tr}^{Si{O}_2}\left(\tau \right)$ is the complex amplitude of the SH fields emitted into SiO₂ (in the transmission direction); $t_{2\omega }^{Si{O}_2}=0.71$ is the gas–SiO₂ Fresnel transmission coefficient of the SH field (generated in the gas before SiO₂); and $R_{}^{Si{O}_2}=0.085$ and $T_{}^{Si{O}_2}=0.915$ are the power reflection and transmission coefficients of SH radiation from the SiO₂ boundary, respectively.

The time-delay profile of the detected SH signal, as follows from Equations (6) and (7), significantly depends on the amplitude, relative phase, and time-delay profiles of the corresponding terms. Below we will assume that the electron nonlinearity is dominant, i.e., we will consider the nonlinear response to be instantaneous both in the samples and gases (see the discussion below). The amplitude of the SH field generated from the entrance facet of the material in the reflection direction $A_{ref}^{samp}\left(\tau \right)$ does not strongly depend on the dynamics of the terahertz field in the media [37]. The main factor is the ratio of the THz penetration depth ($d_{THz}$) and the SH wavelength in the material (${\lambda }_{2\omega }/n_{2\omega }^{samp}$), which affects the phase of $A_{ref}^{samp}\left(\tau \right)$ [34,49]. In our samples, $d_{THz}>>{\lambda }_{2\omega }/n_{2\omega }$, which gives:

$$A_{ref}^{samp}\left(\tau \right)=-4\pi \frac{{\chi }_{eff}^{\left(3\right)}{E}_{THz}\left(\tau \right)E_{\omega }^2}{\left[n_{2\omega }^{samp}\cos ({\beta }_{2\omega }^{samp}\right)+n_{\omega }^{samp}\cos ({\beta }_{\omega }^{samp})]\left[n_{2\omega }^{samp}\cos ({\beta }_{2\omega }^{samp}\right)+\cos (\alpha )]},$$

(8)

where ${\chi }_{eff}^{\left(3\right)}=3/2{\chi }_{samp}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$, $n_{\omega }^{samp}$, and ${\beta }_{\omega }^{samp}$ are the sample refractive index and the transmission angle for the fundamental wavelength; $E_{THz}\left(\tau \right)$ is the transmitted terahertz field related to the corresponding incident fields $E_{THz}^{ins}\left(\tau \right)$ through the Fresnel transmission coefficient $t_{THz}^{samp}$ (note that in general, $E_{THz}\left(\tau \right)$ should not trace $E_{THz}^{ins}\left(\tau \right)$ due to material dispersion); and $E_{\omega }=t_{\omega }^{samp}{E}_{\omega }^{ins}$, where $E_{\omega }^{ins}$ is the amplitude of the incident fundamental optical field and $t_{\omega }^{samp}$ is the Fresnel transmission coefficient.

The amplitude of the SH field generated inside the material $A_{tr}^{samp}\left(\tau \right)$ (in our case this field is propagated only in SiO₂), in contrast, is rather sensitive to the mutual propagation dynamics of the terahertz and optical pulses [31]. In fused quartz, the coherent length of the fundamental and SH radiation (~10 µm) is significantly less than the walk-off length of the optical and terahertz pulses: ~200 µm. This corresponds to the adiabatic regime of SH generation which, with an allowance for the theory developed in Ref. [31] and generalized for the oblique incident terahertz and optical pulses, gives (see also [37]):

$$A_{tr}^{Si{O}_2}\left(\tau \right)=4\pi \frac{{\chi }_{eff}^{\left(3\right)}{E}_{\omega }^2{E}_{THz}\left(\tau \right)}{{(n_{2\omega }^{Si{O}_2})}^2-{(n_{\omega }^{Si{O}_2})}^2}\frac{n_{\omega }^{Si{O}_2}\cos ({\beta }_{\omega }^{Si{O}_2})+\cos (\alpha )}{n_{2\omega }^{Si{O}_2}\cos ({\beta }_{2\omega }^{Si{O}_2})+\cos (\alpha )}\approx 4\pi \frac{{\chi }_{eff}^{\left(3\right)}{E}_{\omega }^2{E}_{THz}\left(\tau \right)}{{(n_{2\omega }^{Si{O}_2})}^2-{(n_{\omega }^{Si{O}_2})}^2}.$$

(9)

Note that Equation (9) is valid only for $\tau$ less than the time delay required for terahertz and optical pulses to overlap on the exit facet of the SiO₂ layer. (For the 2 mm thick SiO₂ layer used in the TFISHG experiment, this condition gives $\tau <2$ ps; for $\tau >2$ ps, there appears to be an SH signal from the exit facet of the SiO₂ layer [31,32].)

The complex amplitudes of the SH field generated in the gas $A_{ins}^{gas}\left(\tau \right)$ are mainly determined by the spatiotemporal dynamics of the terahertz pulse during the joint propagation with the fundamental optical pulse after the parabolic mirror PM2 (see the inset in Figure 1b). Optical and terahertz pulses propagate in gas with approximately the same velocities. However, due to strong terahertz beam-focusing (with NA~0.4) and having a near-single cycle duration, not only the amplitude but also the terahertz field waveforms change significantly (see Figure 4). Thus, $A_{ins}^{gas}\left(\tau \right)$ should not trace the local in space $E_{THz}^{ins}\left(\tau \right)$ measured near the focus of the parabolic mirror PM2 (where the sample was placed). On the other hand, the strongest terahertz field and, as a consequence, a more efficient SH emission occurs in a definite region near the focus. Thus, we can assume that the SH is generated in some effective gas layer, which, according to our estimates, has a thickness $D\approx 1$ cm. Additionally, we can consider average terahertz fields $E_{av}^{ins}\left(\tau \right)$ with a constant profile in the effective gas layer. According to Figure 4, $E_{av}^{ins}\left(\tau \right)$ should be shifted forward for some time delay $\Delta {\tau }_{ins}$ relative to the terahertz field in focus $E_{THz}^{ins}\left(\tau \right)$. This time shift was directly observed in the experiment. As will be shown below, the SH signal measured from the entrance facet of SiO₂ in an air atmosphere $S_{entr}^{Si{O}_2,air}\left(\tau \right)$ has the main contribution from the gas ($S_{entr}^{Si{O}_2,air}\left(\tau \right)~|A_{ins}^{air}\left(\tau \right){|}^2~{(E_{av}^{ins}\left(\tau \right))}^2$). As seen from Figure 3b, $S_{entr}^{Si{O}_2,air}\left(\tau \right)$ is shifted to the right by ~120 fs relative to ${(E_{THz}^{ins}\left(\tau \right))}^2$. This confirms the proposed model. Additionally, it is interesting that the shape of $S_{entr}^{Si{O}_2,air}\left(\tau \right)~{(E_{av}^{ins}\left(\tau \right))}^2$ is rather close to ${(E_{THz}^{ins}\left(\tau \right))}^2$ (with a slight deviation of the lateral maxima). This allows us to assume that $E_{av}^{ins}\left(\tau \right)$ and $E_{THz}^{ins}\left(\tau \right)$ have similar profiles. In the framework of the model, taking into account that $n_{2\omega }^{gas}-n_{\omega }^{gas}<<1$ in gas, we obtain [31]:

$$A_{ins}^{gas}\left(\tau \right)=16\pi {\chi }_{gas}^{\left(3\right)}{E}_{av}^{ins}\left(\tau \right){\left(E_{\omega }^{ins}\right)}^2\frac{L_c^{gas}}{{\lambda }_{\omega }{n}_{2\omega }^{gas}}\exp \left(-i\frac{\pi }{2}\left(1+\frac{D}{L_c^{gas}}\right)\right)\sin \left(\frac{\pi }{2}\frac{D}{L_c^{gas}}\right),$$

(10)

where $L_c^{gas}={\lambda }_{\omega }/\left(4\left(n_{2\omega }^{gas}-n_{\omega }^{gas}\right)\right)$ is coherent length ($L_c^{air}\approx 2.5$ cm in air and $L_c^{He}\approx 60$ cm in helium), and ${\chi }_{gas}^{\left(3\right)}$ is the effective third-order nonlinear susceptibility of gas (${\chi }_{air}^{\left(3\right)}/{\chi }_{He}^{\left(3\right)}\approx 20$ [47,48]). Assuming D~1 cm, the phase of the peak complex amplitude in air and in helium can be estimated to be $\arg (A_{ins}^{air})\approx -1.4\pi /2$ and $\arg (A_{ins}^{He})\approx -\pi /2$, respectively.

Now let us use the proposed model to describe the SH signals shown in Figure 3 and to find ${\chi }_{samp}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$. For the simplicity of our presentation, we will take $C_{entr}=1$ (complex SH amplitudes will be in arb.u). Since helium makes a significantly smaller additional contribution to the SH signal compared to air, the measurements in the helium atmosphere are our major interest. Using Equations (6) and (8), it is possible to establish the relationship between the nonlinear susceptibilities of the studied (tellurite and chalcogenide glasses) and the reference (SiO₂) samples from the corresponding known SH signals. However, contributions of SiO₂ and gas (helium) towards the reference SH field amplitude should be also known. (Note that the absolute value of nonlinear susceptibility ${\chi }_{samp}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ can be also obtained from the magnitude of the SH signal, the terahertz field magnitude and the intensity of the fundamental optical pulses using Equations (6) and (8), with an accounting of the quantum efficiency of the PMT and transmission coefficients of the optical filters (BG39 and BP, see Figure 1b). This was earlier performed in Ref. [34] and close values of ${\chi }_{Si{O}_2}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }_{Si{O}_2}^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ were shown for SiO₂.)

From the comparison of the SH signal magnitudes in Figure 3a,b, we conclude that only the material (SiO₂) makes a substantial contribution to the SH signal from the exit facet of SiO₂ in He (red points in Figure 3a), whereas the contribution from He gas is negligible. This agrees with the fact that the SH signal profile is well fitted by the profile of the terahertz electric field squared (dotted line in Figure 3a). Thus, Equation (7) is simplified to $S_{exit}^{Si{O}_2,He}\left(\tau \right)=C_{exit}^{Si{O}_2}{R}_{}^{Si{O}_2}{T}_{}^{Si{O}_2}{\left|A_{tr}^{Si{O}_2}\left(\tau \right)\right|}^2$, which, taking into account the peak value of the SH signal $S_{exit}^{Si{O}_2}$~3000 counts, gives the peak amplitude $A_{tr}^{Si{O}_2}=146$. After substituting this value into Equation (9) and then using Equation (8), we found the peak SH amplitude generated in the reflection direction $A_{ref}^{Si{O}_2}\approx -0.01A_{tr}^{Si{O}_2}=-1.5$. This value can be considered as a reference in the following calculation.

To find the helium contribution $A_{ins}^{He}\left(\tau \right)$, one should consider $S_{entr}^{Si{O}_2,He}\left(\tau \right)$ which is shown by the red diamonds in Figure 3b. Due to the low photon numbers, $S_{entr}^{Si{O}_2,He}\left(\tau \right)$ has large fluctuations. We averaged several points near the maximum of the SH signal, which gave $S_{entr}^{Si{O}_2,He}\approx 8$ counts. From Equation (6) with known $A_{ref}^{Si{O}_2}$, taking into account Equation (10) (where, similarly to air, $\Delta {\tau }_{ins}$ is assumed to be close to 120 fs and $E_{av}^{ins}\left(\tau \right)$ close to $E_{THz}^{ins}\left(\tau \right)$), we obtained the SH amplitude generated in helium $A_{ins}^{He}\approx 9\exp (-i\pi /2)$.

Now we can calculate complex SH amplitudes from other samples. In the case of the tellurite samples, the measured SH signal is about 10 times higher than that from SiO₂ (see Figure 3c for the TWLA sample), which is associated with the amplification of the SH signal from the sample due to its larger nonlinearity. However, it should be remembered that the reflection from the sample of the SH field generated in helium also increases (in Equation (6) $r_{2\omega }^{TWLA}\approx 1.7r_{2\omega }^{Si{O}_2}$). From the peak value of $S_{entr}^{TWLA,He}~70$ counts, using Equation (6) with known $A_{ins}^{He}$, we estimated $\left|A_{ref}^{TWLB}\right|~9$. Note that the profile of $S_{entr}^{TWLA,He}\left(\tau \right)$ diverges noticeably from ${(E_{THz}^{ins}\left(\tau \right))}^2$. This can be explained by partial interference of $A_{ins}^{He}\left(\tau \right)$ and $A_{ref}^{TWLA}\left(\tau \right)$ due to the appearance in the latter amplitude of a nonzero phase (according to our estimates it is about 0.2–$0.3$ rad; see the dotted curve in Figure 3c). This phase can arise from the complex value of $n_{2\omega }$ (tellurite glass is not transparent for SH) and the complex value of ${\chi }_{TWLA}^{\left(3\right)}$ (the latter is supported by the z-scan measurement; see Table 1). The TWLB sample has an SH signal profile similar to TWLA with an amplitude of about 90 counts, which, following the same approach, gives $\left|A_{ref}^{TWLB}\right|~11$.

In the case of chalcogenide glasses, we observed an SH signal that was more than 200 times stronger than that in tellurite glasses ($S_{entr}^{A{s}_2{S}_3,He}\left(\tau \right)$ in Figure 3d for As₂S₃). The time-delay shapes of the SH signals were close to the squared terahertz electric field ${\left(E_{THz}^{ins}\left(\tau \right)\right)}^2$ (dotted curve in Figure 3d). Thus, in this case, the helium contribution was negligible. From the measured peak SH signals, we obtained $\left|A_{ref}^{A{s}_2{S}_3}\right|\approx 152$, $\left|A_{ref}^{IKS23}\right|\approx 148$ and $\left|A_{ref}^{IKS25}\right|\approx 130$.

Let us make a brief note about the SH signals measured in air. By comparing $|A_{ref}^{Si{O}_2}{|}^2\approx 2$ counts and $S_{entr}^{Si{O}_2,air}\approx 800$ counts (blue points in Figure 3b), we can confirm the above-mentioned statement that $S_{entr}^{Si{O}_2,air}\left(\tau \right)$ primarily has a contribution from gas. Then, taking into account the theoretically estimated phase $-1.4\pi /2$, we have $A_{ins}^{air}\approx 97\exp (-i0.65\pi )$. By substituting $A_{tr}^{Si{O}_2}\left(\tau \right)$ and $A_{air}^{ins}\left(\tau \right)$ into Equation (7) (with allowance for the $\Delta {\tau }_{ins}\approx 120$ fs shift of $E_{av}^{ins}\left(\tau \right)$ and $E_{THz}^{ins}\left(\tau \right)$), we obtain the profile (dash-dotted curve in Figure 3a) which agrees rather well with the experimental signal $S_{exit}^{Si{O}_2,air}\left(\tau \right)$. This confirms our estimated phase of $A_{air}^{ins}$. The SH signal from the TWLA sample in air $S_{entr}^{TWLA,air}\left(\tau \right)$, similarly to SiO₂, mainly has a gas contribution. The difference in the $S_{entr}^{TWLA,air}\left(\tau \right)$ and $S_{entr}^{Si{O}_2,air}\left(\tau \right)$ profiles can be explained by the addition of the SH signal generated in air after the reflection from TWLA ($A_{ref}^{air}\left(\tau \right)$ should be added to Equation (6)) by virtue of large TWLA reflection coefficients in optical and terahertz ranges.

From the obtained values of the SH amplitudes $\left|A_{ref}^{samp}\right|$, using Equation (8) we calculated the relation of the sample and SiO₂ susceptibilities $\left|{\chi }_{samp}^{\left(3\right)}\left(2\omega \right)\left|/\right|{\chi }_{Si{O}_2}^{\left(3\right)}\left(2\omega \right)\right|$ (see Table 1). For tellurite glasses TWLA and TWLB, the ratio is 65 and 80, respectively. For chalcogenide glasses, it is more than 1000. IKS23 and As₂S₃ exhibit equal nonlinear susceptibilities $~1200\left|{\chi }_{Si{O}_2}^{\left(3\right)}\left(2\omega \right)\right|$, which is consistent with their similar compositions. IKS25 has a nonlinearity about 1.5 times higher ($\left|{\chi }_{IKS25}^{\left(3\right)}\left(2\omega \right)\left|\approx 1900\right|{\chi }_{Si{O}_2}^{\left(3\right)}\left(2\omega \right)\right|$), which is due to the stronger nonlinear response of Se contained in the sample [44].

Now we can compare the susceptibility values measured by the TFISHG technique with the data obtained from the z-scan. As mentioned in the Introduction, the nonlinear susceptibilities ${\chi }_{Si{O}_2}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }_{Si{O}_2}^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ for SiO₂ can be considered to be close, according to previous experimental measurements [22,23,24,34]. For tellurite glasses and As₂S₃, our measurements demonstrated that ${\chi }_{samp}^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ is approximately a factor of 1.5 higher than ${\chi }_{samp}^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$. However, taking into consideration the measurement error of both methods, it is a very good agreement. Note that the main error in the TFISHG experiment was the calibration by the fused quartz. This calibration was made for the experimental conditions slightly differing from the main measurements, which could have produced a systematic error. For the IKS23 sample, the nonlinearity measured by the TFISHG method, in contrast, was 1.5 times smaller than that measured by the z-scan. As noted above, when considering z-scan measurements, this seems to be related to the different values of the intensity of the probe optical radiation. The intensity in the z-scan measurements was about 0.8 GW/cm² and was in the region of the increased nonlinear response of the sample, whereas in the TFISHG measurements, the optical intensity was about 25 GW/cm² and corresponded to a minimum saturation of nonlinearity [44]. This may explain the stronger nonlinear response of IKS23 in the z-scan.

Table 1. Linear and nonlinear parameters of the studied samples.

Sample (Thickness)	Linear Parameters			Nonlinear Parameters from z-Scan Measurements (1)					Nonlinear Parameters from TFISHG Measurements (1)
	nω	n2ω	nTHz	n2, 10−15 cm2/W	β, 10−11 cm/W	Reχ (3)(ω), 10−13 esu	Imχ (3)(ω), 10−13 esu	|χ (3)(ω)|/| χ (3)SiO2(ω)|	|χ (3)(2ω)|/| χ (3)SiO2(2ω)|	$\left|{\mathit{A}}_{\mathit{r}\mathit{e}\mathit{f}}^{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}\right|$
SiO2 (0.9 mm or 2 mm)	1.453	1.47	2	0.25 (2)	-	0.13	-	1	1	1.5
TWLA (2.3 mm)	2.2 (3)	2.35 (3)	~5 (4)	5.1	6	6.3	0.47	47	65	9
TWLB (2.3 mm)	2.2 (3)	2.35 (3)	~5 (4)	6.1	8	7.5	0.63	56	80	11
IKS23 (1.6 mm)	2.45	(3—0.5i) (5)	2.9	145 (6)	780 (6)	220	75	1760	1200	148
IKS25 (1.8 mm)	2.9 (5)	(3.3—i) (5)	3.3	-	-	-	-	-	1900	130
As2S3 (6.1 mm)	2.45 (7)	3—0.5i (7)	2.8	65 (8)	~250 (9)	95	~23	730	1200	152

⁽¹⁾ Estimated experimental errors of z-scan and TFISHG measurements are ~20% and ~40%, respectively. ⁽²⁾ Refs. [25,38,50]. ⁽³⁾ Refs. [51,52]. ⁽⁴⁾ Refs. [53,54]. ⁽⁵⁾ Refractive indices for IKS23 and IKS25 were taken close to As₂S₃ and As₂Se₃, Ref. [55], respectively, due to their close compositions. ⁽⁶⁾ Probe optical intensity is ~0.8 GW/cm². ⁽⁷⁾ Ref. [55]. ⁽⁸⁾ Ref. [41]. ⁽⁹⁾ Ref. [42].

Table 1. Linear and nonlinear parameters of the studied samples.

Sample (Thickness)	Linear Parameters			Nonlinear Parameters from z-Scan Measurements (1)					Nonlinear Parameters from TFISHG Measurements (1)
	nω	n2ω	nTHz	n2, 10−15 cm2/W	β, 10−11 cm/W	Reχ (3)(ω), 10−13 esu	Imχ (3)(ω), 10−13 esu	|χ (3)(ω)|/| χ (3)SiO2(ω)|	|χ (3)(2ω)|/| χ (3)SiO2(2ω)|	$\left|{\mathit{A}}_{\mathit{r}\mathit{e}\mathit{f}}^{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}\right|$
SiO2 (0.9 mm or 2 mm)	1.453	1.47	2	0.25 (2)	-	0.13	-	1	1	1.5
TWLA (2.3 mm)	2.2 (3)	2.35 (3)	~5 (4)	5.1	6	6.3	0.47	47	65	9
TWLB (2.3 mm)	2.2 (3)	2.35 (3)	~5 (4)	6.1	8	7.5	0.63	56	80	11
IKS23 (1.6 mm)	2.45	(3—0.5i) (5)	2.9	145 (6)	780 (6)	220	75	1760	1200	148
IKS25 (1.8 mm)	2.9 (5)	(3.3—i) (5)	3.3	-	-	-	-	-	1900	130
As2S3 (6.1 mm)	2.45 (7)	3—0.5i (7)	2.8	65 (8)	~250 (9)	95	~23	730	1200	152

⁽¹⁾ Estimated experimental errors of z-scan and TFISHG measurements are ~20% and ~40%, respectively. ⁽²⁾ Refs. [25,38,50]. ⁽³⁾ Refs. [51,52]. ⁽⁴⁾ Refs. [53,54]. ⁽⁵⁾ Refractive indices for IKS23 and IKS25 were taken close to As₂S₃ and As₂Se₃, Ref. [55], respectively, due to their close compositions. ⁽⁶⁾ Probe optical intensity is ~0.8 GW/cm². ⁽⁷⁾ Ref. [55]. ⁽⁸⁾ Ref. [41]. ⁽⁹⁾ Ref. [42].

Finally, we can consider in brief the possible nuclear contribution to the SH generation. This contribution would evidently change the time-delay profile of the SH signal [56] (it will not trace the squared terahertz electric field). In our experiment, we observed some signal changes for the TWLA and TWLB glasses (see $S_{entr}^{TWLA,He}\left(\tau \right)$ in Figure 3c). We explained these changes by the interference of the SH fields generated in helium and in the sample, which are in reasonable agreement with the experiment. The assumption of a nuclear contribution could also partially explain our experimental results, but it will add a tail in $S_{entr}^{TWLA,He}\left(\tau \right)$, which was not observed in the experiment (Figure 3c). In the chalcogenide glasses, we did not find any deviation of the SH signal profile from the terahertz field squared within the accuracy of our experiment. Thereby, we do not expect that the nuclear contribution is essential in the SH response in the investigated samples.

4. Conclusions

To conclude, the third-order nonlinear susceptibilities ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ and ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ of tellurite and chalcogenide glasses have been measured using the recently developed THISHG and classical z-scan techniques. It has been shown that for correct retrieval of ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ from the THISHG measurements, the influence of the ambient gas on the generated SH signal must be taken into account. In some cases, the gas contribution to the total detected signal can dominate over that of the sample. Using helium instead of an air atmosphere significantly decreases the contribution from gas, but even in this case, this contribution cannot always be ignored, especially for materials with a low nonlinearity comparable to that of fused quartz. Using a series of measurements with a calibrated sample (in our experiments it was fused quartz), it was possible to estimate the contribution of gas and obtain the nonlinear susceptibilities of the samples. Note that the interfering (additional) SHG from the ambient gas can also be beneficial and provide information on the complexity (existence of imaginary components) of the nonlinear susceptibility in the investigated samples. With the help of this procedure, we measured ${\chi }^{\left(3\right)}\left(-2\omega ;\omega ,\omega ,0\right)$ for the tellurite and chalcogenide glasses and found that these susceptibilities are close to ${\chi }^{\left(3\right)}\left(-\omega ;\omega ,-\omega ,\omega \right)$ (measured by the z-scan technique), which is consistent with the dominant role of nonresonant electronic nonlinearity in the investigated samples. This result is significant from both fundamental and applied points of view. From the fundamental point of view, it proves the commonly used assumption of Kleiman symmetry for glasses. From the point of view of application, it confirm the applicability of χ⁽³⁾(−ω;ω,−ω,ω) for estimating the effective second-order nonlinear susceptibility ${\chi }_{eff}^{\left(2\right)}=3{\chi }^{\left(3\right)}{E}_{froz}$ (see the Introduction) used in studies of glass poling.