1 Introduction

The lowest order optical nonlinearity observed in centrosymmetric materials is the third-order nonlinearity. It is responsible for a gamut of phenomenon-like third harmonic generation (THG), laser-induced transient grating (LITG) formation, self-focusing, optical Kerr effect (OKE), and supercontinuum generation. The third-order nonlinear optical response of transparent materials is thus an important parameter from the viewpoint of applications. Hence, glasses with large instantaneous third-order nonlinear coefficients are used in optical communication and all-optical switching applications. Ultrafast all-optical processing and switching capabilities are required in order to achieve high-speed optical communication. The basic switching mechanism involved is third-order optical nonlinearity (nonlinear index coefficient n 2—the Kerr nonlinearity) and, hence, the amplitude and the response time of all the contributions to the third-order optical nonlinearity assume importance in optical design considerations.

The origin of the optical nonlinearity in transparent materials like glasses is attributed to electronic and nuclear contributions for laser pulses shorter than a few hundreds of picoseconds (a consequence of the Born–Oppenheimer approximation). The electronic response is associated with the distortion of the electronic charge cloud around the nuclei while the nuclear response appears due to slower optical-field-induced changes in the motions of nuclei. These two contributions have significantly different relaxation times, electronic response being less than 1 fs whereas nuclear response ranges from several hundred femtoseconds to several picoseconds. The nuclear contribution to the nonlinear refractive index of some glasses is not negligible, being as high as 15–20 % of the total, depending on glass composition [1]. There are some earlier measurements of the nuclear contribution in glasses that have been performed in the femtosecond regime [2–6]. However, these reported values of nuclear contributions are often subject to change since they depend strongly on duration of the optical pulse [6, 7]. The nuclear contributions are, of course, of importance as they may limit the speed of response and induce thermal effects in a device. However in usual glasses, previous experiments performed in the time domain with short pulses (∼100 fs) have failed to demonstrate such a response. It has been suggested that this could be due to a rapid dephasing of localized vibrations in the inhomogeneous and disordered structure of a glass, inducing a short nuclear response that is almost indistinguishable from the instantaneous electronic contribution. In case of non-instantaneous response, earlier reports [8] measured the temporal evolution of the nonlinear susceptibility in standard glasses but did not observe a nuclear contribution, attributing their results to nuclear relaxation times being either too fast or too slow. Kang et al. [2, 9] were able to show evidence of such a contribution using ultrashort pulses (∼35 fs) exciting localized stretching modes of glasses.

In a degenerate non-collinear pump-probe OKE experiment, the optical Kerr signal usually originates from two effects: photoinduced birefringence and self-diffraction that arises from LITG. LITG occurs due to the interference of pump and probe beams; part of the pump beam may be diffracted into the direction of the probe beam by LITG, and may then be detected as a Kerr signal. In the case of photoinduced birefringence, the incident pump beam induces a transient optical anisotropy in the medium, which results in rotation of the probe beam’s polarization, allowing transmission of light through the analyzer and, thus, enabling detection as a Kerr signal. An earlier study has shown that for ultrashort pulses used in OKE experiments’ inorganic molecules [10], Kerr signals result mainly from LITG-induced self-diffraction of the pump beam.

The temporal response of self-diffraction and OKE is different and can help in understanding the self-diffraction effect on ultrafast optical OKE measurements. Recently [11] the influence of self-diffraction on ultrafast optical Kerr measurements has been reported both in CS2 and BI glass. It was observed that there were two components in the Kerr signal of CS2: a fast response attributable to self-diffraction whose contribution could be controlled by varying the pump power, and a slow response that was attributed to photoinduced birefringence. Studies [12–16] have shown that in BaF2, transient grating formation at high intensities gave rise to a diffraction efficiency of better than 10%. Using 45-fs pulses it has been shown that BaF2 also displays remarkable propensity for supercontinuum generation and filamentation [17–20]. In BaF2, cross-polarized wave generation (XPW) was investigated that led to achievement of the highest-reported contrast value of ∼1010 in the temporal domain situated tens of picoseconds before the main femtosecond pulse [21]. All these experiments involved excitation at ∼1.5 eV photon energy that was far from any resonance (the band gap in BaF2 is ∼9 eV); hence, the optical nonlinearity involved would be expected to have an instantaneous response. Is there any contribution of non-instantaneous response to these nonlinear effects in BaF2?

We investigate this question using LITG and OKE, measuring instantaneous and non-instantaneous responses in various transparent solids like glasses (BK7 and fused silica) and crystals (BaF2 and quartz), using 36-fs, 800-nm laser pulses. OKE in BaF2, to the best of our knowledge, has not hitherto been investigated. Furthermore, apart from the electronic response we also observe contribution from the nuclear response when BaF2 is irradiated by ultrashort pulses. These manifest themselves in the form of oscillatory modulations that persist for ∼400 fs. These new measurements on BaF2 are of particular contemporary importance because of the recently established usage of this crystal as contrast enhancer in multi-PW laser systems [22]. BaF2 has also recently found utility as a cross-polarized wave generator [21]. Similarly, the experiments that we report on BK7 glass and fused silica have direct relevance to recent progress achieved in femtosecond laser writing of waveguides embedded within these materials [23, 24].

We also report in the following values of the third-order optical nonlinearity, χ (3), in transparent glasses (BK7 and fused silica) as well as in crystals (BaF2 and quartz); our values lie in the range 1.3–1.7×10-14 esu, in good accordance with theoretical estimates. In the case of BaF2 and BK7 glass, we are not aware of any earlier measurements of χ (3) at 800 nm.

2 Experimental setup

A schematic of the setup used by us to carry out two-beam experiments is shown in Fig. 1(a). In brief, output pulses from a Ti-sapphire laser (wavelength 800 nm, 36 fs pulse duration, 1 kHz repetition rate) are split using a 50:50 beam splitter. The two beams are then steered through two pair of mirrors, with one pair fixed on a manual translation stage (with a 1-{μ}m resolution) and the other pair of mirrors fixed on a motorized translation stage (with a 0.1-{μ}m resolution) to generate a temporal delay. The two beams are focused using a 1-m focal length lens (placed before the beam splitter) onto the sample. A typical temporal profile of the laser pulse measured using spectral phase interferometry for direct electric field reconstruction (SPIDER) technique is shown in Fig. 1(b). The pulse is ensured to possess a flat zero phase. Both the beams are ensured to be vertically polarized. The temporal delay is initially matched using second harmonic generation [25].

Fig. 1
figure 1

Experimental configurations used in the present studies. (a) A two-beam setup using 800-nm, 36-fs, 1-kHz laser pulses in a non-collinear geometry with an angle of 4∘; (b) Laser pulse characterization using the SPIDER technique (see text); (c) Two-beam configuration for the transient grating geometry; and (d) Optical Kerr effect geometry

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With this setup we have carried out two sets of experiments: one set that involved transient grating formation in the self-diffraction geometry, and the other involving the formation of an optical Kerr gate. The transient grating geometry is shown in Fig. 1(c). As can be seen, the two beams are incident on the sample at an angle of 4∘. Due to the interference of the two beams, intensity modulation occurs, which gives rise to spatial intensity variation. This variation in intensity changes the local refractive index. At the location of maxima the refractive index increases. The variation in refractive index, in turn, gives rise to a phase grating which diffracts the incident light beams. The two central spots that are shown are the transmitted fundamental beams along with a diffraction signal. In the self-diffraction geometry typical energies (intensities) used in both the beam paths are in the range of 35–60 μJ (0.6–1.0×1012 W cm-2). The third harmonic spectrum is measured using a fiber-coupled CCD based spectrometer (wavelength range 200–870 nm).

For optical Kerr gate measurements, the two-beam setup is slightly modified. The energy in one of the paths is reduced so that it acts as a probe beam and the other beam acts as a pump beam. In the probe beam path a half-wave (λ/2) plate is placed before the sample and an analyzer is kept after the sample (Fig. 1(d)). The half-wave plate rotates the plane of polarization of the probe beam by 45∘ with respect to the pump beam. The probe beam, after passing through the sample, is crossed using a Glan-laser polarizer (extinction ratio 105:1) at the detector (calibrated silicon photodiode). No additional polarizer was used; the laser polarization contrast was ∼1000:1. Initially, when the pump and the probe beams do not overlap, no signal is detected on the detector. At zero delay, when the two beams overlap, the pump beam causes anisotropy in the sample, inducing birefringence; changing the polarization of the probe beam causes some light to transmit through the analyzer. Typical energies (intensities) used in this geometry for the pump and probe beam are ∼92 μJ (1.65×1012 W cm-2) and ∼6 μJ (1×1011 W cm-2), respectively.

3 Results and discussion

When the two laser beams overlap in both space and time in a transparent material, they interfere and subsequently give rise to an intensity modulation. This variation in intensity changes the local refractive index of the medium due to intensity-dependent refractive index. At locations of maximum intensity the refractive index increases compared to locations of intensity minima. This variation in the refractive index gives rise to a phase grating whose spatial period, Λ, is given by [26]

$$\varLambda = 2\pi /\mathbf{q} = \lambda /[2 \sin (\theta /2)], $$
(1)

where λ is the fundamental wavelength and θ is the angle between the two interfering beams. At an incident angle of 4∘ between the two beams we estimate the grating period to be ∼11 μm. When two beams obtained from an ultrashort laser are incident at small angle on the sample, along with the transmitted beam spots we observe diffracted spots due to self-diffraction from the phase grating (see Fig. 1(c)). If the material has a high nonlinear response or if the intensity of beams is high, higher diffraction order (second and higher) might be observed. Using a dye molecule we have observed higher diffraction orders and intense incoherent amplified spontaneous emission using ps-transient grating geometry [27, 28]. Further, at even higher incident intensities the possibility of supercontinuum (white light) generation cannot be avoided. In our measurements we have used such laser intensity values that they did not generate any supercontinuum, and only first-order diffraction signals were observed.

We have carried out measurements using the self-diffraction geometry in different crystals as well as glasses. The photograph in Fig. 2 shows the transmitted pattern of the laser light observed in case of BaF2 when two ultrashort laser pulses at 800 nm are incident at an angle of 4∘. As can be seen, the two bright spots are the transmitted beams at 800 nm. In between the two bright spots we also observe two blue spots on a white screen. We measured the wavelength of these two spots to be 266 nm, which is the third harmonic of the fundamental. The inset in Fig. 2 shows the measured spectrum which is peaked at 266 nm.

Fig. 2
figure 2

The temporal profile of the third harmonic (TH) signal observed in BaF2 using a transient grating geometry. The solid line is the Gaussian fit to the data points, yielding a pulse duration of 45 fs, a value that is close to the pulse duration of the input pulse. Inset shows a measured TH spectrum peaked at 266 nm. The photograph shows two blue central spots which are due to the absorption of the third harmonic, giving rise to blue fluorescence from paper. The two intense spots on the side are the incident laser beams at fundamental wavelength

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The origin of the third harmonic signal that we observe can be explained as follows [12, 14]. Because of phase mismatch due to dispersion, the fundamental and the third-harmonic beams do not travel at the same speed through the irradiated medium. Therefore, when only one 800-nm beam is propagating, light at 266 nm is hardly detectable. The efficient conversion is a consequence of overcoming phase mismatch due to the existence of a grating. For third harmonic generation in a non-collinear geometry, the phase matching condition has to be satisfied. In our case it is the presence of a phase grating that satisfies the phase matching condition. In the case of a transient grating geometry, if we consider two waves with wave vectors k 1 and k 2, the grating vector q is given as q=±(k 1-k 2). The first of the third harmonic spots may be attributed to the mixing of two photons from the laser pulse propagating along the wave vector k 1 and one photon from the laser pulse propagating along k 2. A similar mechanism may be invoked for the second of the third harmonic spots: mixing of two photons from the laser pulse propagating along the wave vector k 2 and one photon from the laser pulse propagating along k 1. This self-phase matched third harmonic is much more efficiently produced than that which is produced from the individual beams [14, 16]. By changing the delay between the two beams we see a reduction in the THG signal. The THG signal as a function of delay gives a measure of the pulse duration of the fundamental beam and, as shown in Fig. 2 to be ∼45 fs, is obtained by dividing the full-width-at-half-maximum (FWHM) of the Gaussian fit with the third-order correlation factor of 1.22.

Figure 3(a) shows the temporal response of the first-order diffracted signal in the case of BK7, fused silica and, in Fig. 3(b), that for BaF2. In these cases one of the pulses is delayed in time and the diffracted signal intensity is plotted. For comparison we have also plotted the temporal response of the third harmonic signal that is generated due to phase matching of the two incident beams. As is seen, there is fairly good temporal overlap between the THG signal and the signal from various samples. The measured temporal values of the FWHM for the diffracted signal and the first-order diffraction efficiency are summarized in Table 1. In earlier experiments [12–16], the probe energy was kept fixed at 140 μJ and the pump energy was varied over the range 10–200 μJ using a combination of a half-wave plate and a thin-film polarizer as an attenuator, and the diffraction efficiency obtained was in the range of 3–10 %. Even though our measurements are carried out at lower values of pump and probe energy (30–60 μJ), the intensity that is incident on our samples is essentially the same as in earlier experiments. The observation that our values of diffraction efficiency (Table 1) are almost two orders of magnitude lower may be due to the difference in pulse duration (36 fs in our experiments, 100 fs in earlier work [12–16]). We postulate that longer pulse durations (at 1012 W/cm2) may result in higher contributions from nonlinear absorption, with consequent plasma-induced enhancement of refractive index contrast. Further work needs to be done to gain insight into how pulse duration explicitly affects the dynamics.

Fig. 3
figure 3

(a) The temporal response of the first-order diffracted signal from BK7 and fused silica along with the TH signal (for comparison) in a transient grating geometry; (b) The temporal response of the first-order diffracted signal from BaF2 in a transient grating geometry

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Table 1 Diffraction efficiency and response time measured using the LITG (laser-induced transient grating) technique.
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For optical Kerr gate measurements the two-beam setup is slightly modified. The energy in one of the paths is reduced to ∼6 μJ so that this beam path acts as a probe beam and the other beam path acts as a pump beam. In the probe beam path a λ/2 plate is placed before the sample and an analyzer is located after the sample (Fig. 1(d)). The half-wave plate and the analyzer are in a cross geometry. Initially, when the pump and the probe beams are further apart, no signal is observed. At zero delay, when the two beams overlap, the pump beam causes anisotropy due to the optical Kerr effect in the sample, thereby inducing birefringence; changing the polarization of the probe beam causes some light to transmit through the analyzer. Thus, when both the beams arrive simultaneously in the material, there is leakage of light due to induced birefringence. Such behavior is analogous to an optical switch and the switching time is equal to the rise time of the incident ultrashort laser pulse. The ultrafast response arises because of the extremely fast response of the electronic charge cloud. Since the Kerr effect is a third-order process, the signal can be used to determine the third-order optical susceptibility, χ (3), of the materials at a specific wavelength by [29]

$$\chi _{\mathrm{sam}}^{(3)} =\chi _{\mathrm{ref}}^{(3)} \biggl(\frac{I_{\mathrm{sam}} }{I_{\mathrm{ref}} } \biggr)^{1/2}\biggl(\frac{n_{0\mathrm{sam}} }{n_{0\mathrm{ref}} } \biggr)^{2} , $$
(2)

where \({\chi }^{3}_{\mathrm{ref}}\) is the nonlinear susceptibility of the reference material and n 0sam and n 0ref are the respective linear refractive indices of sample and reference, with I sam and I ref being the respective diffraction intensities. The \({\chi }^{3}_{\mathrm{ref}}\) reference value for fused silica is 1.59×10-14 esu [26]. Our measured χ 3 values for different samples are tabulated in Table 2. The n 2 values are estimated following the Boling, Glass and Owyoung (BGO) treatment [30].

Table 2 Third-order electrical susceptibility, nonlinear index coefficient and response time measured using the OKE (optical Kerr effect) technique
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The optical Kerr gate signal for BK7 and BaF2 is shown in Fig. 4. Along with the Kerr gate signal a Gaussian fit to the experimental data is also shown and the fitted width for all the samples is tabulated in Table 2. In case of BaF2, upon increasing the delay between the two pulses beyond 100 fs, we see periodic modulations. Along with the electronic contribution to the nonlinearity in this case there is significant contribution from the nuclear nonlinearity. The nuclear contribution arises due to the T2g Raman-active phonon of the fluorite crystal structure and is observed at 240 cm-1. The width of the Raman line is 8.5 cm-1, according to the treatment of Elliott et al. [31] using an anharmonic potential. The pulse duration of the exciting laser is 36 fs and thus we find that coherent nuclear motion is induced by pulses much shorter than the period of dominant vibrational modes, that is, when the excitation is “impulsive” [32]. From the inset in Fig. 4 it becomes clear that after the pump pulse leaves the sample, at a delay of 155 fs, the Kerr signal that arises is due to the nuclear contribution. The signal intensity is ∼14 % of the signal intensity at t=0. Beyond a delay of 274 fs we observe a second signal followed by a third signal that appears after a delay of 403 fs, and subsequently the signal decays. The oscillations appear at 147, 127 and 119 fs at pump intensity of 1.1×1012 W cm-2, whereas for intensity of 1.5×1012 W cm-2 the oscillations appear at 155, 120 and 128 fs. It should be noted that the oscillations should have appeared at period of 139 fs, the inverse of the phonon frequency corresponding to Raman shift of 240 cm-1. For comparison, we also show the nuclear contribution arising from the E 1 line (127 cm-1) [33] in case of quartz (Fig. 4(c)). In this case the oscillations are observed at 339, 629 and 936 fs at pump intensity of 1.2×1012 W cm-2.

Fig. 4
figure 4

(a) Optical Kerr gate signal measured in BK7 glass. Dashed lines are the measured signal and the solid lines are the Gaussian fits; (b) Kerr gate signal in BaF2 at longer delays. The inset shows oscillatory modulation at two pump intensities; (c) Kerr gate signal in quartz showing oscillatory modulation

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An optical pulse with a frequency bandwidth spanning two or more vibrational levels creates a superposition of vibrational eigenstates, which evolves with frequencies determined by the energy level differences. The motion of the photo-excited wave packet is detected as an oscillatory modulation or “quantum beat” that manifests itself in transient optical properties. Raman-active vibrational modes have been driven and detected in solvents and semiconductor crystals [34] although the details of the excitation and detection processes are different. In case of glasses, vibrational states are often either structureless or exhibit very broad features. Consequently, nuclear contributions are difficult to observe due to rapid dephasing of localized vibrations in the inhomogeneous structure of a glass. In case of lead oxide glasses, failure to observe oscillatory signals in four-wave mixing experiments [35] has, indeed, been attributed to rapid inhomogeneous vibrational dephasing.

4 Summary

In summary, we have measured third-order optical nonlinearity, χ (3), in various transparent solids, such as glasses (BK7 and fused silica) and crystals (BaF2 and quartz), using 36-fs, 800-nm laser pulses. The χ (3) values measured using optical Kerr gate (OKE) technique lie in the range of 1.3 to 1.7 ×10-14 esu; they are in agreement with estimated values. Purely electronic response to the incident ultrashort laser pulse is observed in fused silica and BK7. In the case of BaF2 and quartz, apart from the electronic response, we also observe an oscillatory modulation that is due to the response of the nuclear contribution (Raman-active phonon) to the incident ultrashort pulses. Using LITG, the electronic response of glasses and crystals to incident ultrashort pulses has been measured, with the diffraction efficiency of the first-order grating lying in the range 0.06–0.13 % for the samples under investigation. Third harmonic generation is observed due to the self-phase matching in the transient grating geometry.