Giant enhancement of THz-frequency optical nonlinearity by phonon polariton in ionic crystals

The field of nonlinear optics has grown substantially in past decades, leading to tremendous progress in fundamental research and revolutionized applications. Traditionally, the optical nonlinearity for a light wave at frequencies beyond near-infrared is observed with very high peak intensity, as in most materials only the electronic nonlinearity dominates while ionic contribution is negligible. However, it was shown that the ionic contribution to nonlinearity can be much larger than the electronic one in microwave experiments. In the terahertz (THz) regime, phonon polariton may assist to substantially trigger the ionic nonlinearity of the crystals, so as to enhance even more the nonlinear optical susceptibility. Here, we experimentally demonstrate a giant second-order optical nonlinearity at THz frequency, orders of magnitude higher than that in the visible and microwave regimes. Different from previous work, the phonon-light coupling is achieved under a phase-matching setting, and the dynamic process of nonlinear THz generation is directly observed in a thin-film waveguide using a time-resolved imaging technique. Furthermore, a nonlinear modification to the Huang equations is proposed to explain the observed nonlinearity enhancement. This work brings about an effective approach to achieve high nonlinearity in ionic crystals, promising for applications in THz nonlinear technologies.


Supplementary Note 1: Calculation of the experimental second-order nonlinear susceptibility
The second-order nonlinear susceptibility is calculated based on the coupled-wave equations 1 by considering two special aspects in our system: (I) the mode dispersion of the LN waveguide 2 causes the major difference. On one hand, the THz waves generated by the difference frequency d = # − % are not the eigenmodes, so they are not supported by the LN waveguide. On the other hand, the mode dispersion also causes a walk-off effect in time between THz waves with frequencies % and # ; (II) The absorption of THz waves by the LN material also exerts some influence.
The walk-off effect is caused by the difference in THz group velocities with frequencies % and # , which does not change during the DFG process. Therefore, this effect can be described by simply setting a parameter = 1 − / end , where end = 2.2 mm is the position where the walk-off effect is finished and DFG signal stops generating, whose value is determined by the experiments.
As the DFG signal is not supported by the LN waveguide, the mode attenuation can be calculated by analyzing the phase change during its propagation. Since the THz waves with frequencies % and # are the zero and first order TE modes of the waveguide, the following expression can be obtained 2 where % and # are the wavevectors orthogonal to propagation direction for the two modes, and indicates the phase-shift in the mode equation of the waveguides. In Eq. (S1), the difference of the phase-shift in different modes is ignored.
According to the phase-matching condition, the propagation constants of the THz waves satisfy the condition 3 = # − % , therefore 4 ∝ 4 with =1, 2, and d. Thus, the phasechange = −2 = 0.92 rad can be easily calculated, and the corresponding loss per millimeter is ; 3 = 20.43 ? , where 3 is the electric field of THz waves with frequency ν d . At the same time, THz waves are also absorbed by LN. At room temperature, the absorption coefficients for the THz waves with frequencies of % , # , and 3 are % = 0.3 mm B# , # = 1.5 mm B# , and α d = 0.9 mm B# , respectively 3 .
It is assumed that the THz electric fields for the three frequencies at the position are % ( ), # ( ), and d ( ), respectively. Considering the change in the -field from position to + , we can write the change in the field % and # , which mainly comes from the generation of femtosecond laser and the material absorption, as follows: where M% and M# depend on the effective intensity of the laser pulse due to the properties of impulsive stimulated Raman scattering 4,5 . The function exp(− | − |) with parameters and represents the power change of the pump laser pulses. Here, we assume 0 < < # according to the experimental results.
According to the initial value of % ( = 0) = # ( = 0) = 0, % ( ) and # ( ) can be analytically solved as Supplementary Table 1 The electric fields of THz waves at different frequencies and different positions. (S4) The change in d signal is mainly due to the DFG, material absorption, and the waveguide mode attenuation. Thus where DFG stands for the DFG of the field 3 signal in terms of % and # . Here, we think the contribution of pump laser generation at the DFG frequency is ignorable, and it makes little difference to our results. The initial and boundary conditions are considered to be d ( = 0) = 0;

Supplementary Note 2: Additional results/discussion about pump-power dependence
In order to exclude the influence of pump lasers on the DFG signal and verify the differencefrequency relation during the DFG process, another two experiments are performed. The corresponding dispersion curves and the spectral information of the two new results are provided in Supplementary Fig. 2 and Supplementary Fig. 4, respectively.
Supplementary Figure 2 Experiment under weak pump (about 81% smaller than the maintext value, marked as Experiment 1). a Experimental dispersion curve under a low pump power b The representative spectrum at the position x = 1.14 mm. The DFG signal can be hardly seen.
In these two results, the Experiment 1 is performed under a low-pump power (about 81% smaller than the main-text value), where the field amplitudes of the matched zero-order ( % d ≈ % = 0.34 THz) and first-order ( # d ≈ # = 1.1 THz) waveguide modes have similar frequencies as the results in the main text. However, the field amplitudes of the waveguide modes are much smaller than the values in the main text. In this case, the DFG signal is very weak and hardly to be seen in the spectra, as shown in Supplementary Fig. 2b. This result indicates that the DFG signal does depend on the two waveguide modes nonlinearly, and the nonlinear frequency-mixing process dominates the generation of Ed. Only in the dispersion curve can we see the weak DFG signal, as Supplementary Fig. 2a indicated, and a very small color-bar limit is used here in order to identify and distinguish the DFG signal. While the noise appears in the low-wavevector regime, thus the DFG signal is hardly seen in Supplementary  Fig. 2b. Nevertheless, the noise causes little influence when the THz field is strong enough, as shown in Supplementary Fig. 3a and Supplementary Fig. 2c.
In order to further verify this result, we can calculate and evaluate the expected value amplitude of DFG signal according to the main-text nonlinear susceptibility of χ (]) = 1.58 mm/kV.
According to Supplementary Fig. 2b, we can calculate the field amplitude of E0 and E1 by a Gaussian fitting (the same method as in the main text), and we obtain # ( = 1.14) = 0.22 kV/mm By using the calculated nonlinear susceptibility, one can evaluate the DFG amplitude by % # ≈ 0.1239 kV/mm Here we ignored the propagation of ? for simplicity. Suppose a temporal expansion of 4 ps (similar to the main-text experiment), we can obtain the relative intensity in the frequency domain.
After calculation, we can get the Fourier spectrum of the DFG signal, which is shown in Supplementary Fig. 3a. This value is indeed in the noise level of the Supplementary Fig. 2b, which is shown below as Supplementary Fig. 3b with superimposed data from Supplementary  Fig. 2b for direct comparison (please notice that the two sub figures have different y-axis scales -differs by one order of magnitude).
Supplementary Figure 3 The theoretical evaluation of the Fourier spectrum of difference frequency signal according to the main-text nonlinear susceptibility. a The Fourier spectrum of evaluated Ed. b The calculated Ed was superimposed onto the experimental data for comparison.
As Supplementary Fig. 4 shows, the Experiment 2 is performed under a high-pump power (about 43% larger than the main-text value), where the field amplitudes of the matched waveguide modes are larger than the values in the main text. In order to further verify the difference-frequency relation, we slightly change the wavefront tilt angle .  Considering the material absorption for THz waves at the frequencies % dd , # dd , and ? dd are % dd = 0.35 mm -1 , # dd = 2.1 mm -1 , and ? dd = 1.1 mm -1 respectively, we constructed a similar model as we used in the main text. After calculation, the nonlinear susceptibility shows a value of χ (]) > 1.352 mm/kV. Correspondingly, the theoretical value in these frequencies given by Eq. (4) in the main text is about 8.43 mm/kV. This result shows a good agreement with that in the main-text, although the nonlinear susceptibility and the material absorption are slightly different in higher frequencies.
In summary, we can conclude that the measured signal indeed comes from the DFG nonlinear process, rather than from the pump laser directly by the comparison of Experiment 1, Experiment 2, and the main-text experiment. Furthermore, in the theoretical view, the influence of pump laser on the DFG signals can also be excluded from the following aspects: First, THz waves generated by pump laser (ISRS) have a broadband spectrum, which is centered at about 0.5 THz under the same condition, seen in our previous work 6 . However, in the tilted-wavefront generation, only the frequency components that match the waveguide modes remain, as shown in Fig. 4b, where even the energy distribution at about 0.5 THz is still very low. If the pump laser could generate THz waves at the difference frequency, it must also generate THz waves at the frequency range between the two waveguide modes, especially at 0.5 THz, rather than only at the difference frequency point.
Second, even if we assume ISRS could generate THz waves at the difference frequency point, it is different from the other two frequencies that match with the waveguide modes, where the generated THz waves could have a continual increase as a function of position x. Considering the general generation efficiency 5,6 , a 400 μJ pump laser beam could generate THz waves with a maximum amplitude of 0.4 kV/mm focused by a cylindrical lens to a line of about 0.1 mm. Here we consider an extreme condition that the wavefront-tilted pump laser excites THz waves simultaneously in the LiNbO3 waveguide, since the length of the pump laser on LiNbO3 is much larger (about 0.9 mm) than that in our previous work (about 0.1 mm), the field amplitude of the pump is about 1/9 of that value 6 . Accordingly, the THz intensity it generated is only 1/9 2 of the value. Considering our practical setup, where the wavefront-tilted pump does not generate THz waves in a very short time span and for a very large waveguide mode attenuation, the practical THz waves at difference frequency point are far smaller, which is negligible compared with the DFG generated value.
Additionally, one can also see that the second harmonic generation (SHG) signal of ν % is absent in all of our experiments, while the second-order nonlinear susceptibility seems much larger than that in the previous reported results 7 . Here in our experiment, the different frequency component is selectively enhanced (compared with that from SHG) and the nonlinear susceptibility for the DFG process was calculated. The missing (or weak appearance at high pump power) of the SHG signal in our experiment is a complicated question, which could depend on several factors that include the phase-match selection, the transient effect, and the anisotropic subwavelength waveguide modes in the experiments. In addition, an unexpected enhancement of SFG signal at high-pump power case can also be observed. This puzzling behavior of the SFG in the high-pump power case seems to be caused by a false signal, since where the frequency of the SFG signal is nearly the resolution limit of our experimental system. Table   Supplementary Table 2 The comparison of second-order nonlinear susceptibility between our results and the common nonlinear optical materials. Asterisk marks the material with relevant parameters that have been used for THz wave generation 6 . DAST: (4-N, N-dimethylamino-4'-N'-methyl-stilbazolium tosylate); DSTMS: (4-N, N-dimethylamino-4'-N'-methyl-stilbazolium-2,4,6-trimethylbenzenesulfonate); OH1: (2-(3-(4-hydroxystyryl)-5,5-dimethylcyclohex-2-enylidene)malononitrile).