Third-order optical nonlinearities play a vital role for the generation and characterization of ultrashort optical pulses. One particular characterization method is frequency-resolved optical gating, which can be based on a large variety of third-order nonlinear effects. Any of these variants presupposes an instantaneous temporal response, as it is expected off resonance. In this paper we show that resonant excitation of the third harmonic gives rise to surprisingly large decay times, which are on the order of the duration of the shortest oscillator pulses generated to date. To this end, we measured interferometric third-harmonic frequency-resolved optical gating traces in and , corroborating polarization decay times up to 6.5 fs in . This effect is among the fastest effects observed in ultrafast spectroscopy. Numerical solutions of the time-dependent Schrödinger equation are in excellent agreement with experimental observations. Our work (experiments and simulations) corroborates that a noninstantaneous polarization decay may appear in the presence of a 3-photon resonance. In turn, pulse generation and characterization in the ultraviolet may be severely affected by this previously unreported effect.
© 2015 Optical Society of America
Third-order optical nonlinearities are a ubiquitous effect in dielectric materials. Under nonresonant conditions outside absorption resonances, they stem from the interaction with valence electrons in the material. An oscillating electric field induces a dipole moment inside the material, which gives rise to a polarization , with the dielectric constant and the susceptibility . In an amorphous medium and under nonresonant conditions, the susceptibility can be written as a series of odd orders of the electric field . Assuming an oscillating electric field one therefore expects Fourier components of at the fundamental frequency and all odd harmonics thereof. Using a 780 nm driver pulse, for example, third-harmonic (TH) radiation appears in the ultraviolet at 260 nm. Apart from harmonic generation, the real part of the nonlinearity also gives rise to a self-effect for the fundamental component, namely self-refraction. This reactive nonlinearity results in a power-dependent phase shift at the driver wavelength. Self-phase modulation is the dominant nonlinear effect in optical fibers. In bulk media, self-refraction additionally induces a spatial lensing effect at high intensities, which is named self-focusing. Both self-phase modulation and self-focusing play a vital role in characterization [1–5] and generation methods [6,7] for ultrashort pulses. The pulse generation method of Kerr-lens mode-locking relies on the quasi-instantaneous action of the self-focusing mechanism and additive-pulse mode-locking on that of self-phase modulation. A femtosecond relaxation time of this nonlinearity would severely limit obtainable pulse durations [8,9] from such laser sources. Similarly, one expects incorrect results from any pulse characterization method if a persistence appears in the nonlinear response.
2. MEASUREMENTS AND RETRIEVAL PROCEDURE
In our work, we used an interferometric variant of TH frequency-resolved optical gating (FROG [2,3,10]); see Fig. 1. The input pulses from a Ti:sapphire laser (7.5 fs pulse duration, 780 nm center wavelength) are split into two beam paths, experience a relative delay, and are then collinearly recombined and focused into the material by an off-axis parabolic mirror. Dispersion of the beam splitter setup and the air paths is precompensated by chirped mirrors. The generated TH light is recollimated by a UV microscope objective. Fundamental light is rejected by two interference filters, which transmit a 40 nm bandwidth at 266 nm center wavelength. The light is then spectrally dispersed in a spectrograph equipped with an electron multiplier charge-coupled device (EMCCD). The experiments employed titania () thin-film samples that are deposited on silica () substrates. We exclusively used titania layers of about 180 nm (one-half wave) thickness, as these layers previously showed a maximum efficiency . These samples were oriented with the titania side toward the concave mirror. Reference measurements were done in the same geometry, using the surface TH generation of silica substrates. Our measurements indicate the practical absence of a TH signal unless the beam waist is separated by less than the confocal parameter from either one of the surfaces . This effect has been explained by destructive interference of the generated TH radiation when the focus is located in the bulk of the material . The relative strength of the two TH peaks allows for verification of the correct orientation of the thin-film samples.
FROG techniques generally measure the spectrally resolved autocorrelation traces of the pulse under test , i.e., in this case the TH autocorrelation. The FROG method is fully self-referenced, and no well characterized external reference pulse is required. As interferometric FROG variants allow for a collinear beam path, tight focusing at high numerical apertures enables the use of nanojoule pulses with uncompressed durations of two to three optical cycles as they are directly available from mode-locked oscillators [8,9]. The measured interferometric third-harmonic interferometric (THI) FROG traces are of the form2(a) and 2(e). Using Fourier analysis, this measured FROG trace can be decomposed into three subtraces that show sinusoidal variation with , and a fourth subtrace () that does not rapidly vary with delay . Examples for and 1 are shown in Figs. 2(b) and 2(f), and 2(c) and 2(g), respectively. In principle, one can retrieve the complex-valued from any of the three subtraces () whereas the trace for contains only a spectral interferometry signal. Moreover, given the fairly weak signals, we could practically only use the subtraces for and 1 for retrieval of . To this end, a specialized retrieval software was written that computes via Eq. (1) for a given initial guess for , separates the relevant subtraces , and employs the Nelder–Mead optimization algorithm  to iteratively minimize the difference between measured and computed FROG trace. In this iterative reconstruction, the delay range encompassed 80 fs with a total of 50 points to represent . This corresponds to a temporal resolution of 1.6 fs, i.e., 70% of the optical cycle at 780 nm. Retrieval was seeded with assumption of an unchirped hyperbolic secant pulse shape, with a width that was matched to the autocorrelation width of the component. Using the redundancy in the THIFROG traces, a total of three independent retrieval attempts were made for each data set, employing either the or the component as well as a combination of the two. These independent retrievals were employed for an estimation of the robustness of the retrieved pulse shapes.
3. DISCUSSION OF EXPERIMENTAL RESULTS
We used the retrieval procedure on measured THIFROG data that was obtained with two different nonlinear optical materials, namely silica and titania . Experimental results are displayed in Fig. 2. Even without pulse retrieval, one can immediately see that the titania-based FROG trace is significantly wider than the one measured in silica. This broadening effect becomes even more pronounced in a comparison of the reconstructed electric-field profiles in Figs. 2(d) and 2(h). Despite an otherwise completely identical setup, a pulse duration of 10.1 fs (FWHM) is retrieved for silica, whereas the use of titania yields 15.7 fs. Deviations between the retrieved pulse durations and the input pulse duration of 7.5 fs may partially stem from imperfections in the dispersion compensation as well as from unavoidable spatial aberrations of the tight focusing geometry . The redundant information in the THIFROG traces enabled the estimation of retrieval errors, as indicated by the error bars in Figs. 2(d) and 2(h). This analysis indicates meaningful reconstruction in the central around zero delay whereas the trailing hump structure in Fig. 2(h) is probably a retrieval artifact. In the following analysis, we therefore only consider the meaningfully retrieved part of the intensity profiles.
For further analysis, we followed the deconvolution approach described in Ref. , iteratively reconstructing the response function that minimizes the functional difference2) is based on intensity envelopes, the resulting decay times are readily converted to those of field envelopes by multiplying them with two. For Figs. 2(d) and 2(h), was chosen. The intensities , , are deduced from the interferometric FROG measurements at the silica and titania samples, respectively. The response function is given as single-sided exponential , with defining the polarization decay time. The deconvolution is computed in a forward fashion, starting with an initial guess for . Numerically minimizing Eq. (2), we use again Nelder–Mead optimization for the deconvolution procedure. This deconvolution procedure yields for [Fig. 3]. This effect of a noninstantaneous response of titania was reproduced in a series of measurements on different samples that were manufactured by different deposition methods. In contrast, within experimental accuracy, we observe an instantaneous response when switching to samples with a band gap which is chosen such that 3-photon absorption cannot occur anymore. Moreover, we can also safely rule out Fabry–Perot effects from multiple reflections within the sample. The cavity lifetime of the thin film amounts to 2.6 fs, which is 2.5 times shorter than the measured .
The peak intensity of the 780 nm pulses reaches values of the order of on the samples. As a result, free electrons and holes are excited via 2- and 3-photon interband transitions in the titania sample. From the respective absorption coefficients reported in Ref. , one estimates a free-carrier density of several . Under such conditions, both nonlinear inter- and intraband polarizations occur and in general the related (imaginary or real) inter- and intraband currents represent source terms for generating electric fields at new frequencies. An analysis of this scenario requires in-depth calculations of the nonlinear response which are presented next.
4. TIME-DEPENDENT SCHRöDINGER EQUATION
In order to gain insight into the nature of the resonant process we solve the 1D time-dependent Schrödinger equation (TDSE)18] 19]. The macroscopic current density is then computed as the sum over all -states and all valence bands. It is important to understand that this model focuses on the electronic response and neglects any lattice contribution, i.e., phonons. In the following, we therefore restricted ourselves to a comparison with experimental data in the central range around zero delay.
For silica, we use the values from Ref. , i.e., , , , and , yielding a bandgap and an effective mass in the lowest conduction band. For modeling the effective mass and band gap of the anatase modification of , a simple functional dependence in Eq. (4) was found insufficient. Adding a term and using , , and , we obtain a good match for the pseudopotential of titania, yet with a slightly smaller effective mass than experimentally reported. In the numerical simulations, a excitation pulse shape  with 6.5 fs FWHM was chosen to temporally localize the response. (1.55 eV) was assumed as central excitation wavelength. It is important to note that solution of the TDSE models the response of a single atom. Propagation effects, e.g., reabsorption of the generated THG light, are not included.
Figure 4 shows the extracted nonlinear phase contribution as well as the TH polarization amplitude in for a range of intensities from 0.1 to . To extract the nonlinear phase, we carried out an additional calculation at a much smaller intensity of to subtract a reference phase. Both the nonlinear phase contribution as well as the simulated THG response exhibit an increasing polarization decay constant with growing intensity, as confirmed by a deconvolution analysis [Fig. 4(c)]. The THG response shows relatively little variation () over the simulation range. Best agreement with experimental results is obtained for the highest intensities, which already lie above the expected damage threshold of . This discrepancy may be attributed to a surface modification of the nonlinear response, which was not considered in our simulations. Moreover, the self-refraction response exhibits a significantly higher onset of noninstantaneous effects, which are accompanied by an oscillatory behavior. This results from a quantum beat between the populated first and second conduction bands, whose energy spacing () matches the observed oscillation period of 2.8 fs. Lengthening of self-refraction response therefore appears to be related to population of the conduction bands via 3-photon absorption, similar as one would expect from the Kramers–Kronig relation [21,22]. It is important to understand that the Kramers–Kronig relation does not immediately couple the THG response and self-refraction, which explains the different dynamics. Nevertheless, very similar relaxation time constants appear for the two effects.
The TDSE was used to model interband transitions from the valence band into the first 13 conduction bands as well as intra-conduction band transitions. For an estimation of the prevalent mechanism, all intraband currents were summed up and compared to the total interband current [23,24], clearly indicating that the nonlinear response is dominated by nonlinear interband polarization whereas intraband contributions play a minor role; see Fig. 5(d). In titania, the intraband current amplitudes are typically 1 order of magnitude smaller than the interband contributions. Given the bandgap of 9 eV, the third harmonic of the 1.5 eV excitation is still off-resonant in silica. In this case, the polarization follows the rapidly oscillating fundamental field [Fig. 5(a)]. Polarization contributions at the third harmonic are comparatively weak [Fig. 5(e)]. For titania with a bandgap of , the third harmonic becomes resonant with the quantum mechanical dipole, cf. Fig. 5(c). This gives rise to an apparent build-up of an oscillation of at the TH frequency [Fig. 5(a)]. The spectral overlap causes a continuation of the dipole oscillation for several cycles beyond the excitation pulse. Deconvolving the computed titania response with a single-sided exponential, we again determine a relaxation time constant [Fig. 5(e)], close to the measured values. Isolating the fundamental frequency contribution of the polarization and comparing its phase to the driver field, one can also extract the self-refraction part of the response. In this case one observes a qualitatively similar behavior as for the third harmonic. Quite generally, once there is a response in resonance with the quantum mechanical dipole, self-phase modulation and self-focusing will not immediately follow the envelope of the driver pulse anymore, but show a persistence of several femtoseconds beyond [Fig. 5(f)]. For a given bandgap, the effective mass in the pseudopotential plays a decisive role for the length of the observed relaxation time constant, i.e., the relaxation behavior on few-femtosecond time scales may be used as a local probe for the effective mass of the valence electrons inside the material. Moreover, we repeated the simulations assuming hydrogen-like atomic potentials  and observed qualitatively the same behavior. In these simulations, we adjusted the ionization potential of the model atom to match the band gap of our material. Once there is spectral overlap between the third harmonic and the conduction band (or the continuum state in an atomic system), a prominent persistence of the nonlinear susceptibility is observed.
In principle, some of our findings can also be qualitatively understood in a semi-classical picture, as previously discussed for nano-plasmonic systems . Driving a harmonic oscillator far off resonance, the forced oscillation will immediately disappear if the driver is switched off. On resonance, however, the forced oscillation will persist for several cycles, until the oscillating system dephases. In this scenario, the intrinsic single-sided exponential decay of the system gives rise to Lorentzian line shapes. Moreover, in a weakly anharmonic oscillator, there may be a build-up of a resonant oscillation if the system is driven at a suitable subharmonic. Quickly switching off the driver, one then also observes a persistence of the resonant oscillation. Despite the intriguing accessibility of such models, however, semi-classical pictures fail to correctly predict the sharp transition between resonant and off-resonant excitation, which is only correctly modeled in the full quantum mechanical simulations.
Our findings have important implications for applications of nonlinear effects, in particular for short-pulse characterization techniques. Specifically, limitations arise when the third harmonic approaches the band gap of the nonlinear material. While this appears straightforward for TH generation, adverse effects also appear at the fundamental wavelength for self-refraction as shown in Fig. 4(a). The dynamics of this effect is obviously not identical to the observed noninstantaneity of the THG, but similar relaxation time constants of a few femtoseconds appear and the appearance of this effect is equally related to a 3-photon resonance with the band gap. This modified response could, in principle, limit Kerr-lens mode-locking, even though we are not aware of such lasers being operated above one-third of the bandgap [26–28]. More importantly, our findings impose limitations on -based pulse measurement techniques in the visible and ultraviolet range. Using sapphire or silica as the medium, limitations may appear below 600 nm. There is an increasing trend to employ near-resonant thin-film samples [10,29,30], as they provide substantially enhanced conversion efficiencies. This favorable behavior may come with a significant loss of temporal resolution, particularly for pulses in the few-cycle regime. Finally, even using the widest bandgap material LiF, sensible based characterization appears to be restricted to wavelengths above 320 nm.
In conclusion, we present a novel approach of using frequency-resolved optical gating as a tool to resolve the temporal response of nonlinear optical processes at a few-cycle time scale. This response shows a threshold-like dependence on the bandgap of the material. The reported polarization decay times of belong to the fastest noninstantaneous effects that were ever reported at near-visible wavelengths. Our findings may have important effects in ultrafast optics, imposing the avoidance of resonant 3-photon effects for sensible characterization and passive mode-locking techniques in the few-cycle regime. Given that this issue is apparently not restricted to solid-state materials, there may also be consequences for attosecond pulse characterization techniques . In a somewhat simplifying fashion, particular care appears advisable whenever odd harmonics become resonant with resonant transitions and if pulse durations are below 10 cycles of the carrier field. Considering that the shortest attosecond pulses to date  are approaching the single-cycle regime, there may be an incentive to prefer spectral-interferometry-based techniques over correlation techniques in this regime. Despite all the apparent possible limitations, our advanced method opens a new avenue for accessing fundamental parameters of a dielectric material by an optical probe technique.
Deutsche Forschungsgemeinschaft (DFG) (BR 4654/1, GR 1782/12-2, MO 850/16, STE 762/9).
1. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993). [CrossRef]
2. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2002).
3. T. Tsang, M. A. Krumbügel, K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical-gating measurements of ultrashort pulses using surface third-harmonic generation,” Opt. Lett. 21, 1381–1383 (1996). [CrossRef]
4. D. Meshulach, Y. Barad, and Y. Silberberg, “Measurement of ultrashort optical pulses by third-harmonic generation,” J. Opt. Soc. Am. B 14, 2122–2125 (1997). [CrossRef]
5. A. Major, F. Yoshino, J. S. Aitchison, and P. W. E. Smith, “Wide spectral range third-order autocorrelator based on ultrafast nonresonant nonlinear refraction,” Opt. Lett. 29, 1945–1947 (2004). [CrossRef]
6. E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am. B 6, 1736–1745 (1989). [CrossRef]
7. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991). [CrossRef]
8. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in ultrashort pulse generation: pushing the limits in linear and nonlinear optics,” Science 286, 1507–1512 (1999). [CrossRef]
9. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999). [CrossRef]
10. S. K. Das, C. Schwanke, A. Pfuch, W. Seeber, M. Bock, G. Steinmeyer, T. Elsaesser, and R. Grunwald, “Highly efficient THG in TiO2 nanolayers for third-order pulse characterization,” Opt. Express 19, 16985–16995 (2011). [CrossRef]
11. T. Y. F. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52, 4116–4125 (1995). [CrossRef]
12. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
13. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13, 2617–2626 (2005). [CrossRef]
14. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Computer J. 7, 308–313 (1965).
15. W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by two-dimensional Fourier transform spectral interferometry,” Opt Lett. 31, 2927–2929 (2006).
16. A. Anderson, K. S. Deryckx, X. G. Xu, G. Steinmeyer, and M. B. Raschke, “Few-femtosecond plasmon dephasing of a single metallic nanostructure from optical response function reconstruction by interferometric frequency resolved optical gating,” Nano Lett. 10, 2519–2524 (2010).
17. C. C. Evans, J. D. B. Bradley, E. A. Martí-Panameño, and E. Mazur, “Mixed two- and three-photon absorption in bulk rutile (TiO2) around 800 nm,” Opt. Express 20, 3118–3128 (2012). [CrossRef]
18. T. Paasch-Colberg, A. Schiffrin, N. Karpowicz, S. Kruchinin, Ö. Sağlam, S. Keiber, O. Razskazovskaya, S. Mühlbrandt, A. Alnaser, M. Kübel, V. Apalkov, D. Gerster, J. Reichert, T. Wittmann, J. V. Barth, M. I. Stockman, R. Ernstorfer, V. S. Yakovlev, R. Kienberger, and F. Krausz, “Solid-state light-phase detector,” Nature Photon. 8, 214–218 (2014).
19. M. Korbman, S. Y. Kruchinin, and V. S. Yakovlev, “Quantum beats in the polarization response of a dielectric to intense few-cycle laser pulses,” New J. Phys. 15, 013006 (2013). [CrossRef]
20. H. Tang, K. Prasad, R. Sanjinès, P. E. Schmid, and F. Lévy, “Electrical and optical properties of TiO2 anatase thin films,” J. Appl. Phys. 75, 2042–2047 (1994). [CrossRef]
21. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991). [CrossRef]
22. B. Borchers, C. Brée, S. Birkholz, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect in solids,” Opt. Lett. 37, 1541–1543 (2012). [CrossRef]
23. V. Juvé, M. Holtz, F. Zamponi, M. Woerner, T. Elsaesser, and A. Borgschulte, “Field-driven dynamics of correlated electrons in LiH and NaBH4 revealed by femtosecond X-ray diffraction,” Phys. Rev. Lett. 111, 217401 (2013). [CrossRef]
24. C. Somma, K. Reimann, C. Flytzanis, T. Elsaesser, and M. Woerner, “High-field terahertz bulk photovoltaic effect in lithium niobate,” Phys. Rev. Lett. 112, 146602 (2014). [CrossRef]
25. H. G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys. 9, 138–148 (1999).
26. S. Uemura and K. Torizuka, “Generation of 12-fs pulses from a diode-pumped Kerr-lens mode-locked Cr:LiSAF laser,” Opt. Lett. 24, 780–782 (1999). [CrossRef]
27. H. Zhao and A. Major, “Powerful 67 fs Kerr-lens mode-locked prismless Yb:KGW oscillator,” Opt. Express 21, 31846–31851 (2013). [CrossRef]
28. N. Tolstik, E. Sorokin, and I. T. Sorokina, “Kerr-lens mode-locked Cr:ZnS laser,” Opt. Lett. 38, 299–301 (2013). [CrossRef]
29. A. B. Djurišić and Y. H. Leung, “Optical properties of ZnO nanostructures,” Small 2, 944–961 (2006). [CrossRef]
30. A. Chen, G. Yang, H. Long, F. Li, Y. Li, and P. Lu, “Nonlinear optical properties of laser deposited CuO thin films,” Thin Solid Films 517, 4277–4280 (2009). [CrossRef]
31. M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nature Photon. 8, 178–186 (2014).