Open Access
Add to BibSonomy
Abstract
It is commonly assumed that for low-intensity short optical pulses far from resonance, the third-order optical nonlinear response is instantaneous. We solve the three-dimensional time-dependent Schrödinger equation for the hydrogen atom and show that this is not the case: the polarization is not simply proportional to the cube of the electric field even at low intensities. We analyze the fundamental-frequency and third-harmonic nonlinear susceptibilities of hydrogen, investigate their dependence on intensity, and find that the delays in the Kerr response rapidly approach the femtosecond time-scale at higher intensities, while the delays in the third harmonic generation remain much lower. We also propose an experimental scheme to detect and characterize the above effects.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
The nonlinear optical response of materials is ubiquitous in optics, and constitutes the core of key photonic technologies, from frequency conversion and interferometry, to pulse characterization, metrology and attosecond science (see e.g., [1–3]). Significant efforts were dedicated to characterize experimentally and to understand theoretically such fundamental nonlinear optical processes as self-phase modulation, frequency conversion, high-harmonic generation, high-order Raman effect etc.
In particular, the nonlinear effects in gases have attracted a significant interest since the advent of the nonlinear optics, as gases are a flexible, easy-to-handle nonlinear optical medium which can sustain significant intensities of the incident light.
At high intensities, ionization of gases and subsequent electron dynamics leads to effects such as high-harmonic generation [4]. However, many important effects happen in gases already at lower intensities. Examples include such fundamental effects as self- and cross-phase modulation and third-harmonic generation due to the third-order nonlinear response, as well as nonlinear processes of order five, seven etc. Experimentally, the third-order nonlinear susceptibility responsible for the self-phase modulation is well-characterized for a number of gases, including in particular noble gases. Reliable experimental data on higher-order susceptibilities of gases are scarce and mostly refer to the fifth-order susceptibility of noble gases such as argon [5–17]. From the theoretical perspective, extensive discussion was devoted to the origin and analysis of the so-called higher-order Kerr effect [6,18–25]. Less attention was given to the third-order nonlinearity, particularly its frequency dependence for highly non-resonant infrared drivers.
A general expression for the third-order nonlinear polarization at a time moment $t$ is $P(t)=\epsilon _0\iiint \chi _{3}(\tau _1,\tau _2,\tau _3)E(t-\tau _1)E(t-\tau _1-\tau _2)E(t-\tau _1-\tau _2-\tau _3)d\tau _1d\tau _2d\tau _3$, which can be reformulated in the frequency domain using $\chi _{3}(\omega _0;\omega _1,\omega _2,\omega _3)$, with $\omega _0=\omega _1+\omega _2+\omega _3$. For pump frequencies far from resonances, the instantaneous response $P(t)=\epsilon _0\chi _{3}E^3(t)$ is commonly assumed, which is equivalent to an $\omega$-independent $\chi _{3}$ in the frequency domain. The frequency dependence of $\chi _{3}$ is ignored in the vast majority of investigations; to the best of our knowledge, it was only once mentioned in Ref. [26] for isolated silver atoms in the high-intensity regime. We are not aware of any publication, either theoretical or experimental, that explicitly considers a non-instantaneity of the third-order response in simulating nonlinear pulse propagation, calculating conversion efficiencies, characterizing nonlinear losses, etc.
In this paper, we show that the frequency dependence of $\chi _{3}$ is not negligible even at very low intensities, even at relatively low frequencies, and even far from resonance, and therefore, the instantaneous relation $P(t)=\epsilon _0\chi _{3}E(t)^3$ does not hold.
We perform a first-principle investigation of atomic hydrogen and, for comparison, an artificial atom with the Yukawa potential with the same ionization potential, but without the long-range Coulomb tail. We analyze the intensity dependence of the nonlinear response at the fundamental and third-harmonic frequencies and show significant (up to a quarter period and beyond) delays in the response as the intensity grows. Such delays imply a contribution which corresponds purely to nonlinear loss, without a nonlinear modification of the real part of the refractive index, or even (for delays around half a period) to inversion of the nonlinearity sign. Finally, we propose an experimental interferometric scheme to characterize the predicted effects.
We have selected atomic hydrogen for the investigation of the nonlinear effects. Our choice allows straightforward and accurate first-principle investigations, since the exact atomic potential of atomic hydrogen is known.
We simulate the electron dynamics using the three-dimensional time-dependent Schrödinger equation (TDSE):
After the time-dependent solution $\Psi (\hat {r},t)$ is found, the time-dependent dipole moment, which is oriented along the $z$ axis, can be determined from $d(t)=-\langle \Psi (\hat {r},t)|z|\Psi (\hat {r},t)\rangle$. The nonlinear part of the dipole was obtained by subtracting its linear part, which is determined using the time-dependent dipole $d_{\textrm{low}}(t)$ for a pulse with a very low maximum electric field $E_{\textrm{max,low}}$: $d_{\textrm{NL}}(t)=d(t)-E_{\textrm{max}}/E_{\textrm{max,low}}d_{\textrm{low}}(t)$. The fundamental-frequency and third-order parts of the nonlinear dipole, $d_{{\textrm{NL}},\omega }(t)$ and $d_{{\textrm{NL}},3\omega }(t)$, were determined by transforming $d_{\textrm{NL}}(t)$ into the frequency domain, applying band-pass filters (FWHM of the pump frequency) around the corresponding frequencies, and transforming back into the time domain. After that, the in-phase [$d_{{\textrm{NL}},\omega,i}(t)$ and $d_{{\textrm{NL}},3\omega,i}(t)$] as well as out-of-phase [$d_{{\textrm{NL}},\omega,o}(t)$ and $d_{{\textrm{NL}},3\omega,o}(t)$] parts of the fundamental-frequency and third-harmonic contributions were determined as the symmetric and asymmetric parts of $d_{\textrm{NL}}(t)$, since the considered pump field has a carrier-envelope offset of zero and is thus symmetric in time. The fundamental-frequency contribution is determined by the (generally speaking, complex-valued) quantities $\chi _3(\omega ;\omega,\omega,-\omega )=\chi _3(\omega ;\omega,-\omega,\omega )=\chi _3(\omega ;-\omega,\omega,\omega )$, while the third-harmonic dipole is determined by $\chi _3(3\omega ;\omega,\omega,\omega )$. Note that the susceptibilities $\chi _3(\omega ;\omega,\omega,-\omega )=\chi _3(\omega ;\omega,-\omega,\omega )=\chi _3(\omega ;-\omega,\omega,\omega )$ contribute equally to the fundamental-frequency nonlinear polarization, and henceforth we discuss only $\chi _3(\omega ;\omega,\omega,-\omega )$.
To determine the real and imaginary parts of $\chi _3$, we have taken the corresponding contributions to the dipole, filtered out the temporal regions outside of the pump pulse, and calculated their energy, i.e., the integral of the square dipole. The obtained energy was divided by the energy of the reference dipole $d_{\textrm{ref}}$ which corresponds to $\chi _3=1$ m$^2$/V$^2$:
In Fig. 1(a), the dependence of the nonlinear dipole $d_{{\textrm{NL}},\omega }(t)+d_{{\textrm{NL}},3\omega }(t)$ on time is shown (blue curve) together with the electric field (red dotted curve) and the cube of the electric field (green dashed curve). If the instantaneous relation $d_{\textrm{NL}}(t)=\epsilon _0\chi _3E^3(t)$ between the dipole and the field were true, the blue and green curves would coincide. However, this is clearly not the case, showing that the polarization of hydrogen is not an instantaneous function of the electric field, even at low intensities. We expect this conclusion to be true also for other gases, as well as for liquid and solid-state materials. We stress that higher-order nonlinear effects (5$^{\textrm {th}}$-order, 7$^{\textrm {th}}$-order, etc.) do not contribute to the non-instantaneity illustrated in Fig. 1(a) at the intensity of 20 TW/cm$^2$; this is confirmed by Fig. 1(b) where the values of the third-order susceptibility at 20 TW/cm$^2$ do not differ significantly from those at close to 0 TW/cm$^2$.
Fig. 1. The nonlinear dipole momentum (a) and the nonlinear susceptibilities (b). In (a), the electric field (red dotted curve) as well as the cube of the electric field (green dashed curve) are shown in addition to the nonlinear dipole momentum (blue curve). In (b), the $|\chi _3(\omega ;\omega,\omega,-\omega )|$ (thick red curve) and $|\chi _3(3\omega ;\omega,\omega,\omega )|$ (thick green dashed curve) are shown along with the delays of the fundamental-frequency (thin red curve) and third-harmonic responses (thin green dashed curve). In (a), we consider an 8-fs FWHM, 20 TW/cm$^2$ pulse at 830 nm.
To elucidate the mechanisms of the non-instantaneity we note that the electric field induces resonant excitations via the Freeman resonances. These resonances both have field-induced lifetimes associated with their depletion by the field and also lead to very different – and extended – electron orbits. We associate the delay in the nonlinear response to the electron motion during its transitions within the manifold of the excited states.
In Fig. 1(b), the dependencies of the absolute values of the fundamental-frequency and third-harmonic susceptibilities on intensity are shown, indeed indicating different values even for relatively low intensities below 5 TW/cm$^2$, where higher-order effects play no role. As the intensity grows, both susceptibilities start to vary, indicating the contribution from higher-order effects. In addition, by the thin curves, we present the delays between the fundamental-frequency polarization and the field, as well as between the third harmonic polarization and the field. The delays are zero for low intensities, but at about 60 TW/cm$^2$, the delay of the fundamental-frequency response grows to values around 0.7 fs. This value corresponds to one quarter of the optical cycle (2.76 fs), which means that above 60 TW/cm$^2$, the nonlinear response corresponds entirely to loss, and the third-order susceptibility $\chi _3(\omega ;\omega,\omega,-\omega )$ is almost purely imaginary. At higher intensities, the delay of the fundamental-frequency response grows to values around 1.3 fs, indicating a real but negative susceptibility, while $\chi _3(3\omega ;\omega,\omega,\omega )$ remains positive. For increasing intensity, besides the contribution from ionization, the increased role and number of the involved excited states leads to increased delays due to the associated field-induced lifetimes of resonances.
To explain the origin of the purely imaginary loss, we compare the intensity dependence of the polarization derivative which arises due to the instantaneous part of Kerr effect, $\partial P_{Kerr}/\partial t = \partial [\epsilon _0 \chi _{3} E^{3}(t)]/\partial t$, and due to ionization, as shown in Fig. 2, for a pump wavelength of 830 nm. Ionization leads to additional nonlinear polarization contributions due to two effects: Brunel harmonics [28] and electron displacement immediately after the ionization [29]. In particular the latter effect, which is equivalent to energy absorption during the ionization, described by $\partial P_{ion}/\partial t = \partial [I_p \Gamma (t) N_0 E^{-1}(t)]/\partial t$ ($I_p$ is the ionization potential, $N_0$ is the particle density, $\Gamma (t)$ is the ionization rate), is responsible for strong optical loss at higher intensities and can overcome the bound-state contribution as the intensity grows. As one can see in Fig. 2, this happens around an intensity of 60 TW/cm$^2$. This is in excellent agreement with the results shown in Fig. 1(b), where the delay becomes a quarter of the optical cycle (corresponding to a purely lossy nonlinear response) at the same intensity. Note that for the Keldysh parameter $\gamma \sim 1$ or slightly above, multiphoton ionization transitions to tunnel ionization, which is associated with the Brunel harmonics and electron displacement harmonics. Here, the $\gamma \sim 1$ range corresponds to an intensity of 105 TW/cm$^2$ or slightly below, again in very good agreement with our conclusions. Thus, analysis of the time delay of the nonlinear response provides the characterization of the relative contributions of Kerr and ionization effects, and allows determination of the intensity at which the ionization contribution to the polarization predominates.
Fig. 2. The intensity dependence of the polarization time derivative due to the Kerr effect (green dashed curve) and ionization (red solid curve) at the center of the pulse.
To understand the observed effect and to elucidate the role of the atomic potential shape, we have repeated the simulations using the wavelengths 400 nm and 1300 nm, as well as a Yukawa potential instead of the hydrogen potential, where the ionization potential is kept the same. The Yukawa potential is characterized by the absence of a Coulomb tail for large $r$ and supports only a single bound state. In Fig. 3, the dependence of the susceptibilities $\chi _3(\omega ;\omega,\omega,-\omega )$ and $\chi _3(3\omega ;\omega,\omega,\omega )$ on intensity is shown for the hydrogen atom [(a), (b), and (c), i.e., top panels] and the Yukawa potential [(d), (e), and (f), i.e., bottom panels] for wavelengths of 400 nm, 830 nm, and 1300 nm (from left to right, as indicated in the caption).
Fig. 3. The nonlinear susceptibilities $|\chi _3(\omega ;\omega,\omega,-\omega )|$ (thick red curve) and $|\chi _3(3\omega ;\omega,\omega,\omega )|$ (thick green curve), along with the delays of the fundamental-frequency (thin red curve) and third-harmonic responses (thin green curve). Results are presented for the hydrogen atom in (a), (b), and (c) and for the Yukawa potential in (d), (e), and (f), for wavelengths of 400 nm (a),(d), 830 nm (b),(e), and 1300 nm (c),(f). In (e),(f) the intensities are limited to 40 TW/cm$^2$ due to the excessive numerical effort at large wavelengths and intensities.
One can recognise several tendencies in the presented results. First, susceptibilities are larger for the hydrogen atom than for the Yukawa potential, due to the stronger localization of the electrons in the latter case. Similarly, the delays are larger for hydrogen atoms, since the role of ionization and "laser-dressed" bound states is much weaker for the Yukawa potential, for which the susceptibility remains almost real (although for strong fields Kramers-Henneberger states and associated delays can start to appear also for the Yukawa potential). Comparing results for different wavelengths, one can see, in particular for the hydrogen atom, that with decreasing wavelength (increasing photon energy) the susceptibility grows, and the higher-order effects set in at lower intensity. We attribute these effects to the excitation of bound states and to the onset of ionization, which happens at lower intensities for higher energies of the pump photons.
Also, the difference between the fundamental- and third-order susceptibilities drops with increasing wavelength, and the corresponding curves become close at low wavelength in Fig. 3(c) – but not quite equal. We conclude that the relation $P(t)=\epsilon _0\chi _3E^3(t)$ does not hold even for this case, at low intensities, long wavelength, and far from any resonance. Similar tendencies are observed when comparing results for the Yukawa potential for 400 nm and 830 nm [Figs. 3(d) and (e), respectively]. The case represented in Fig. 3(f) is an exception; despite the large wavelength, there is a significant difference between $\chi _3(\omega ;\omega,\omega,-\omega )$ and $\chi _3(3\omega ;\omega,\omega,\omega )$, indicating non-instantaneous response and corroborating our expectation that non-instantaneity is a universal feature.
We note that at very long wavelength and correspondingly very low frequencies (quasi-dc-field), the relation $P(t)=\epsilon _0\chi _3E^3(t)$ must be recovered. This will happen as soon as the frequency is far below $\omega ^*=1/\tau ^*$, where $\tau ^*$ is the longest of the response times of the system. In the case of 1300 nm and the Yukawa potential, the frequency $\omega ^*$ is not yet reached.
The above findings indicate that the approach to the simulation of third-order nonlinearity should be revised in many cases. It is possible to keep the standard description (using a single value of $\chi _3$) for the cases when either only self-phase-modulation-like effects, or only third-harmonic generation effects, are of interest. If both fundamental-frequency response and third-harmonic response need to be described accurately and simultaneously, frequency dependence of the nonlinear susceptibility should be included.
A simple way to experimentally detect and characterize the delay of polarization is to split the pump into two pulses of different intensity, let the pulses propagate through the gas medium, and then observe the interference pattern between the third-harmonic radiation as a function of intensity and delay between the two pulses. In Fig. 4, we modelled the output of such an experiment as a function of intensity of one of the pulses, while the other pulse is fixed at 20 TW/cm$^2$. One can see the clear shift of the interference pattern with intensity, indicating the growing delay. Note that the interference shift does not directly follow the time shift of the third-harmonic polarization, as indicated by the thin green curve shown in Fig. 1(b), since the interference shift is intensity-dependent and averaged over the pulse profile.
Fig. 4. The interference of two third-harmonic pulses as a function of delay and intensity of one of the pulses. The intensity of the second pulse is fixed at 20 TW/cm$^2$.
In conclusion, by a first-principle study of the nonlinear dipole moment, we have established the counter-intuitive non-instantaneous nature of the third-order polarization in atoms even at low intensities and for long wavelengths far from resonances. We have also shown significant delays of the nonlinear polarization resulting from a strong contribution of nonlinear loss and excited states. We have proposed an experimental scheme to observe and characterize the predicted effects.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Springer, 2000).
2. R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85(11), 2264–2267 (2000). [CrossRef]
3. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]
4. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]
5. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429 (2009). [CrossRef]
6. P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-Order Kerr Terms Allow Ionization-Free Filamentation in Gases,” Phys. Rev. Lett. 104(10), 103903 (2010). [CrossRef]
7. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental Tests of the New Paradigm for Laser Filamentation in Gases,” Phys. Rev. Lett. 106(15), 153902 (2011). [CrossRef]
8. M. Kolesik, D. Mirell, J.-C. Diels, and J. V. Moloney, “On the higher-order Kerr effect in femtosecond filaments,” Opt. Lett. 35(21), 3685–3687 (2010). [CrossRef]
9. O. Kosareva, J.-F. Daigle, N. Panov, T. Wang, S. Hosseini, S. Yuan, G. Roy, V. Makarov, and S. L. Chin, “Arrest of self-focusing collapse in femtosecond air filaments: higher order Kerr or plasma defocusing?” Opt. Lett. 36(7), 1035–1037 (2011). [CrossRef]
10. P. Béjot, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, and O. Faucher, “From higher-order Kerr nonlinearities to quantitative modeling of third and fifth harmonic generation in argon,” Opt. Lett. 36(6), 828–830 (2011). [CrossRef]
11. M. Petrarca, Y. Petit, S. Henin, R. Delagrange, P. Béjot, and J. Kasparian, “Higher-order Kerr improve quantitative modeling of laser filamentation,” Opt. Lett. 37(20), 4347–4349 (2012). [CrossRef]
12. D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-Order Nonlinearities Revisited and Their Effect on Harmonic Generation,” Phys. Rev. Lett. 114(9), 093901 (2015). [CrossRef]
13. M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140(21), 214316 (2014). [CrossRef]
14. M. Tarazkar, D. A. Romanov, and R. J. Levis, “High-order nonlinear refractive indices for He, Ne, Kr, and Xe atoms,” Phys. Rev. A 90(6), 062514 (2014). [CrossRef]
15. P. Béjot, E. Hertz, J. Kasparian, B. Lavorel, J. -P. Wolf, and O. Faucher, “Transition from Plasma-Driven to Kerr-Driven Laser Filamentation,” Phys. Rev. Lett. 106(24), 243902 (2011). [CrossRef]
16. B. Borchers, C. Brée, S. Birkholz, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect in solids,” Opt. Lett. 37(9), 1541–1543 (2012). [CrossRef]
17. J. M. Brown, E. M. Wright, J. V. Moloney, and M. Kolesik, “On the relative roles of higher-order nonlinearity and ionization in ultrafast light-matter interactions,” Opt. Lett. 37(10), 1604–1606 (2012). [CrossRef]
18. P. Béjot, E. Cormier, E. Hertz, B. Lavorel, J. Kasparian, J.-P. Wolf, and O. Faucher, “High-Field Quantum Calculation Reveals Time-Dependent Negative Kerr Contribution,” Phys. Rev. Lett. 110(4), 043902 (2013). [CrossRef]
19. C. Köhler, R. Guichard, E. Lorin, S. Chelkowski, A. D. Bandrauk, L. Berge, and S. Skupin, “Saturation of the nonlinear refractive index in atomic gases,” Phys. Rev. A 87(4), 043811 (2013). [CrossRef]
20. C. Bree, A. Demircan, and G. Steinmeyer, “Saturation of the All-Optical Kerr Effect,” Phys. Rev. Lett. 106(18), 183902 (2011). [CrossRef]
21. M. Nurhuda, A. Suda, and K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. 10(5), 053006 (2008). [CrossRef]
22. A. Teleki, E. M. Wright, and M. Kolesik, “Microscopic model for the higher-order nonlinearity in optical filaments,” Phys. Rev. A 82(6), 065801 (2010). [CrossRef]
23. A. Spott, A. Jaron-Becker, and A. Becker, “Time-dependent susceptibility of a helium atom in intense laser pulses,” Phys. Rev. A 96(5), 053404 (2017). [CrossRef]
24. A. Spott, A. Jaron-Becker, and A. Becker, “Ab initio and perturbative calculations of the electric susceptibility of atomic hydrogen,” Phys. Rev. A 90(1), 013426 (2014). [CrossRef]
25. M. Richter, S. Patchkovskii, F. Morales, O. Smirnova, and M. Ivanov, “The role of the Kramers–Henneberger atom in the higher-order Kerr effect,” New J. Phys. 15(8), 083012 (2013). [CrossRef]
26. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. 94(7), 519–524 (2011). [CrossRef]
27. S. Patchkovskii and H. G. Muller, “Computer Physics Communications Simple, accurate, and efficient implementation of 1-electron atomic time-dependent Schrödinger equation in spherical coordinates,” Comput. Phys. Commun. 199, 153–169 (2016). [CrossRef]
28. F. Brunel, “Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit,” J. Opt. Soc. Am. B 7(4), 521–526 (1990). [CrossRef]
29. P. Jürgens, B. Liewehr, B. Kruse, C. Peltz, D. Engel, A. Husakou, T. Witting, M. Ivanov, M. J. J. Vrakking, T. Fennel, and A. Mermillod-Blondin, “Origin of strong-field-induced low-order harmonic generation in amorphous quartz,” Nat. Phys. 16(10), 1035–1039 (2020). [CrossRef]
